What would it take to build a thermodynamically reversible Universal Turing machine? Computational and thermodynamic constraints in a molecular design
WWhat would it take to build a thermodynamically reversible Universal Turingmachine? Computational and thermodynamic constraints in a molecular design
Rory A. Brittain, ∗ Nick S. Jones, and Thomas E. Ouldridge Department of Mathematics, Imperial College London, London, SW7 2AZ, UK Centre for Synthetic Biology and Department of Bioengineering,Imperial College London, London, SW7 2AZ, UK
We outline the construction of a molecular system that could, in principle, implement a thermody-namically reversible Universal Turing Machine (UTM). By proposing a concrete—albeit idealised—design and operational protocol, we reveal fundamental challenges that arise when attempting to im-plement arbitrary computations reversibly. Firstly, the requirements of thermodynamic reversibilityinevitably lead to an intricate design. Secondly, thermodynamically reversible UTMs, unlike simplerdevices, must also be logically reversible. Finally, implementing multiple distinct computations inparallel is necessary to take the cost of external control per computation to zero, but this approachis complicated the distinct halting times of different computations.
Computational operations use physical substrates, andthus have physical consequences [1]. The equivalence ofthe Shannon and the non-equilibrium thermodynamic en-tropies [2] provides a concrete link between information-processing and thermodynamics; how the resultant ther-modynamic costs might be minimised is an importantopen question. Many analyses have focused on specifictasks, such as erasing and copying bits [3, 4]; recent stud-ies have extended these ideas to the the processing ofstrings [5–8]. Others have focused instead on computa-tion as an arbitrary process in which an input state isconverted into an output. Within this framework, Ben-nett argued that thermodynamically reversible comput-ers must use reversible logic, wherein the input state canbe inferred unambiguously from its output [9]. However,it has since been shown that any input-output map—whether logically reversible or not—can be implementedin a thermodynamically reversible fashion [10], providedthat the distribution of input states is known [11].Turing machines are a model of arbitrary computation[12]. Traditionally, the machine’s memory is a tape: asequence of symbols from a finite alphabet. The tapeis processed by a head with a finite state set that readsthe symbol at a position on the tape. Then, based ona machine-dependent set of transition rules, the headwrites a new symbol at that tape site, moves either leftor right to an adjacent site, and the head state changes.Sufficiently complex machines can be ‘Turing complete’:they can compute any computable function. These ma-chines are called universal Turing machines (UTM).The thermodynamics of abstract UTMs has been ex-plored in the context of algorithmic information theory[13]. However, it is also illuminating to consider ac-tual physical designs of low thermodynamic-cost UTMs[8, 13, 14]. By making even an idealised design explicit,one can avoid hidden violations of the second law, andreveal fundamental challenges in constructing a device. ∗ [email protected] Molecular systems are promising substrates for com-puting [15]; the similarity of tape-processing ribosomesand TMs is tantalising. One approach to molecular com-putation uses macroscopic quantities of molecules foreach computation, and either repeatedly intervening todirect the computational steps [16–20] or allowing thesolution to relax to equilibrium and perform a compu-tation [21, 22]. In both cases, the bulk scale makes thethermodynamic cost per computation large.An alternative is to use a small number of computa-tional molecules, coupled to large baths of ancillary fuel.Bennett sketched a UTM of this kind [9], and more re-cently Qian, Soloveichick and Winfree [23] conceived aspecific DNA-based realisation of a stack machine, a closeanalogue of a UTM. In both designs, the overall free-energy change per step in the forward direction, − (cid:15) , isconstant during a computation. For (cid:15) >
0, the computa-tion proceeds irreversibly forwards with a finite entropyproduction (or thermodynamic cost) per step [24]. For (cid:15) →
0, the computation undergoes unbiased diffusion,and the output computational state will not dominatethe ensemble even at infinite time. Moreover, the totalentropy production is positive even in this unbiased limit,due to the irreversible spreading of the system’s proba-bility distribution over its computational states [24].We outline a thermodynamically reversible,sequentially-operated molecular UTM in which theunderlying computational logic is implemented by anexplicit model of a surface-localised chemical reactionnetwork. We first outline the general design principles,before describing the device in more detail. We thenillustrate key challenges that arise when implementingthermodynamically reversible UTMs, as opposed tosimpler operations. Firstly, even a minimal designis intricate, emphasising how challenging it would beto actually construct a thermodynamically reversible,sequentially-operated molecular UTM. Secondly,thermodynamically reversible, sequentially-operatedUTMs—unlike simpler devices—must be logicallyreversible. Finally, running distinct computations inparallel to minimise the costs of control is complicated a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b by the distinct halting times of different computations. Control protocols for thermodynamically reversibleUTMs . A reversible UTM must be subject to time-dependent external control, since any free evolution of athermodynamic system is necessarily irreversible [25, 26].However, systematic evolution of the control variable it-self implies entropy generation [25]. In traditional ther-modynamics, controls are applied to large systems: costsassociated with evolving the control variable itself, andimplementing any logic therein, can be made small rela-tive to changes in the system [25]. The same is not truefor controlling a single computation. The same controlmust therefore be applied to multiple UTMs in paral-lel [8], so that any thermodynamic costs of applying acontrol protocol are negligible when divided by the num-ber of computations. Controls cannot, therefore, includefeedback on the state of individual computations, sincethe costs arising cannot be spread over multiple UTMs.
Ideal molecular reactions provide an explicit model sys-tem from which to construct a thermodynamically re-versible UTM.
We assume that it is possible to designchemical reaction networks with reactions of the form (cid:88) { i } X i + (cid:88) { j } Y j + F ∗ (cid:42)(cid:41) (cid:88) { i } X (cid:48) i + (cid:88) { j } Y j + F . (1)Here, X { i } is a set of substrate species converted intoproducts X (cid:48){ i } by the action of catalysts Y { j } . Each reac-tion is driven by the turnover of a fuel molecule F ∗ (cid:42)(cid:41) F.While such reactions are not elementary, compound cat-alytic action occurs naturally in, for example, transcrip-tion and translation, and can be engineered using DNAnanotechnology [21, 23, 27]. These reactions need not ap-proximate mass-action kinetics [28], but the overall sto-ichiometry must be tightly observed, with the reactionsonly occurring if all catalysts and fuel are present.We consider many UTMs in a single reaction volume(Fig. 1 (a)), all coupled to the same large baths of fuel(the external control) via a semi-permeable membrane.By varying the concentration of these baths [29–31], re-actions of the form of Eq. 1, involving the UTM’s species,can be quasistatically driven in either direction, or turnedon and off [8]. Our design of a molecular UTM is sequen-tial: it has clocked control cycles involving a single up-date of the head, tapes and the head position. All cyclesare driven by the same sequence of baths, which can bearranged in a circle as shown in Fig. 1(a). One turn ofthe circle corresponds to one step of each UTM.This setup has several conceptual advantages [14].Firstly, providing the control is simple; we cycle a se-ries of buffers past a reaction volume, without adjustingto the system’s response; the control is mechanical butthe computation is chemical. Secondly, the baths bothcontrol the reaction and act as a reservoir of chemicalwork; there is no ambiguity about how work is storedand transferred. Thirdly, the reaction in Eq. 1 allows theY { j } and X { i } species to influence the evolution of theX { i } species. However, the reaction only occurs if the ReactionvolumeOperatorturns railSteppingphase UpdatephaseRail ofbuffers Scaffold AD M TMolecularmotorX X X (a)(b) Stepping phase Update phaseMTD AM (cid:48) T (cid:48) D (cid:48) A (cid:48) G ∗ GD=RF ∗ F Or (c)X X X D=LF ∗ F X X X FIG. 1.
Overview of a molecular UTM. (a) A reaction volumecontaining many UTMs. A protocol of fuel concentrations isapplied by connecting the volume to a series of large, fuel-containing baths. The baths are arranged in a circle so thesame protocol can be applied repeatedly. Individual UTMsact on two tape polymers with evenly spaced molecules (M,T)to represent their state. The state of the UTM head is en-coded in the state of molecules (A,D) attached to a scaffoldsuch as DNA origami. The head is attached to the tapes bymolecular motors that move the head from one symbol to thenext. (b) Stepping is powered by fuel molecules F i . The di-rectional molecule D on the head catalyses stepping in theappropriate direction. (c) During the update phase, statesof the head (A), tape molecules (M,T), and the directionalmolecule (D) are updated, powered by a fuel molecule G . fuel is present; quasistatic re-wiring can be performed bychanging which fuel molecules are present [8, 14]. Design and operation of a molecular UTM.
The headof a UTM in our design consists of two or more molecu-lar species localised to a scaffold surface (Fig. 1(a)). Thehead interacts with two tapes, a main tape and a mem-ory tape, represented by molecules attached to polymersat regular intervals. The head is attached to the poly-mers by molecular motors. At any one time, the headmolecules can interact with each other and at most onepair of sites on the tapes. Since we are concerned onlywith the thermodynamics of converting inputs to out-puts, we assume that the machines and tapes are pro-vided in their starting configurations.The UTM cycle has distinct stepping and updatephases. During the stepping phase (Fig. 1(b)), The headmust step selectively in either direction along the maintape without relying on the discretion of an operator.This directionality is implemented through a ‘direction T i m e s t e p T i m e s t e p Symbol and statePosition 0.000.050.100.150.20a b c da a a a a b b b b b c c c c c d dd d d0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5FIG. 2.
UTM operation generates both a complex distributionof head and tape states at each step, and correlations betweenstates on the tape.
We consider a 4-state, 6-symbol machine(Appendix C 2) (a) Evolution of the probability distributionof the head and active tape site at the stare of each updatephase. The head is initially in state A and positioned in themiddle of 5 non blank symbols in an infinite tape of blanksymbols. The machine is run for 5 steps with the states ofthe 5 non-blank symbols randomly chosen for each of 100,000iterations. (b) The mutual information between the symbolat position 6, the initial position of the head, and the symbolsat the neighbouring positions at the start of each update. Thehead is initially in state A and at position 6 in the middle of 13non-blank symbols in an infinite tape of blank symbols. Themachine is run for 5 steps with the states of the 13 symbolsrandomly chosen for each of 100,000 iterations. molecule’ D, which catalyses the transition of the motorbetween its configurations when the appropriate fuels arepresent. The direction molecule can either be in the left,L, or the right, R, state, specifying the direction of thenext step. The direction molecule also has a halting state H that does not catalyse stepping on the main tape.If the stepping of the motor involves at least three dif-ferent substates, X , X and X (Fig. 1(b)), fuels can beadded that drive X → X and X → X ; X → X andX → X ; and X → X and X → X . If these fuelsare introduced and removed quasistatically and sequen-tially, stepping will be deterministic and thermodynami-cally reversible; one step is shown in Fig. 1(b) and furtherdetails are given in Appendix A. The motor attached tothe memory tape always takes a single step in the samedirection during the stepping phase.The head state, the symbols on the tapes and direc-tion molecule are updated during the update phase viareactions of the form of Eq. 1 (Fig. 1(c)). The detailsdepend on the specific set of rules chosen for the UTM,and how the states of the head are encoded in the molec-ular species. In the conceptually simplest case, a singlemolecule A represents the head state, along with the di- rection molecule. In this case, each update rule of a UTMcan be expressed through reactions of the formA + L + T + M + G ∗ (cid:42)(cid:41) A (cid:48) + D (cid:48) + T (cid:48) + M i + G , A + R + T + M + G ∗ (cid:42)(cid:41) A (cid:48) + D (cid:48) + T (cid:48) + M i − + G , (2)where M is the default initial state of the memory tapeand i numbers each transition rule; the memory tapetherefore makes a record of the transition taken. A andA (cid:48) , T and T (cid:48) , and L/R and D (cid:48) represent the initial andfinal states of the head, the main tape and the direc-tion molecule, respectively. The protocol involves simplymodulating fuel concentrations in the following manner[G] : 0 → g → g → → ∗ ] : 0 → → (cid:29) g → (cid:29) g → , (3)where g is a large concentration. This protocol drivesthe reactions in Eq. 2 reversibly from left to right, thenswitches them off before stepping.In this set-up, each transition rule of the UTM needstwo reactions. Each rule also requires two different mem-ory states, in addition to M . The directional moleculehas three states. The number of head molecule and tapestates follow from the specific UTM. It is possible to usefewer memory species at the expense of increased con-ceptual complexity; the reactions and species requiredfor some example machines are shown in Appendix C.It is apparent that even the simplest design is sophisti-cated, with molecular species involved in several highlyspecific reactions. The need to perform precisely the rightseries of sequential reactions, and couple those reactionstightly to motion along a polymer, makes this complexityunavoidable. Without even considering the challenges ofapplying quasistatic manipulation of the baths, this com-plexity far exceeds what is currently implementable inpractice. This complexity is easily overlooked in modelsthat do not represent every stage explicitly [15]. An ensemble of thermodynamically reversible UTMsrequires an ensemble of distinct input programs.
Fig. 1shows an ensemble of UTMs operating in parallel. If thetask was to erase or extract work from inputs, then itwould be reasonable to act in parallel on identical inputs.For each additional system, something more is achieved.However, if the goal is to answer a computational ques-tion, then acting in parallel on identical inputs does notreduce the cost per answer. Spreading the costs of ex-ternal control over many computations requires paralleloperation on multiple different inputs.The first consequence of performing distinct compu-tations in parallel is that a record of each computationmust be unambiguously associated with each tape. If not,there would be no way to match the answers to questionsasked. This record could be physically attached to thetape. Note that the record is a specification (eg. “divide135 by 5 and output the answer in a specific way”), soneed not be a copy of the program itself.
Sequentially-operated, thermodynamically-reversibleUTMs must have logically-reversible update rules.
Thememory tape makes the computation locally logicallyreversible: at each stage the input state of head andtape molecules can be unambiguously inferred fromtheir final values. Without the memory, multiple inputswould be mapped to the same output by the reactionsof Eq. 2. This mixing defines logical irreversibility.Consider A + L + T and A (cid:48) + L + T being converted tothe same output. At some instant, systems that startedin A + L + T will transition to the states occupied bysystems that started in A (cid:48) + L + T, and vice versa. Atthis time, if a net flow of trajectories occurs in eitherdirection, the process is thermodynamically irreversible[26]. For logical states to mix reversibly, they must beoffset in free energy to avoid net transitions from themore populated state to to the less populated one. Asa result, a thermodynamically reversible protocol for alogically irreversible input-output map must be tuned tothe distribution of input states [11].Tuning to the input distribution is far more problem-atic for sequentially-operated UTMs than for simpler op-erations like erasing or copying of bits. Firstly, sequentialUTMs implement complex computations through a seriesof simpler steps. To be thermodynamically reversible, aprotocol would need to be optimised to the initial distri-bution of the head and tape at each step [32]. As illus-trated in Fig. 2 (a), this initial distribution has complexbehaviour even for a simple UTM; re-applying the samecyclic fuel protocol, as in Fig. 1 (a), would be impossible.Instead, we’d need to encode a priori knowledge of howthe UTM would process a distribution of input tapes intoa long, non-cyclic array of fuel baths. We’d effectivelyneed to know the output of the relevant computationsalready just to build the UTM.Secondly, although the update rules of UTMs are lo-cal, statistical information will exist between the locallyactive region of the UTM and other sites in the tape, aswell as the record of the computation. As demonstratedin Fig. 2 (b), UTMs generate this information even if itdoes not initially exist. A local update of a subsystemwhen information exists between that subsystem and therest of the system typically causes “modularity dissipa-tion” [33, 34], wherein the non-equilibrium free energystored in this information is wasted irreversibly. To avoidmodularity dissipation, either: (a) no information withthe rest of the system must be lost; or (b) the free en-ergy stored in this information must be extracted. If thelocal update is logically reversible, (a) is trivially satis-fied since the update simply permutes the occupancy ofstates. For logically irreversible updates, the mixing ofdistinct inputs will tend to reduce information with therest of the system. To compensate, the rest of the sys-tem would have to be involved in the update—not toset the output, but to influence the protocol experiencedby the active subsystem and thereby recover the storedfree energy. For the the design in Fig. 1, the rest of thetape would have to catalytically couple the head and ac-tive site to well-tuned fuel baths for information to beexploited. Doing so would be even more challenging— both mechanistically and computationally—than build-ing a logically reversible device.
The interplay of halting and thermodynamics.
To com-pute, a UTM must halt. In simpler contexts, when thesame operation can be performed in parallel on multipleidentical inputs, all operations will halt simultaneously.For UTMs performing arbitrary and distinct computa-tions in parallel, however, halting times may be widelydistributed; individual halting times may not be known a priori , and some computations may not halt at all.This additional complexity raises three challenges.The first is that halted UTMs must stay in a valid haltedstate as the external protocol continues to be applied.This feature is designed into our molecular UTMs: thehead and main tape do not evolve once the halt state isreached. The stepping is halted on the main tape butcontinues on the memory tape. Secondly, the controlcycle must eventually stop, and it cannot simply intuitwhen all computations are halted. If an upper limit onthe time of the computations in question is known, orif the operator is prepared to accept that some compu-tations may not have halted or that unnecessary stepsmay be taken, the protocol can be stopped after a fixednumber of steps with a cost that scales sub-linearly withthe number of computations [25].Alternatively, although measurement and feedback onindividual UTMs would violate the parallel nature of theprotocol, the minimal thermodynamic cost of answeringand acting upon the binary question of whether all ma-chines have halted scales sub-linearly with the numberof UTMs. However, the mechanistic challenges are high.For example, a single molecule E that can undergo fuel-powered conversion catalysed by L or R, E L → E (cid:48) orE R → E (cid:48) , would reach the state E (cid:48) via a single reactionif and only if at least one of the UTMs had not halted.However, the operator would have to wait for a singleprobe to interact with all UTMs.Thirdly, and most deeply, as the faster computationshalt, the protocol is applied to fewer active UTMs. Thestrategy of spreading the control cost over many compu-tations is then questionable. In our case, control cost isthe cost of ensuring that the baths move systematicallypast the reaction volume for the desired number of cycles;it is separate from the efficiency of the chemical compu-tation. Machta argued that each unidirectional cycle of acontrol parameter has a minimal entropy generation as-sociated with it [25]; we therefore assume that the cost ofexternal control C is linear in the number of protocol cy-cles t implemented, C ( M, t ) = α ( M ) t + C ( M ), with C representing initiation and termination costs and M thenumber of distinct computations under control, a proxyfor the system size. We allow for a dependence of C on M : Machta found α ( M ) ∼ M . [25].If all M computations in a set S M halt in a time t M ,the minimal control cost per computation is C ( S M ) /M = α ( M ) t M /M + C ( M ) /M . If the M computations aredrawn from a set S ⊃ S M of interest, we can ask whetherit is always possible to reduce C ( S M ) /M by using a sub-set with a larger M = |S M | . Assuming C ( M ) scalessub-linearly with M [25], C ( M ) /M → M → ∞ .However, as we increase M, newly-chosen inputs mayhave a longer halting time and hence t M will grow. Ifthe halting times of computational problems of interestare sparsely distributed, α ( M ) t M /M may grow with M ,implying that C ( S M ) /M cannot be made negligible.For α ( M ) ∼ M . , we require that subsets S M can befound such that t M grows more slowly than M . . Con-sider an algorithm with a halting time that is linear inthe length N of the input. If there are N ! problems ofinterest of length ≤ N , sets S M can be found satisfying t M ∼ M/M ! and C ( S M ) /M can easily be made negligi-ble. If, by contrast, there is only one problem of interestat each length, t M ∼ M and C ( S M ) /M grows with M .Requiring that C ( S m ) /M → S still exist for which the normalisedcost of control grows with M for S M ⊂ S .We have shown that the construction of a sequentially-operated, thermodynamically reversible molecular UTMhas challenges that arise from the very purpose of a UTM:to perform complex computations. To circumvent someof these challenges, one could consider a machine thatoperates in a single control step. For example, the stackmachine of Qian et al. [23] could migrate from the in-put to the output state if the driving from ancillary fuelmolecules was slowly increased over time, avoiding theissue of choosing when to stop. However, doing so wouldraise a new problem: the later states would rapidly tran-sition from being exponentially suppressed to exponen-tially favoured relative to the early states as the drivingforce was varied. For calculations of unknown (and ar-bitrary) length, it would be extremely challenging to im-plement a protocol that manages this change reversibly.In fact, in cases where the halting state is never reached,the computation would undergo an uncontrolled and irre-versible growth as the driving force passed through zero. [1] R. Landauer. Phys. lett. A , 217(4-5):188–193, 1996.[2] M. Esposito and C. Van den Broeck.
EPL , 95(4):40004,2011.[3] R. Landauer.
IBM J. Res. Dev. , 5(3):183–191, 1961.[4] L. Szilard.
Z. Phys. , 53(11-12):840–856, 1929.[5] D. Mandal and C. Jarzynski.
Proc. Natl. Acad. Sci.U.S.A. , 109(29):11641–11645, 2012.[6] A. B. Boyd, D. Mandal, and J. P. Crutchfield.
New J.Phys. , 18(2):023049, 2016.[7] E. Stopnitzky, S. Still, T. E. Ouldridge, and L. Altenberg.
Phys. Rev. E , 99:042115, 2019.[8] R. A. Brittain, N. S. Jones, and T. E. Ouldridge.
NewJournal of Physics , 21(6):063022, 2019.[9] C. H. Bennett.
IBM J. Res. Dev. , 17(6):525–532, 1973.[10] J. A. Owen, A. Kolchinsky, and D. H. Wolpert.
New J.Phys. , 2018.[11] A. Kolchinsky and D. H. Wolpert.
J. Stat. Mech. TheoryExp. , 2017:083202, 2017.[12] A. M. Turing.
Proc. London Math. Soc. , 2(1):230–265,1937.[13] A. Kolchinsky and D. H. Wolpert. arXiv preprintarXiv:1912.04685 , 2019.[14] T. E. Ouldridge, R. A. Brittain, and P. R. ten Wolde.In D. H. Wolpert, C. P. Kempes, P. Stadler, andJ. Grochow, editors,
The Interplay of Thermodynamicsand Computation in Natural and Artificial Systems . SFIPress, 2018.[15] C. H. Bennett.
Int. J. Theor. Phys. , 21(12):905–940,1982.[16] L. M. Adleman.
Science , 266(5187):1021–1024, 1994.[17] D. Beaver. In L. F. Landweber and E. B. Baum, editors,
DNA Based Computers , pages 29–36. American Mathe-matical Society, 1995.[18] P. W. K. Rothemund. In L. F. Landweber and E. B.Baum, editors,
DNA Based Computers , pages 75–119.American Mathematical Society, 1995.[19] W. D. Smith.
DNA based computers , 27:121–186, 1995. [20] A. Currin, K. Korovin, M. Ababi, K. Roper, D. B.Kell, P. J. Day, and R. D. King.
J. R. Soc. Interface ,14(128):20160990, 2017.[21] Y.-J. Chen, N. Dalchau, N. Srinivas, A. Phillips,L. Cardelli, D. Soloveichik, and G. Seelig.
Nat. Nan-otechnol. , 8(10):755, 2013.[22] L. Cardelli, M. Kwiatkowska, and M. Whitby. In
In-ternational Conference on DNA-Based Computers , pages67–81. Springer, 2016.[23] L. Qian, D. Soloveichik, and E. Winfree. In
Interna-tional Workshop on DNA-Based Computers , pages 123–140. Springer, 2010.[24] P. Strasberg, J. Cerrillo, G. Schaller, and T. Brandes.
Phys. Rev E , 92(4):042104, 2015.[25] B. B. Machta.
Phys. Rev. Lett. , 115:260603, 2015.[26] T. E. Ouldridge.
Nat. Comput. , 17(1):3–29, 2018.[27] N. Srinivas, J. Parkin, G. Seelig, E. Winfree, andD. Soloveichik.
Science , 358(6369):eaal2052, 2017.[28] T. Plesa. arXiv preprint arXiv:1811.02766 , 2018.[29] T. Schmiedl and U. Seifert.
J. Chem. Phys. ,126(4):044101, 2007.[30] R. Rao and M. Esposito.
Phys. Rev. X , 6(4):041064,2016.[31] T. E. Ouldridge and P. R. ten Wolde.
Phys. Rev. Lett. ,118(15):158103, 2017.[32] D. H. Wolpert. arXiv preprint arXiv:1508.05319 , 2015.[33] A. B. Boyd, D. Mandal, and J. P. Crutchfield.
Phys. Rev.X , 8(3):031036, 2018.[34] D. H. Wolpert. arXiv preprint arXiv:2001.02205 , 2020.[35] A. Smith.
Complex Syst. , 2007.[36] D. Woods and T. Neary.
Theor. Comput. Sci. , 410(4-5):443–450, 2009.[37] Y. Rogozhin.
Theor. Comput. Sci. , 168(2):215–240, 1996.
FIG. 3. The stepping of the molecular motor.
Appendix A: Stepping
We assume that the head changing position is an elementary transition that can happen in a chemical reaction: weassume that the molecular motor can rotate a third of a step while simultaneously switching which segment of themotor is bound to the polymer in a single step without the polymer and motor becoming unbound at any point. Thisis a strong simplifying assumption so a version of stepping without this assumption is given in Appendix B.The tape has a repeating structure of sites where the molecular motor can bind. There are three different typesof site: X, Y and Z that can, respectively, bind to the X, Y and Z regions of the molecular motor. This is shown infigure 3. The X sites are the positions on the tape where the symbol molecule on the tape lines up with the tetheredhead molecules (these are the only positions that correspond to positions in the abstract view of the machine) andthe Y and Z are intermediate positions to which the head can attach.To move right the reactions F ∗ + R + X i (cid:42)(cid:41) R + Y i + F , F ∗ + R + Y i (cid:42)(cid:41) R + Z i + F , F ∗ + R + Z i (cid:42)(cid:41) R + X i +1 + F , (A1)are used and to move left the reactions F ∗ + L + X i (cid:42)(cid:41) L + Z i − + F , F ∗ + L + Z i − (cid:42)(cid:41) L + Y i − + F , F ∗ + L + Y i − (cid:42)(cid:41) L + X i − + F , (A2)are used. If the head contains the direction molecule in the R state then the head can be quasistatically moved fromX i to Y i to Z i to X i +1 and if the direction molecule is in the L state then the head can be quasistatically moved fromX i to Z i − to Y i to X i − . This is done using the protocol for three separate bit flips starting from a definite state.For the left motion we could, alternatively, use three different pairs of fuel molecules, F , F and F , from the rightmotion. This could make the mechanism easier to understand because the mechanisms for the two directions areindependent. We adopt the approach in equations A2 and A1 as reusing the same species reduces the total numberof fuel molecule required.The protocol of concentrations for the fuel molecules is shown in figure 4. The work extracted from the buffers isthe change in free energy of the head when it is attached to the different positions. We assume that the free energyis the same when the head is attached to any of the X i . The free energy when attached to either of the intermediatestates could be different but that is unimportant since the work done to move from X i to X i +1 or X i − is zero. F F F F F F i to Y i if rightor Xi to Z i-1 if leftStep 2Y i to Z i if rightor Z i-1 to Y i-1 if leftStep 3Z i to X i+1 if rightor Y i-1 to X i-1 if left FIG. 4. The protocol of concentrations of the fuel molecules to implement one step, which is left or right depending on thestate of the direction molecule. While apparently complicated in fact the same approach is repeated three times. f1, f2 and f3need only be a large concentration and in fact the same label could have been used in all cases. X' i Y' X' i+1
X YX' i Y' X' i+1
X Y X' i Y' X' i+1
X Y X' i Y' X' i+1
XYX' i Y' X' i+1
XYX' i Y' X' i+1
XYRelabelX i+1 to X i FIG. 5. Overview of more realistic molecular motor. The arrows represent the direction of the quasistatic change when takinga rightwards step. Of course, each step is microscopically reversible and when moving left the arrows are all in the oppositedirections. The X (cid:48) is meant to show that that is a place where the X foot can bind.
Appendix B: Alternative Molecular motor
It is more realistic if the motor can detach from the tape. Consider an actin/myosin style set-up where the motorhas two feet. An overview is shown in figure 5.This is more complicated than the previous set-up because the motor has more possible states but there are only5 different reactions required. To move right there are the reactionsF ∗ + R + X attached (cid:42)(cid:41) R + X unattched + F F ∗ + R + Y attached (cid:42)(cid:41) R + Y unattached + F F ∗ + R + XY (cid:42)(cid:41) R + YX + F (B1)where XY and YX represent the configurations where either the X foot is on the left of the motor or the Y foot is onthe left of the motor. To move left the L state catalyses the same equations but the fuels that cause the attachingand detaching of the X and Y are swapped and the same reaction to change the conformation of the walker is usedF ∗ + L + X attached (cid:42)(cid:41) L + X unattched + F F ∗ + L + Y attached (cid:42)(cid:41) L + Y unattached + F (B2)However, a more complicated protocol of fuel concentrations is required. This protocol is shown in figure 6. Thisprotocol requires twice as many steps as the previous one. Appendix C: Examples1. 2-state 3-symbol Machine
This machine is claimed to be the smallest possible Turing machine [35] but it is only Turing complete if the tapeinitially has a complex encoding of an infinite string. The usual definition of a Turing machine requires that the tapeis initialised with an infinitely repeating symbol containing only a finite string of different symbols [36].The molecules on the tape can be in three states: 0, 1 and 2. The ‘state molecule’ of the head (like the ‘memorymolecule’) has two states: A and B. The transitions for this machine are shown in table I.As mentioned in the main text, the most straightforward way to convert this list of rules to an invertible map isfor the memory molecule to be initially in the state M and to convert it to a state labelled by the number of the rule A B0 B1R A2L1 A2L B2R2 A1L A0RTABLE I. The rules for Wolfram’s 2 state 3 symbol machine. that is used as below: A + L + 0 + M + G ∗ (cid:42)(cid:41) B + R + 1 + M + GA + R + 0 + M + G ∗ (cid:42)(cid:41) B + R + 1 + M + GA + L + 1 + M + G ∗ (cid:42)(cid:41) A + L + 2 + M + GA + R + 1 + M + G ∗ (cid:42)(cid:41) A + L + 2 + M + GA + L + 2 + M + G ∗ (cid:42)(cid:41) A + L + 1 + M + GA + R + 2 + M + G ∗ (cid:42)(cid:41) A + L + 1 + M + GB + L + 0 + M + G ∗ (cid:42)(cid:41) A + L + 2 + M + GB + R + 0 + M + G ∗ (cid:42)(cid:41) A + L + 2 + M + GB + L + 1 + M + G ∗ (cid:42)(cid:41) B + R + 2 + M + GB + R + 1 + M + G ∗ (cid:42)(cid:41) B + R + 2 + M + GB + L + 2 + M + G ∗ (cid:42)(cid:41) A + R + 0 + M + GB + R + 2 + M + G ∗ (cid:42)(cid:41) A + R + 0 + M + G . (C1)We only need one pair of fuel molecules because each reaction is otherwise unique. The protocol is to start with[G] = [G ∗ ] = 0, then increase [G] up to some value then increase [G ∗ ] up to a value much greater than [G] thendecrease [G] to zero, then decrease [G ∗ ] to zero.There are no reactions that convert the directional molecule to the H state in equation C1 because this machinedoes not have a halting state. This lack of a halting state is another way in which this machine differs from thestandard definition of a Turing machine.This scheme uses a state molecule with two states, a directional molecule with two states, a symbol molecule foreach position on the main tape with three states, and a memory molecule at each position on the memory tape with13 states. One pair of fuel molecule species is also needed (not including the stepping).This naive scheme uses more memory states than necessary. Figure 7 shows that the maximal in degree of anystate is four so only M to M are needed to distinguish the previous state of the machine.Therefore, we only need five states for the memory molecule and we can use the reactions:A + L + 0 + M + G ∗ (cid:42)(cid:41) B + R + 1 + M + GA + R + 0 + M + G ∗ (cid:42)(cid:41) B + R + 1 + M + GA + L + 1 + M + G ∗ (cid:42)(cid:41) A + L + 2 + M + GA + R + 1 + M + G ∗ (cid:42)(cid:41) A + L + 2 + M + GA + L + 2 + M + G ∗ (cid:42)(cid:41) A + L + 1 + M + GA + R + 2 + M + G ∗ (cid:42)(cid:41) A + L + 1 + M + GB + L + 0 + M + G ∗ (cid:42)(cid:41) A + L + 2 + M + GB + R + 0 + M + G ∗ (cid:42)(cid:41) A + L + 2 + M + GB + L + 1 + M + G ∗ (cid:42)(cid:41) B + R + 2 + M + GB + R + 1 + M + G ∗ (cid:42)(cid:41) B + R + 2 + M + GB + L + 2 + M + G ∗ (cid:42)(cid:41) A + R + 0 + M + GB + R + 2 + M + G ∗ (cid:42)(cid:41) A + R + 0 + M + G . (C2)This scheme uses a state molecule with two states, a directional molecule with two states, a symbol molecule foreach position on the main tape with three states, a memory molecule at each position on the memory tape with fivestates, and one pair of fuel molecule species.Instead of a single molecule with five states we can use three molecules each with two states. Of course, this leavesmultiple unused states. Use one molecule for direction of previous step (M L /M R ), one to distinguish the 2-to-1 rules0(M /M ) and one to give reaction direction/take place of hidden states (M i /M f ). We are free to choose the initialstate of the (M L /M R ) and (M /M ) molecules so we arbitrarily choose M L and M .Therefore, reactions become:A + L + 0 + M i + M L + M + G ∗ (cid:42)(cid:41) B + R + 1 + M f + M L + M + GA + R + 0 + M i + M L + M + G ∗ (cid:42)(cid:41) B + R + 1 + M f + M R + M + GA + L + 1 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 2 + M f + M L + M + GA + R + 1 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 2 + M f + M R + M + GA + L + 2 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 1 + M f + M L + M + GA + R + 2 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 1 + M f + M R + M + GB + L + 0 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 2 + M f + M L + M + GB + R + 0 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 2 + M f + M R + M + GB + L + 1 + M i + M L + M + G ∗ (cid:42)(cid:41) B + R + 2 + M f + M L + M + GB + R + 1 + M i + M L + M + G ∗ (cid:42)(cid:41) B + R + 2 + M f + M R + M + GB + L + 2 + M i + M L + M + G ∗ (cid:42)(cid:41) A + R + 0 + M f + M L + M + GB + R + 2 + M i + M L + M + G ∗ (cid:42)(cid:41) A + R + 0 + M f + M R + M + G (C3)Clearly, our trade-off for reducing the number of states is to have reactions of six molecules.The situation can be improved because we do not need all of the molecules in all of the reactions. We only needthe M /M reaction in the rule that needs it. Therefore, the reactions are:A + L + 0 + M i + M L + G ∗ (cid:42)(cid:41) B + R + 1 + M f + M L + GA + R + 0 + M i + M L + G ∗ (cid:42)(cid:41) B + R + 1 + M f + M R + GA + L + 1 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 2 + M f + M L + M + GA + R + 1 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 2 + M f + M R + M + GA + L + 2 + M i + M L + G ∗ (cid:42)(cid:41) A + L + 1 + M f + M L + GA + R + 2 + M i + M L + G ∗ (cid:42)(cid:41) A + L + 1 + M f + M R + GB + L + 0 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 2 + M f + M L + M + GB + R + 0 + M i + M L + M + G ∗ (cid:42)(cid:41) A + L + 2 + M f + M R + M + GB + L + 1 + M i + M L + G ∗ (cid:42)(cid:41) B + R + 2 + M f + M L + GB + R + 1 + M i + M L + G ∗ (cid:42)(cid:41) B + R + 2 + M f + M R + GB + L + 2 + M i + M L + G ∗ (cid:42)(cid:41) A + R + 0 + M f + M L + GB + R + 2 + M i + M L + G ∗ (cid:42)(cid:41) A + R + 0 + M f + M R + G . (C4)
2. 4-state 6-symbol Machine
There is a trade-off between the number of head states and the number of tape symbols [36]. The machine withthe smallest number of different transition rules, which correspond to chemical reactions, is the four state six symbolmachine from Rogozhin [37]. The transitions for this machine are shown in table II. A is the initial head state and 4is the blank symbol.
A B C D0 A3L B4R C0R D4R1 A2R C2L D3R B5L2 A1L B3R C1R D3R3 A4R B2L H H4 A3L B0L A5R B5L5 D4R B1R A0R D1RTABLE II. The rules for Rogozhin’s universal 4 state 6 symbol machine (with relabelled states and symbols).The letters arethe states and the numbers are the symbols. F F F F F F FIG. 6. The protocol of fuel concentrations for the molecular motor that is inspired by actin myosin. A0LB1R A0RB0LA2LB0R A1LA1RB1L B2R A2RB2L
FIG. 7. The transitions in a single step of the 2-state 3-symbol machine. The largest in degree is four. A0LA3LA0R B0LB4RB0RC0LC0RD0LD4RD0R A1LA2RA1RB1LC2LB1R C1LD3RC1RD1LB5LD1R A2LB2LB3RB2R C2R D2LD2RA4R A3RB3L C3L h C3RD3LA4L B4LC4LA5RC4R D4LA5L B5RC5LC5R D5LD5R
FIG. 8. Graph showing one step of Rogozhin’s 4-state 6-symbol machine. There are 5 nodes with an in degree of 4.
5A + 0 + L + M + M + G ∗ (cid:42)(cid:41) A + 3 + L + M + GA + 0 + R + M + M + G ∗ (cid:42)(cid:41) A + 3 + L + M + GB + 0 + L + M + M + G ∗ (cid:42)(cid:41) B + 4 + R + M + GB + 0 + R + M + M + G ∗ (cid:42)(cid:41) B + 4 + R + M + GC + 0 + L + M + M + G ∗ (cid:42)(cid:41) C + 0 + R + M + GC + 0 + R + M + M + G ∗ (cid:42)(cid:41) C + 0 + R + M + GD + 0 + L + M + M + G ∗ (cid:42)(cid:41) D + 4 + R + M + GD + 0 + R + M + M + G ∗ (cid:42)(cid:41) D + 4 + R + M + GA + 1 + L + M + M + G ∗ (cid:42)(cid:41) A + 2 + R + M + GA + 1 + R + M + M + G ∗ (cid:42)(cid:41) A + 2 + R + M + GB + 1 + L + M + M + G ∗ (cid:42)(cid:41) C + 2 + L + M + GB + 1 + R + M + M + G ∗ (cid:42)(cid:41) C + 2 + L + M + GC + 1 + L + M + M + G ∗ (cid:42)(cid:41) D + 3 + R + M + GC + 1 + R + M + M + G ∗ (cid:42)(cid:41) D + 3 + R + M + GD + 1 + L + M + M + G ∗ (cid:42)(cid:41) B + 5 + L + M + GD + 1 + R + M + M + G ∗ (cid:42)(cid:41) B + 5 + L + M + GA + 2 + L + M + M + G ∗ (cid:42)(cid:41) A + 1 + L + M + GA + 2 + R + M + M + G ∗ (cid:42)(cid:41) A + 1 + L + M + GB + 2 + L + M + M + G ∗ (cid:42)(cid:41) B + 3 + R + M + GB + 2 + R + M + M + G ∗ (cid:42)(cid:41) B + 3 + R + M + GC + 2 + L + M + M + G ∗ (cid:42)(cid:41) C + 1 + R + M + GC + 2 + R + M + M + G ∗ (cid:42)(cid:41) C + 1 + R + M + GD + 2 + L + M + M + G ∗ (cid:42)(cid:41) D + 3 + R + M + GD + 2 + R + M + M + G ∗ (cid:42)(cid:41) D + 3 + R + M + GA + 3 + L + M + M + G ∗ (cid:42)(cid:41) A + 4 + R + M + GA + 3 + R + M + M + G ∗ (cid:42)(cid:41) A + 4 + R + M + GB + 3 + L + M + M + G ∗ (cid:42)(cid:41) B + 2 + L + M + GB + 3 + R + M + M + G ∗ (cid:42)(cid:41) B + 2 + L + M + GC + 3 + L + M + M + G ∗ (cid:42)(cid:41) C + 3 + H + M + GC + 3 + R + M + M + G ∗ (cid:42)(cid:41) C + 3 + H + M + GD + 3 + L + M + M + G ∗ (cid:42)(cid:41) D + 3 + H + M + GD + 3 + R + M + M + G ∗ (cid:42)(cid:41) D + 3 + H + M + GA + 4 + L + M + M + G ∗ (cid:42)(cid:41) A + 3 + L + M + GA + 4 + R + M + M + G ∗ (cid:42)(cid:41) A + 3 + L + M + GB + 4 + L + M + M + G ∗ (cid:42)(cid:41) B + 0 + L + M + GB + 4 + R + M + M + G ∗ (cid:42)(cid:41) B + 0 + L + M + GC + 4 + L + M + M + G ∗ (cid:42)(cid:41) A + 5 + R + M + GC + 4 + R + M + M + G ∗ (cid:42)(cid:41) A + 5 + R + M + GD + 4 + L + M + M + G ∗ (cid:42)(cid:41) B + 5 + L + M + GD + 4 + R + M + M + G ∗ (cid:42)(cid:41) B + 5 + L + M + GA + 5 + L + M + M + G ∗ (cid:42)(cid:41) D + 4 + R + M + GA + 5 + R + M + M + G ∗ (cid:42)(cid:41) D + 4 + R + M + GB + 5 + L + M + M + G ∗ (cid:42)(cid:41) B + 1 + R + M + GB + 5 + R + M + M + G ∗ (cid:42)(cid:41) B + 1 + R + M + GC + 5 + L + M + M + G ∗ (cid:42)(cid:41) A + 0 + R + M + GC + 5 + R + M + M + G ∗ (cid:42)(cid:41) A + 0 + R + M + GD + 5 + L + M + M + G ∗ (cid:42)(cid:41) D + 1 + R + M + GD + 5 + R + M + M + G ∗ (cid:42)(cid:41) D + 1 + R + M + G (C5)6 Appendix D: Cost per time complexity
Instead of counting the number of different computations done we could instead measure the total amount of timespent doing the computations. This is how long all of the computations would take if run in series on a single machine.By ‘time’ we mean not the physical time but the number of steps the Turing machines take.In the set of computations of interest S each computation i has a time to halt of t i . As in the main text all M computations in set S halt within a time of t M so t i < t M for the i in this set.As in the main text the cost is C ( M, t ) = α ( M ) t + C ( M ), with C representing initiation and termination costs.The amount of time spent doing the computations in the set S M is simply the sum of the times to halt of each ofthe computations A ( M ) = (cid:88) S M t i . (D1)i.e. this the time multiplied by the number of machines but not counting the time each machine spends in the haltstate.So now the quantity to be considered is the cost per amount of time spent doing the computations C ( M, t ) /A ( M ).This is the cost per useful step of a machine. We want to know if this quantity can be made arbitrarily small byincreasing the number of computations.The computations take at least one step of the Turing machine so the total amount of time spent computing islower bounded by the number of machines, A ( M ) ≥ M , so the upper bound on the cost per amount of time spenddoing the computation is the cost per computation, C ( M, t ) /A ( M ) ≤ C ( M, t ) /M . Therefore, the constraint on theset of computations is easier to satisfy than he constraint in the main text that S M must be such that t M grows moreslowly than M . .The constraint in this case is that (cid:80) S M t i t M must grow more slowly than M .5