Scalable, Time-Responsive, Digital, Energy-Efficient Molecular Circuits using DNA Strand Displacement

We propose a novel theoretical biomolecular design to implement any Boolean circuit using the mechanism of DNA strand displacement. The design is scalable: all species of DNA strands can in principle be mixed and prepared in a single test tube, rather than requiring separate purification of each species, which is a barrier to large-scale synthesis. The design is time-responsive: the concentration of output species changes in response to the concentration of input species, so that time-varying inputs may be continuously processed. The design is digital: Boolean values of wires in the circuit are represented as high or low concentrations of certain species, and we show how to construct a single-input, single-output signal restoration gate that amplifies the difference between high and low, which can be distributed to each wire in the circuit to overcome signal degradation. This means we can achieve a digital abstraction of the analog values of concentrations. Finally, the design is energy-efficient: if input species are specified ideally (meaning absolutely 0 concentration of unwanted species), then output species converge to their ideal concentrations at steady-state, and the system at steady-state is in (dynamic) equilibrium, meaning that no energy is consumed by irreversible reactions until the input again changes. Drawbacks of our design include the following. If input is provided non-ideally (small positive concentration of unwanted species), then energy must be continually expended to maintain correct output concentrations even at steady-state. In addition, our fuel species - those species that are permanently consumed in irreversible reactions - are not "generic"; each gate in the circuit is powered by its own specific type of fuel species. Hence different circuits must be powered by different types of fuel. Finally, we require input to be given according to the dual-rail convention, so that an input of 0 is specified not only by the absence of a certain species, but by the presence of another. That is, we do not construct a "true NOT gate" that sets its output to high concentration if and only if its input's concentration is low. It remains an open problem to design scalable, time-responsive, digital, energy-efficient molecular circuits that additionally solve one of these problems, or to prove that some subset of their resolutions are mutually incompatible.Comment: version 2: the paper itself is unchanged from version 1, but the arXiv software stripped some asterisk characters out of the abstract whose purpose was to highlight words. These characters have been replaced with underscores in version 2. The arXiv software also removed the second paragraph of the abstract, which has been (attempted to be) re-inserted. Also, although the secondary subject is "Soft Condensed Matter", this classification was chosen by the arXiv moderators after submission, not chosen by the authors. The authors consider this submission to be a theoretical computer science paper

Noise-assisted classical adiabatic pumping in a symmetric periodic potential

We consider a classical overdamped Brownian particle moving in a symmetric periodic potential. We show that a net particle flow can be produced by adiabatically changing two external periodic potentials with a spatial and a temporal phase difference. The classical pumped current is found to be independent of the friction and to vanish both in the limit of low and high temperature. Below a critical temperature, adiabatic pumping appears to be more efficient than transport due to a constant external force.Comment: six pages, 3 figure

DNA hybridization catalysts and catalyst circuits

Practically all of life's molecular processes, from chemical synthesis to replication, involve enzymes that carry out their functions through the catalysis of metastable fuels into waste products. Catalytic control of reaction rates will prove to be as useful and ubiquitous in DNA nanotechnology as it is in biology. Here we present experimental results on the control of the decay rates of a metastable DNA "fuel". We show that the fuel complex can be induced to decay with a rate about 1600 times faster than it would decay spontaneously. The original DNA hybridization catalyst [15] achieved a maximal speed-up of roughly 30. The fuel complex discussed here can therefore serve as the basic ingredient for an improved DNA hybridization catalyst. As an example application for DNA hybridization catalysts, we propose a method for implementing arbitrary digital logic circuits

Inattainability of Carnot efficiency in the Brownian heat engine

We discuss the reversibility of Brownian heat engine. We perform asymptotic analysis of Kramers equation on B\"uttiker-Landauer system and show quantitatively that Carnot efficiency is inattainable even in a fully overdamping limit. The inattainability is attributed to the inevitable irreversible heat flow over the temperature boundary.Comment: 5 pages, to appear in Phys. Rev.

Deterministic ratchets: route to diffusive transport

The rectification efficiency of an underdamped ratchet operated in the adiabatic regime increases according to a scaling current-amplitude curve as the damping constant approaches a critical threshold; below threshold the rectified signal becomes extremely irregular and eventually its time average drops to zero. Periodic (locked) and diffusive (fully chaotic) trajectories coexist on fine tuning the amplitude of the input signal. The transition from regular to chaotic transport in noiseless ratchets is studied numerically.Comment: 9 pages, 5 figures, to be published in Phys. Rev.

Leaderless deterministic chemical reaction networks

This paper answers an open question of Chen, Doty, and Soloveichik [1], who showed that a function f:N^k --> N^l is deterministically computable by a stochastic chemical reaction network (CRN) if and only if the graph of f is a semilinear subset of N^{k+l}. That construction crucially used "leaders": the ability to start in an initial configuration with constant but non-zero counts of species other than the k species X_1,...,X_k representing the input to the function f. The authors asked whether deterministic CRNs without a leader retain the same power. We answer this question affirmatively, showing that every semilinear function is deterministically computable by a CRN whose initial configuration contains only the input species X_1,...,X_k, and zero counts of every other species. We show that this CRN completes in expected time O(n), where n is the total number of input molecules. This time bound is slower than the O(log^5 n) achieved in [1], but faster than the O(n log n) achieved by the direct construction of [1] (Theorem 4.1 in the latest online version of [1]), since the fast construction of that paper (Theorem 4.4) relied heavily on the use of a fast, error-prone CRN that computes arbitrary computable functions, and which crucially uses a leader.Comment: arXiv admin note: substantial text overlap with arXiv:1204.417

Hopping motion of lattice gases through nonsymmetric potentials under strong bias conditions

The hopping motion of lattice gases through potentials without mirror-reflection symmetry is investigated under various bias conditions. The model of 2 particles on a ring with 4 sites is solved explicitly; the resulting current in a sawtooth potential is discussed. The current of lattice gases in extended systems consisting of periodic repetitions of segments with sawtooth potentials is studied for different concentrations and values of the bias. Rectification effects are observed, similar to the single-particle case. A mean-field approximation for the current in the case of strong bias acting against the highest barriers in the system is made and compared with numerical simulations. The particle-vacancy symmetry of the model is discussed.Comment: 8 pages (incl. 6 eps figures); RevTeX 3.

Soliton ratchets

The mechanism underlying the soliton ratchet, both in absence and in presence of noise, is investigated. We show the existence of an asymmetric internal mode on the soliton profile which couples, trough the damping in the system, to the soliton translational mode. Effective soliton transport is achieved when the internal mode and the external force are phase locked. We use as working model a generalized double sine-Gordon equation. The phenomenon is expected to be valid for generic soliton systems.Comment: 4 pages, 4 figure

Depinning of kinks in a Josephson-junction ratchet array

We have measured the depinning of trapped kinks in a ratchet potential using a fabricated circular array of Josephson junctions. Our ratchet system consists of a parallel array of junctions with alternating cell inductances and junctions areas. We have compared this ratchet array with other circular arrays. We find experimentally and numerically that the depinning current depends on the direction of the applied current in our ratchet ring. We also find other properties of the depinning current versus applied field, such as a long period and a lack of reflection symmetry, which we can explain analytically.Comment: to be published in PR

Lyapunov Potential Description for Laser Dynamics

We describe the dynamical behavior of both class A and class B lasers in terms of a Lyapunov potential. For class A lasers we use the potential to analyze both deterministic and stochastic dynamics. In the stochastic case it is found that the phase of the electric field drifts with time in the steady state. For class B lasers, the potential obtained is valid in the absence of noise. In this case, a general expression relating the period of the relaxation oscillations to the potential is found. We have included in this expression the terms corresponding to the gain saturation and the mean value of the spontaneously emitted power, which were not considered previously. The validity of this expression is also discussed and a semi-empirical relation giving the period of the relaxation oscillations far from the stationary state is proposed and checked against numerical simulations.Comment: 13 pages (including 7 figures) LaTeX file. To appear in Phys Rev.A (June 1999