Flashing ratchet model with high efficiency

Run-and-tumble motion in a linear ratchet potential: Analytic solution, power extraction, and first-passage properties

We explore the properties of run-and-tumble particles moving in a piecewise-linear "ratchet" potential by deriving analytic results for the system's steady-state probability density, current, entropy production rate, extractable power, and thermodynamic efficiency. The ratchet's broken spatial symmetry rectifies the particles' self-propelled motion, resulting in a positive current that peaks at finite values of the diffusion strength, ratchet height, and particle self-propulsion speed. Similar nonmonotonic behavior is also observed for the extractable power and efficiency. We find the optimal apex position for generating maximum current varies with diffusion and that entropy production can have nonmonotonic dependence on diffusion. In particular, for vanishing diffusion, entropy production remains finite when particle self-propulsion is weaker than the ratchet force. Furthermore, power extraction with near-perfect efficiency is achievable in certain parameter regimes due to the simplifications afforded by modeling "dry" active particles. In the final part, we derive mean first-passage times and splitting probabilities for different boundary and initial conditions. This work connects the study of work extraction from active matter with exactly solvable active particle models and will therefore facilitate the design of active engines through these analytic results.

n-Type Ionic-Organic Electronic Ratchets for Energy Harvesting

Ionic-organic ratchets are three-terminal electronic devices with asymmetric conductivity of the active layer.  These devices  are capable of generating useful direct current electrical power by converting electromagnetic noise signals available in any environment. In this work, we demonstrate for the first time an n-type ionic-organic ratchet which can generate a current of up to 7.29 μA and power up to 12.5 μW that exceed the values reported for many of the presently state-of-the-art, p-type organic electronic ratchets. We show that n-type ratchets require elimination of electron traps at the SiO₂ surface, which is not required in p-type devices. This can be achieved by using a trap-free passivation layer such as benzocyclobutene, where the traditional silane treatment is insufficient. Chemical doping is employed to further fill electron traps in the channel and increase carrier concentration and mobilities. Scanning Kelvin probe force microscopy studies provide evidence of a pn-like rectifying junction in the n-type ratchets fabricated in this work, which inherently differs from the rectification mechanism of previous ionic-organic p-type ratchets.

Perfect anomalous transport of subdiffusive cargos by molecular motors in viscoelastic cytosol

Multiple experiments show that various submicron particles such as magnetosomes, RNA messengers, viruses, and even much smaller nanoparticles such as globular proteins diffuse anomalously slow in viscoelastic cytosol of living cells. Hence, their sufficiently fast directional transport by molecular motors such as kinesins is crucial for the cell operation. It has been shown recently that the traditional flashing Brownian ratchet models of molecular motors are capable to describe both normal and anomalous transport of such subdiffusing cargos by molecular motors with a very high efficiency. This work elucidates further an important role of mechanochemical coupling in such an anomalous transport. It shows a natural emergence of a perfect subdiffusive ratchet regime due to allosteric effects, where the random rotations of a "catalytic wheel" at the heart of the motor operation become perfectly synchronized with the random stepping of a heavily loaded motor, so that only one ATP molecule is consumed on average at each motor step along microtubule. However, the number of rotations made by the catalytic engine and the traveling distance both scale sublinearly in time. Nevertheless, this anomalous transport can be very fast in absolute terms.

Performance optimization in two-dimensional Brownian rotary ratchet models

With a model for two-dimensional (2D) Brownian rotary ratchets being capable of producing a net torque under athermal random forces, its optimization for mean angular momentum (L), mean angular velocity (ω), and efficiency (η) is considered. In the model, supposing that such a small ratchet system is placed in a thermal bath, the motion of the rotor in the stator is described by the Langevin dynamics of a particle in a 2D ratchet potential, which consists of a static and a time-dependent interaction between rotor and stator; for the latter, we examine a force [randomly directed dc field (RDDF)] for which only the direction is instantaneously updated in a sequence of events in a Poisson process. Because of the chirality of the static part of the potential, it is found that the RDDF causes net rotation while coupling with the thermal fluctuations. Then, to maximize the efficiency of the power consumption of the net rotation, we consider optimizing the static part of the ratchet potential. A crucial point is that the proposed form of ratchet potential enables us to capture the essential feature of 2D ratchet potentials with two closed curves and allows us to systematically construct an optimization strategy. In this paper, we show a method for maximizing L, ω, and η, its outcome in 2D two-tooth ratchet systems, and a direction of optimization for a three-tooth ratchet system.

Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion

The discovery of anomalous diffusion of larger biopolymers and submicron tracers such as endogenous granules, organelles, or virus capsids in living cells, attributed to the viscoelastic nature of the cytoplasm, provokes the question whether this complex environment equally impacts the active intracellular transport of submicron cargos by molecular motors such as kinesins: does the passive anomalous diffusion of free cargo always imply its anomalously slow active transport by motors, the mean transport distance along microtubule growing sublinearly rather than linearly in time? Here we analyze this question within the widely used two-state Brownian ratchet model of kinesin motors based on the continuous-state diffusion along microtubules driven by a flashing binding potential, where the cargo particle is elastically attached to the motor. Depending on the cargo size, the loading force, the amplitude of the binding potential, the turnover frequency of the molecular motor enzyme, and the linker stiffness we demonstrate that the motor transport may turn out either normal or anomalous, as indeed measured experimentally. We show how a highly efficient normal active transport mediated by motors may emerge despite the passive anomalous diffusion of the cargo, and study the intricate effects of the elastic linker. Under different, well specified conditions the microtubule-based motor transport becomes anomalously slow and thus significantly less efficient.

Anomalous transport of subdiffusing cargos by single kinesin motors: the role of mechano-chemical coupling and anharmonicity of tether

Here we generalize our previous model of molecular motors trafficking subdiffusing cargos in viscoelastic cytosol by (i) including mechano-chemical coupling between cyclic conformational fluctuations of the motor protein driven by the reaction of ATP hydrolysis and its translational motion within the simplest two-state model of hand-over-hand motion of kinesin, and also (ii) by taking into account the anharmonicity of the tether between the motor and the cargo (its maximally possible extension length). It is shown that the major earlier results such as occurrence of normal versus anomalous transport depending on the amplitude of binding potential, cargo size and the motor turnover frequency not only survive in this more realistic model, but the results also look very similar for the correspondingly adjusted parameters. However, this more realistic model displays a substantially larger thermodynamic efficiency due to a bidirectional mechano-chemical coupling. For realistic parameters, the maximal thermodynamic efficiency can transiently be about 50% as observed for kinesins, and even larger, surprisingly also in a novel strongly anomalous (sub)transport regime, where the motor enzymatic turnovers become also anomalously slow and cannot be characterized by a turnover rate. Here anomalously slow dynamics of the cargo enforces anomalously slow cyclic kinetics of the motor protein.

Inertial effects in adiabatically driven flashing ratchets

We study analytically the effect of a small inertial correction on the properties of adiabatically driven flashing ratchets. Parrondo's lemma [J. M. R. Parrondo, Phys. Rev. E 57, 7297 (1998)] is generalized to include the inertial term so as to establish the symmetry conditions allowing directed motion (other than in the overdamped massless case) and to obtain a high-temperature expansion of the motion velocity for arbitrary potential profiles. The inertial correction is thus shown to enhance the ratchet effect at all temperatures for sawtooth potentials and at high temperatures for simple potentials described by the first two harmonics. With the special choice of potentials represented by at least the first three harmonics, the correction gives rise to the motion reversal in the high-temperature region. In the low-temperature region, inertia weakens the ratchet effect, with the exception of the on-off model, where diffusion is important. The directed motion adiabatically driven by potential sign fluctuations, though forbidden in the overdamped limit, becomes possible due to purely inertial effects in neither symmetric nor antisymmetric potentials, i.e., not for commonly used sawtooth and two-sinusoid profiles.

How molecular motors work in the crowded environment of living cells: coexistence and efficiency of normal and anomalous transport

Recent experiments reveal both passive subdiffusion of various nanoparticles and anomalous active transport of such particles by molecular motors in the molecularly crowded environment of living biological cells. Passive and active microrheology reveals that the origin of this anomalous dynamics is due to the viscoelasticity of the intracellular fluid. How do molecular motors perform in such a highly viscous, dissipative environment? Can we explain the observed co-existence of the anomalous transport of relatively large particles of 100 to 500 nm in size by kinesin motors with the normal transport of smaller particles by the same molecular motors? What is the efficiency of molecular motors in the anomalous transport regime? Here we answer these seemingly conflicting questions and consistently explain experimental findings in a generalization of the well-known continuous diffusion model for molecular motors with two conformational states in which viscoelastic effects are included.

Adiabatically driven Brownian pumps

We investigate a Brownian pump which, being powered by a flashing ratchet mechanism, produces net particle transport through a membrane. The extension of the Parrondo's approach developed for reversible Brownian motors [Parrondo, Phys. Rev. E 57, 7297 (1998)] to adiabatically driven pumps is given. We demonstrate that the pumping mechanism becomes especially efficient when the time variation of the potential occurs adiabatically fast or adiabatically slow, in perfect analogy with adiabatically driven Brownian motors which exhibit high efficiency [Rozenbaum et al., Phys. Rev. E 85, 041116 (2012)]. At the same time, the efficiency of the pumping mechanism is shown to be less than that of Brownian motors due to fluctuations of the number of particles in the membrane.

Symmetry interplay in Brownian photomotors: From a single-molecule device to ensemble transport

Unlike most of Brownian motor models in which the state of a point particle is described by a single scalar fluctuating parameter, we consider light-induced dichotomic fluctuations of electron density distributions in an extended molecule moving in the electrostatic periodic potential of a polar substrate. This model implies that the potential energy profiles of two motor states differ substantially and their symmetry is dictated by the interplay between the symmetries of the substrate potential and of molecular electronic states. As shown, a necessary condition for the occurrence of directed motion, the asymmetry of the potential energy profiles, is satisfied for (i) symmetric electron density distributions in molecules on asymmetric substrates and (ii) asymmetric electron density distributions in molecules on symmetric substrates. In the former case, the average velocity of directed motion is independent of molecular orientations and the ensemble of molecules moves as a whole, whereas in the latter case, oppositely oriented molecules move counterdirectionally thus causing the ensemble to diffuse. Using quantum chemical data for a specific organic-based photomotor as an example, we demonstrate that the behavior of molecular ensembles is controllable by switching on/off resonance laser radiation: they can be transported as a whole or separated into differently oriented molecules depending on the ratio of symmetric and antisymmetric contributions to the substrate electrostatic potential and to the molecular electron density distributions.

Adiabatically slow and adiabatically fast driven ratchets

We revisit two known models of deterministically driven ratchets, which exhibit high energetic efficiency, with the goal to uncover similarities and differences in the principles of their operation. Both the models rely on adiabaticity of the potential change process, however, the adiabaticity that we deal with in the two cases is of different types, slow and fast. It is shown that in the former (latter) case the drift velocity is an even (odd) functional of the potential, with the notable consequence that for the adiabatically slow driven ratchet the necessary symmetry breaking occurs only due to time-dependent parametric perturbations, while the spatial asymmetry of the potential is a mandatory condition for the adiabatically fast driven ratchet to operate. To treat energetic characteristics, the models are restated in terms of traveling potential ratchets. With such an approach, we find that in these cases (i) the conditions of high energetic efficiency to be reached are similar, and (ii) the symmetry properties of the kinetic coefficients are different. Based on our results, a strategy for designing efficient Brownian motors is suggested.

Friction in carborane-based molecular rotors driven by gas flow or electric field: classical molecular dynamics

Friction in molecular rotors is examined by classical molecular dynamics simulations for grid-mounted azimuthal dipolar molecular rotors, whose rotation is either allowed to decay freely or is driven at GHz frequencies by a flow of rare gas or by a rotating electric field. The rotating parts (rotators) are propeller-shaped. Their two to six blades consist of condensed aromatic rings and are attached to a deltahedral carborane hub, whose antipodal carbons carry [n]staffane axles mounted on a square molecular grid. The dynamic friction constant η has been derived in several independent ways with similar results. Analysis of free rotation decay yields η as a continuous exponentially decreasing function of rotor frequency. The calculated dependence of friction torque on frequency resembles the classical macroscopic Stribeck curve. Its relation to rotational potential energy barriers and the key role of the rate of intramolecular vibrational redistribution (IVR) of energy and angular momentum from rotator rotation to other modes are considered in two limiting regimes. (i) In the strongly overdamped regime, rotation is much slower than IVR, and effective friction can be expressed through potential barriers to rotation. (ii) In the strongly underdamped regime, rotation is much faster than IVR, whose rate then determines friction.

Quasiequilibrium directed hopping in a time-dependent two-well periodic potential

We consider the directed motion of a Brownian particle in a two-well periodic potential with time-varying barriers and wells described by arbitrary periodic functions of time, v(t) and u(t), alternating with the period τ. In the framework of the low-temperature kinetic approach, we obtain explicit formulas for the probabilities of finding the particle in potential wells, average velocity of directed motion, input energy P(in) and useful work P(out) against additionally introduced stationary load force f. These formulas are considerably simplified by the assumption of the quasiequilibrium regime of motion corresponding to small values of u(t) and f. It is shown that depending on the same or opposite parity of the functions v(t) and u(t) with respect to time reversal, the motion direction of a Brownian particle is retained or reversed under the reversal of the direction of movement along the (v-u) loop in the phase space of the functions v(t) and u(t), and the nondiagonal kinetic coefficients are mutually symmetric or antisymmetric. In the adiabatic limit τ→∞, the average velocity is proportional to τ(-1) in two cases: (i) the above loop has a nonzero area, (ii) the functions v(t) and u(t) are proportional to each other (zero loop area) and include intervals of fast changes with small durations τ(0) on the period τ of their variations. In both of these cases, the efficiency of energy conversion, η=P(out)/P(in), tends to unity at large variations of the barriers v(t). In the second case, the deviation of η from unity can be split into two contributions: The former decreases exponentially with increasing amplitude v(0) of v(t), while the latter is a small nonadiabatic correction proportional to v(0)(-3/2). It is the nonadiabatic correction that limits high efficiencies at large variations of barriers.

Brownian motor with competing spatial and temporal asymmetry of potential energy

A Brownian motor is considered which operates due to asymmetric dichotomic fluctuations of the spatially periodic asymmetric potential energy. As shown, the motion direction and stopping points of this motor are dictated by the competition between the spatial and temporal asymmetry of the potential energy (or solely by temporal asymmetry in the case that the potential energy sign fluctuates). For an asymmetric sawtooth potential, the Brownian-particle average velocity is calculated numerically as a function of certain parameters of the model, whereas the low-frequency and low-energy approximations allow the corresponding analytical relationships to be derived for an arbitrarily shaped potential profile. It is shown that temporal asymmetry is not necessary for stopping point occurrence provided that the potential profile fluctuates not only in amplitude but in shape as well. This inference is illustrated by photoinduced fluctuations of the potential energy for a number of substituted arylpyrene molecules on a substrate with symmetrically distributed charge density.

Allosteric model of an ion pump

We present a simple model of a free-energy transducer made of allosterically coupled two ratchet subsystems. Each of the subsystems transports particles from one particle reservoir to another. The coupling of the subsystems imposes correlated transitions of the potential profiles of the two subsystems. As a result, a downhill flux in one subsystem with higher chemical-potential difference drives an uphill flux in the other subsystem with lower chemical-potential difference. The direction of the driven flux inverts depending on the direction of the driving flux. The ratio between the fluxes conveyed by the two subsystems is variable and nonstoichiometric. By selecting appropriate parameters, the maximum ratio of the driven flux to driving flux and maximum free-energy transducing efficiency reaches some 90 and 40%, respectively. At a stalled state, the driven flux vanishes while the driving flux remains finite. The allosteric model enables explicit analysis of the timing between binding-unbinding of particles and transitions of potential profile. The behavior of the model is similar to but different from that of the alternate access model, which is a biochemical model for active transport proteins. Our model works also as a regulatory system. We suggest that the correlated transitions of the subsystems (subunits or domains) through allosteric interaction are the origin of the diverse functions of the protein machineries.

Brownian dipole rotator in alternating electric field

The study addresses the azimuthal jumping motion of an adsorbed polar molecule in a periodic n -well potential under the action of an external alternating electric field. Starting from the perturbation theory of the Pauli equation with respect to the weak field intensity, explicit analytical expressions have been derived for the time dependence of the average dipole moment as well as the frequency dependences of polarizability and the average angular velocity, the three quantities exhibiting conspicuous stochastic resonance. As shown, unidirectional rotation can arise only provided simultaneous modulation of the minima and maxima of the potential by an external alternating field. For a symmetric potential of hindered rotation, the average angular velocity, if calculated by the second-order perturbation theory with respect to the field intensity, has a nonzero value only at n=2 , i.e., when two azimuthal wells specify a selected axis in the system. Particular consideration is given to the effect caused by the asymmetry of the two-well potential on the dielectric loss spectrum and other Brownian motion parameters. When the asymmetric potential in a system of dipole rotators arises from the average local fields induced by an orientational phase transition, the characteristics concerned show certain peculiarities which enable detection of the phase transition and determination of its parameters.

Photoinduced molecular transport in biological environments based on dipole moment fluctuations

Consideration is given to the possibility of a molecule moving unidirectionally in an electric field of a polar periodic substrate as a result of the fluctuations of molecular dipole moment occurring on the photoexcitation of the molecule. As estimated for such motion, molecules with sufficiently long fluorescence and strongly differing dipole moments in the ground and excited states can move with an average velocity of the same order as that typical of protein motors such as kinesin. This effect results from the mutual compensation of two opposite factors acting in dipole photomotors, namely, a lower energy of interaction with the substrate relative to that for protein motors and a shorter excited-state lifetime as compared with the duration of the hydrolytic splitting of adenosinetriphosphate in protein motors.

Conventional and generalized efficiencies of flashing and rocking ratchets: analytical comparison of high-efficiency limits

We consider two basic types of Brownian motors which generate directed motion in a periodic asymmetric piecewise-linear potential as a result of random half-period shifts of the potential relief (flashing ratchets) or due to a temporally asymmetric unbiased force applied to the system (rocking ratchets). Analytical relationships have been derived which enable the comparison of the upper limits for the conventional and generalized energy conversion efficiencies in these motors. As found, the increasing amplitude of a sawtooth potential (or the decreasing temperature) makes the conventional efficiency tend to the unity limit faster for a rocking ratchet (in the absence of temporal asymmetry) than for a flashing ratchet. The inverse is true for the generalized efficiency. The potential amplitude being the same, the generalized efficiency is always less than the conventional efficiency. A decreased asymmetry of the potential always results in the reduction of both efficiencies. The temporal asymmetry of an unbiased force has an opposite effect on the conventional and generalized efficiencies: the former rises and the latter drops as the positive signal component becomes shorter in time and larger in amplitude.

On a pulsating Brownian motor and its characterization

As a model of Brownian motor we consider the jump diffusion motion of a particle in the presence of an asymmetric periodic potential with a unique minimum and subject to half-period space shifts at the instants of occurrence of two Poisson processes. The relevant quantities, i.e., probability current, effective driving force, stall force, power and efficiency of the motor are explicitly calculated as averages of certain functions of the random variable representing the particle position.

A Markovian engine for a biological energy transducer: the catalytic wheel

The molecular machines in biological cells are made of proteins, DNAs and other classes of molecules. The structures of these molecules are characteristically "soft", highly flexible, and yet their interactions with other molecules or ions are specific and selective. This chapter discusses a prevalent form, the catalytic wheel, or the energy transducer of cells, examines its mechanism of action, and extracts from it a set of simple but general rules for understanding the energetics of the biomolecular devices. These rules should also benefit design of manmade nanometer scale machines such as rotary motors or track-guided linear transporters. We will focus on an electric work that, by matching system dynamics and then enhancing the conformational fluctuation of one or several driver proteins, converts stochastic input of energy into rotation or locomotion of a receptor protein. The spatial (or barrier) and temporal symmetry breakings required for selected driver/receptor combinations are examined. This electric ratchet consists of a core engine that follows the Markovian dynamic, alleviates difficulties encountered in rigid mechanical model, and tailors to the soft-matter characteristics of the biomolecules.

The enhancement of the energetic efficiency by the cooperation of low-efficient flashing ratchets

The energetic efficiency of the classical flashing ratchet model is much lower (around 1%) than the experimental results (around 50%). Jülicher and Prost have provided a prototype of the cooperative ratchet model [Jülicher, F., Prost, J., 1995. Cooperative molecular motors. Phys. Rev. Lett. 75, 2618-2621], whose energetic efficiency is of the order of 50%. However, it remains an unsettled question how the low-efficient flashing ratchets cooperate with each other and achieve higher energetic efficiency. In this paper, we investigate two energy dissipation rates as well as the energetic efficiency for the single and the cooperative flashing ratchet model, close to and far from the equilibrium, based on the stochastic energetics. By comparing these quantities between the single and the cooperative ratchet, we provide an interpretation of the efficiency enhancement by the cooperation of low-efficient flashing ratchets.

Flux reversal in a simple random-walk model on a fluctuating symmetric lattice

A rather simple random-walk model on a one-dimensional lattice is put forward. The lattice as a whole switches randomly between two possible states which are spatially symmetric. Both lattice states are identical, but translated by one site with respect to each other, and consist of infinite arrays of absorbing sites separated by two nonabsorbing sites. Exact explicit expressions for the long-time velocity and the effective diffusion coefficient are obtained and discussed. In particular, it is shown that the direction of the steady motion can be reversed by conveniently varying the values of either the mean residence times in the lattice states or the transition rates to the absorbing and nonabsorbing sites.

Acceleration of diffusion in randomly switching potential with supersymmetry

We investigate the overdamped Brownian motion in a supersymmetric periodic potential switched by Markovian dichotomous noise between two configurations. The two configurations differ from each other by a shift of one-half period. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential. For the sawtooth potential, we obtain the exact expression for the effective diffusion coefficient, which is valid for the arbitrary mean rate of potential switchings and arbitrary intensity of white Gaussian noise. We find the acceleration of diffusion in comparison with the free diffusion case and a finite net diffusion in the absence of thermal noise. Such a potential could be used to enhance the diffusion over its free value by an appropriate choice of parameters.

Two approaches toward a high-efficiency flashing ratchet

For a flashing ratchet with periodic potentials fluctuating via random shifts of one-half period, a high efficiency is shown to result from two mechanisms. The previously reported one [Yu. A. Makhnovskii, Phys. Rev. E 69, 021102 (2004); V. M. Rozenbaum, JETP Lett. 79, 388 (2004)] is realized in the near-equilibrium region and implies, first, the presence of a high barrier V0 blocking the reverse movement of a Brownian particle and, second, identical, though energy-shifted, portions of the asymmetric flat potential profile on both half periods. We report another mechanism acting far from equilibrium, typical of strongly asymmetric potentials which are shaped identically on both half periods with a large energetic shift DeltaV . The two mechanisms exhibit radically different limiting behavior of the maximum possible efficiency: eta(m) approximately 1-exp (-beta V0 /2) for the former and eta(m) approximately 1-ln (2betaDeltaV) /betaDeltaV for the latter ( beta being the reciprocal temperature in energy units). The flux and the efficiency for a Brownian motor with a piecewise-linear potential are calculated using the transfer matrix method; an exact analytical solution can thus be obtained for an extremely asymmetric sawtooth potential, the simplest example of the second high-efficiency mechanism. As demonstrated, the mechanisms considered are also characteristic of a two-well periodic potential treated in terms of the kinetic approach.

Brownian rectifiers in the presence of temporally asymmetric unbiased forces

The efficiency of energy transduction in a temporally asymmetric rocked ratchet is studied. Time asymmetry favors current in one direction and suppresses it in the opposite direction due to which large efficiency approximately 50% is readily obtained. The spatial asymmetry in the potential together with system inhomogeneity may help in further enhancing the efficiency. Fine tuning of system parameters considered leads to multiple current reversals even in the adiabatic regime.