Subdiffusive rocking ratchets in viscoelastic media: transport optimization and thermodynamic efficiency in overdamped regime

Memory-induced current reversal of Brownian motors

The kinetics of biological motors such as kinesin or dynein is notably influenced by a viscoelastic intracellular environment. The characteristic relaxation time of the cytosol is not separable from the colloidal timescale and therefore their dynamics is inherently non-Markovian. In this paper, we consider a variant of a Brownian motor model, namely, a Brownian ratchet immersed in a correlated thermal bath, and we analyze how memory influences its dynamics. In particular, we demonstrate the memory-induced current reversal effect and explain this phenomenon by applying the effective mass approximation as well as uncovering the memory-induced dynamical localization of the motor trajectories in the phase space. Our results reveal new aspects of the role of memory in microscopic systems out of thermal equilibrium.

Roughness induced current reversal in fractional hydrodynamic memory

The existence of a corrugated surface is of great importance and ubiquity in biological systems, exhibiting diverse dynamic behaviors. However, it has remained unclear whether such rough surface leads to the current reversal in fractional hydrodynamic memory. We investigate the transport of a particle within a rough potential under external forces in a subdiffusive media with fractional hydrodynamic memory. The results demonstrate that roughness induces current reversal and a transition from no transport to transport. These phenomena are analyzed through the subdiffusion, Peclet number, useful work, input power, and thermodynamic efficiency. The analysis reveals that transport results from energy conversion, wherein time-dependent periodic force is partially converted into mechanical energy to drive transport against load, and partially dissipated through environmental absorption. In addition, the findings indicate that the size and shape of ratchet tune the occurrence and disappearance of the current reversal, and control the number of times of the current reversal occurring. Furthermore, we find that temperature, friction, and load tune transport, resonant-like activity, and enhanced stability of the system, as evidenced by thermodynamic efficiency. These findings may have implications for understanding dynamics in biological systems and may be relevant for applications involving molecular devices for particle separation at the mesoscopic scale.

Negative rectification and anomalous diffusion in nonlinear substrate potentials: Dynamical relaxation and information entropy

We numerically investigate the rectification of the probability flux and dynamical relaxation of particles moving in a system with and without noise. The system, driven by two external forces, consists of two substrate potentials that have identical shapes and different potential barriers with different friction coefficients. The deterministic model exhibits the perfect rectification of the probability flux, ratchet effect, and the dependence of the unpredictability of the dynamics on basin of attraction. In contrast, the stochastic model displays that the rectification is sensitive to the temperature and an external bias. They can induce kinetic phase transitions between no transport and a finite net transport. These transitions lead to an unexpected phenomenon, called negative rectification. The results are analyzed through the corresponding time-dependent diffusion coefficient, information entropy (IE), etc. At a low temperature, anomalous diffusions occur in system. For the occurrence of the flux in certain parameter regimes, the larger the diffusion is, the smaller the corresponding IE is, and vice versa. We also present the selected parameter regimes for the emergence of the rectification and negative rectification. Additionally, we study the rectification of the interacting particles in the system and find that the flux may depend on the coupling strength and the number of the interacting particles, and that collective motions occur for the forward flux. Our work provides not only a way of the rectification for the transport of various particles (e.g., ions, electrons, photons, phonons, molecules, DNA chains, nanoswimmers, dust particles, etc.) in physics, chemistry, biology, and material science, but also a design of various circuits.

Nonequilibrium pattern formation in circularly confined two-dimensional systems with competing interactions

We numerically investigate the nonequilibrium behaviors of classic particles with competing interactions confined in a two-dimensional logarithmic trap. We reveal a quench-induced surprising dynamics exhibiting rich dynamic patterns depending upon confinement strength and trap size, which is attributed to the time-dependent competition between interparticle repulsions and attractions under a circular confinement. Moreover, in the collectively diffusive motions of the particles, we find that the emergence of dynamic structure transformation coincides with a diffusive mode transition from superdiffusion to subdiffusion. These findings are likely useful in understanding the pattern selection and evolution in various chemical and biological systems in addition to modulated systems, and add a new route to tailoring the morphology of pattern-forming systems.

Fractional electron transfer kinetics and a quantum breaking of ergodicity

The dissipative curve-crossing problem provides a paradigm for electron-transfer (ET) processes in condensed media. It establishes the simplest conceptual test bed to study the influence of the medium's dynamics on ET kinetics both on the ensemble level, and on the level of single particles. Single electron description is particularly important for nanoscaled systems like proteins, or molecular wires. Especially insightful is this framework in the semiclassical limit, where the environment can be treated classically, and an exact analytical treatment becomes feasible. Slow medium's dynamics is capable of enslaving ET and bringing it on the ensemble level from a quantum regime of nonadiabatic tunneling to the classical adiabatic regime, where electrons follow the nuclei rearrangements. This classical adiabatic textbook picture contradicts, however, in a very spectacular fashion to the statistics of single electron transitions, even in the Debye, memoryless media, also named Ohmic in the parlance of the famed spin-boson model. On the single particle level, ET always remains quantum, and this was named a quantum breaking of ergodicity in the adiabatic ET regime. What happens in the case of subdiffusive, fractional, or sub-Ohmic medium's dynamics, which is featured by power-law decaying dynamical memory effects typical, e.g., for protein macromolecules, and other viscoelastic media? Such a memory is vividly manifested by anomalous Cole-Cole dielectric response in such media. We address this question based both on accurate numerics and analytical theory. The ensemble theory remarkably agrees with the numerical dynamics of electronic populations, revealing a power-law relaxation tail even in a profoundly nonadiabatic electron transfer regime. In other words, ET in such media should typically display fractional kinetics. However, a profound difference with the numerically accurate results occurs for the distribution of residence times in the electronic states, both on the ensemble level and the level of single trajectories. Ergodicity is broken dynamically even in a more spectacular way than in the memoryless case. Our results question the applicability of all the existing and widely accepted ensemble theories of electron transfer in fractional, sub-Ohmic environments, on the level of single molecules, and provide a real challenge to face, both for theorists and experimentalists.

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.

Viscoelastic subdiffusion in a random Gaussian environment

Viscoelastic subdiffusion governed by a fractional Langevin equation is studied numerically in a random Gaussian environment modeled by stationary Gaussian potentials with decaying spatial correlations. This anomalous diffusion is archetypal for living cells, where cytoplasm is known to be viscoelastic and a spatial disorder also naturally emerges. We obtain some first important insights into it within a model one-dimensional study. Two basic types of potential correlations are studied: short-range exponentially decaying and algebraically slow decaying with an infinite correlation length, both for a moderate (several k_BT, in the units of thermal energy), and strong (5-10k_BT) disorder. For a moderate disorder, it is shown that on the ensemble level viscoelastic subdiffusion can easily overcome the medium's disorder. Asymptotically, it is not distinguishable from the disorder-free subdiffusion. However, a strong scatter in single-trajectory averages is nevertheless seen even for a moderate disorder. It features a weak ergodicity breaking, which occurs on a very long yet transient time scale. Furthermore, for a strong disorder, a very long transient regime of logarithmic, Sinai-type diffusion emerges. It can last longer and be faster in the absolute terms for weakly decaying correlations as compared with the short-range correlations. Residence time distributions in a finite spatial domain are of a generalized log-normal type and are reminiscent also of a stretched exponential distribution. They can be easily confused for power-law distributions in view of the observed weak ergodicity breaking. This suggests a revision of some experimental data and their interpretation.

Molecular machines operating on the nanoscale: from classical to quantum

The main physical features and operating principles of isothermal nanomachines in the microworld, common to both classical and quantum machines, are reviewed. Special attention is paid to the dual, constructive role of dissipation and thermal fluctuations, the fluctuation-dissipation theorem, heat losses and free energy transduction, thermodynamic efficiency, and thermodynamic efficiency at maximum power. Several basic models are considered and discussed to highlight generic physical features. This work examines some common fallacies that continue to plague the literature. In particular, the erroneous beliefs that one should minimize friction and lower the temperature for high performance of Brownian machines, and that the thermodynamic efficiency at maximum power cannot exceed one-half are discussed. The emerging topic of anomalous molecular motors operating subdiffusively but very efficiently in the viscoelastic environment of living cells is also discussed.

Modeling magnetosensitive ion channels in the viscoelastic environment of living cells

We propose and study a model of hypothetical magnetosensitive ionic channels which are long thought to be a possible candidate to explain the influence of weak magnetic fields on living organisms ranging from magnetotactic bacteria to fishes, birds, rats, bats, and other mammals including humans. The core of the model is provided by a short chain of magnetosomes serving as a sensor, which is coupled by elastic linkers to the gating elements of ion channels forming a small cluster in the cell membrane. The magnetic sensor is fixed by one end on cytoskeleton elements attached to the membrane and is exposed to viscoelastic cytosol. Its free end can reorient stochastically and subdiffusively in viscoelastic cytosol responding to external magnetic field changes and can open the gates of coupled ion channels. The sensor dynamics is generally bistable due to bistability of the gates which can be in two states with probabilities which depend on the sensor orientation. For realistic parameters, it is shown that this model channel can operate in the magnetic field of Earth for a small number (five to seven) of single-domain magnetosomes constituting the sensor rod, each of which has a typical size found in magnetotactic bacteria and other organisms or even just one sufficiently large nanoparticle of a characteristic size also found in nature. It is shown that, due to the viscoelasticity of the medium, the bistable gating dynamics generally exhibits power law and stretched exponential distributions of the residence times of the channels in their open and closed states. This provides a generic physical mechanism for the explanation of the origin of such anomalous kinetics for other ionic channels whose sensors move in a viscoelastic environment provided by either cytosol or biological membrane, in a quite general context, beyond the fascinating hypothesis of magnetosensitive ionic channels we explore.

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.

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.

Time asymmetry of the Kramers equation with nonlinear friction: fluctuation-dissipation relation and ratchet effect

We show by numerical simulations that the presence of nonlinear velocity-dependent friction forces can induce a finite net drift in the stochastic motion of a particle in contact with an equilibrium thermal bath and in an asymmetric periodic spatial potential. In particular, we study the Kramers equation for a particle subjected to Coulomb friction, namely a constant force acting in the direction opposite to the particle's velocity. We characterize the nonequilibrium irreversible dynamics by studying the generalized fluctuation-dissipation relation for this ratchet model driven by Coulomb friction.