Piecewise Linear Potential Research Articles

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.

Minimal model for spatial coherence resonance

We show that a planar medium, locally modeled by a simple one-dimensional excitable system with a piece-wise linear potential, can serve as a minimal model for spatial coherence resonance. Via an analytical treatment of the spatially extended system, we derive the dependence of the resonant wave number on several crucial system parameters, ranging from the diffusion coefficient to the local excursion time of constitutive excitable units. Thus, we provide vital insights into mechanisms that enable the emergence of exclusively noise-induced spatial periodicity in excitable media.

MolDock: A New Technique for High-Accuracy Molecular Docking

In this article we introduce a molecular docking algorithm called MolDock. MolDock is based on a new heuristic search algorithm that combines differential evolution with a cavity prediction algorithm. The docking scoring function of MolDock is an extension of the piecewise linear potential (PLP) including new hydrogen bonding and electrostatic terms. To further improve docking accuracy, a re-ranking scoring function is introduced, which identifies the most promising docking solution from the solutions obtained by the docking algorithm. The docking accuracy of MolDock has been evaluated by docking flexible ligands to 77 protein targets. MolDock was able to identify the correct binding mode of 87% of the complexes. In comparison, the accuracy of Glide and Surflex is 82% and 75%, respectively. FlexX obtained 58% and GOLD 78% on subsets containing 76 and 55 cases, respectively.

Diffusion in the time-dependent double-well potential

We investigate the one-dimensional diffusion of a particle in a piecewise linear W-shaped potential, on which a harmonically modulated discontinuity situated at the central tip is superimposed. The simplified description of the external driving enables an exact analysis of the emerging non-linear dynamics. The response is represented by the occupation difference between the regions of attraction of the right and the left minima of the potential profile. We discuss the time-asymptotic and time-averaged occupational differences as a function of the temperature, the amplitude and the frequency of the driving. We compare the analysis with the corresponding results based on the popular two-state description of the underlying resonance effects. The comparison reveals the fundamental role of the intra-well dynamics within the space-continuous formulation.

Single-polymer Brownian motor: A simulation study

Numerical simulation is used to study a single polymer chain in a flashing ratchet potential to determine how the mechanism of this Brownian motor system is affected by the presence of internal degrees of freedom. The polymer is modeled by a freely jointed chain with N monomers in which the monomers interact via a repulsive Lennard-Jones potential and neighboring monomers on the chain are connected by finite extensible nonlinear elastic bonds. Each monomer is acted upon by a 1D asymmetric, piecewise linear potential of spatial period L comparable to the radius of gyration of the polymer. This potential is also characterized by a localization time, t(on), and by a free diffusion time, t(off). We characterize the average motor velocity as a function of L, t(off), and N to determine optimal parameter ranges, and we evaluate motor performance in terms of finite dispersion, Peclet number, rectification efficiency, stall force, and transportation of a load against a viscous drag. We find that the polymer motor performs qualitatively better than a single particle in a flashing ratchet: with increasing N, the polymer loses velocity much more slowly than expected in the absence of internal degrees of freedom, and the motor stall force increases linearly with N. To understand these cooperative aspects of motor operation, we analyze relevant Rouse modes. The experimental feasibility is analyzed and the parameters of the model are scaled to those of lambda-DNA. Finally, in the context of experimental realization, we present initial modeling results for a 2D flashing ratchet constructed using an electrode array, and find good agreement with the results of 1D simulations although the polymer in the 2D potential sometimes briefly "detaches" from the electrode surface.

ACCELERATION OF DIFFUSION IN SUBCRITICALLY TILTED PERIODIC POTENTIALS

We study the enhancement of effective diffusion coefficient of an overdamped Brownian particle in tilted periodical potential profile. For analysis we use the model piece-wise linear potential, where independent variation of the potential profile parameters (such as height and width of potential barriers, width of potential wells and period of potential profile) is possible. This approach allows to investigate the influence of potential profile shape on the conditions for the diffusion acceleration, maximal value of the diffusion coefficient, values of noise intensity and the tilt for which the maximum occurs. For the large barriers and subcritical tilt the results are generalized for arbitrary shape of potential profile.

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.

Enhancement of stability in randomly switching potential with metastable state

The overdamped motion of a Brownian particle in randomly switching piece-wise metastable linear potential shows noise enhanced stability (NES): the noise stabilizes the metastable system and the system remains in this state for a longer time than in the absence of white noise. The mean first passage time (MFPT) has a maximum at a finite value of white noise intensity. The analytical expression of MFPT in terms of the white noise intensity, the parameters of the potential barrier, and of the dichotomous noise is derived. The conditions for the NES phenomenon and the parameter region where the effect can be observed are obtained. The mean first passage time behaviours as a function of the mean flipping rate of the potential for unstable and metastable initial configurations are also analyzed. We observe the resonant activation phenomenon for initial metastable configuration of the potential profile.

Quantum mechanical tunneling phenomena in metal–semiconductor junctions

The impact of the quantum mechanical tunneling effect on the operation of MESFET device structure has been investigated. Due to the presence of a Schottky barrier in a highly doped semiconductor, the depletion region is so narrow that electrons can tunnel through the barrier and contribute to the gate leakage current. This, in turn, facilitates current gain of the Schottky junction transistor (SJT) in the subthreshold region. In a simulation of a SJT we have used 2D Monte Carlo particle-based simulations. Quantum mechanical tunneling effects have been accounted for by using the Airy function transfer matrix approach, valid for piecewise linear potential barriers.

Replication ratchets: polymer transport enhanced by complementarity

We show that a rudimentary model of two complementary polymer chains confined to a quasi-one-dimensional geometry, each one interacting separately with a fixed particle via a ratchet potential, can be reformulated in terms of one polymer subject to a ratchet potential produced by its interaction with a dumbbell (or effective dimer). In the over-dampened regime, we look at transport properties when the system is under zero mean periodic external forcing. For a piece-wise linear asymmetric potential we exhibit the general mechanism by which the average mean velocity of a single polymer is enhanced by complementarity. We show that same behavior holds for a continuous ratchet potential. We relate this modelling to primitive molecular replication machines operating in outer space origin of life scenarios and argue that complementarity is not only essential to transcription but may also contribute to the efficiency of the dynamics involved in the replication process.

Escape from a metastable state with fluctuating barrier

We investigate the escape of a Brownian particle from fluctuating metastable states. We find the conditions for the noise enhanced stability (NES) effect for periodical driving force. We obtain general equations useful to calculate the average escape time for randomly switching potential profiles. For piece-wise linear potential profile we reveal the noise enhanced stability (NES) effect, when the height of “reverse” potential barrier of metastable state is comparatively small. We obtain analytically the condition for the NES phenomenon and the average escape time as a function of parameters, which characterize the potential and the driving dichotomous noise.

Nonequilibrium steady-state distributions in randomly switching potentials

We derive the stationary probability density for nonlinear dynamical system driven by white Gaussian noise and Markovian dichotomous noise. General results are applied to analysis of Brownian motion in two switching piece-wise linear potential profiles, namely in a rectangular potential well and in a bistable system.

A Finite Element Method with Lagrange Multipliers for Low-Frequency Harmonic Maxwell Equations

The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics. This system of partial differential equations is a model for the so-called eddy currents problem. After writing this problem in terms of the magnetic field, it is discretized by Nédélec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained if the curl-free condition is imposed on the elements in the dielectric domain. Then, the curl-free condition is imposed, at a discrete level, by introducing a piecewise linear multivalued potential. The resulting problem is shown to be a discrete version of other continuous formulation in which the magnetic field in the dielectric part of the domain has been replaced by a magnetic potential. Moreover, this approach leads to an important saving in computational effort. Problems related to the topology are also considered in that the possibility of having a nonsimply connected dielectric domain is taken into account. Implementation issues are discussed, including an amenable procedure to impose the boundary conditions by means of a Lagrange multiplier. Finally, the method is applied to solve a three-dimensional model problem: a cylindrical electrode surrounded by dielectric.

Effective Potential and Efficiency for a Single Brownian Motor

In this paper, the two-state model is extended to a general multi-state fluctuation model. By adopting a two-state piecewise linear potential, we calculate the current of the directed motion and the energy conversion efficiency of a Brownian particle. The multi-state potentials are equivalent to an effective potential that has a nonzero average slope on a large scale. The incline of the effective potential leads to the directed motion. It is noticed that there exists a maximal efficiency when appropriate values are chosen for parameters such as the noise strength and external force.

Noise-enhanced stability of periodically driven metastable states.

We study the effect of noise-enhanced stability of periodically driven metastable states in a system described by piecewise linear potential. We find that the growing of the average escape time with the intensity of the noise is depending on the initial condition of the system. We analytically obtain the condition for the noise enhanced stability effect and verify it by numerical simulations.

Improved scoring of ligand-protein interactions using OWFEG free energy grids.

A new approach to rapidly score protein-ligand interactions is tested on several protein-ligand systems. Results using this approach - the OWFEG free energy grid - are quite promising and are generally in better agreement with experiment (in some cases much better) than those obtained employing scoring techniques currently in wide use. The OWFEG free energy grid is generated from a one-window free energy perturbation MD simulation (Pearlman, D. A. J. Med. Chem. 1999, 42, 4313-4324). The OWFEG approach is applied to three protein systems: IMPDH, MAP kinase p38, and HIV-1 aspartyl protease. OWFEG scores are compared to experimental K(i) and IC50 data in each case. Empirical scoring functions applied to the same systems for comparison include ChemScore, Piecewise Linear Potential (PLP), and Dock energy score.

Current reversals in ratchets driven by trichotomous noise

The colored three-level Markovian noise-driven nonequilibrium dynamics of overdamped Brownian particles in a spatially periodic asymmetric potential (ratchet) is investigated. An explicit second-order linear ordinary differential equation for the stationary probability density distribution is obtained for the process. In the case of a piecewise linear potential with an additive three-level (trichotomous) noise the exact formula for the stationary current is presented. The dependence of the current reversals on the noise parameters is investigated in detail and illustrated by a phase diagram. Asymptotic formulas for the current for various limits of the noise parameters are found and compared with the results of other authors. Applications to the fluctuation-induced separation of particles are also discussed.

Simulation of the Influence of Interface Charge on Electron Emission

ABSTRACTSeveral materials are promising candidates for electron sources. For diamond, a tunneling interface at the back contact limits injecting charge into the conduction band, but a purely geometric model of internal field emission is inadequate to explain experimental data. The presence of a defect, modeled by a coulomb charge, within the tunneling barrier region significantly enhances transmission and, in concert with a geometrical model, may better account for observed current levels. Charge has been suggested to play a similar role in the SiO2 covering on a single tip silicon field emitter to explain experimental data. The tunneling theory in both cases is similar. In the present work, a general method for estimating electron transport and energy distributions through potential profiles, which describe both semiconductor interfaces and field emission potential barriers when a charged particle modifies the tunneling barrier, is developed. While the model is intended for treating a metal-semiconductor interface, it is cast here in terms of in a thin SiO2 coating over a silicon field emitter tip to enable qualitative comparisons with experimental data. Tunneling probabilities are found by numerically solving Schrödinger's Equation for a piece-wise linear potential using an Airy Function approach. A qualitative comparison to experimental energy distribution findings is possible by utilizing an analytical model of the field emitter tip from which current-voltage relations may be found.

Steady-state currents in sharp stochastic ratchets.

We develop and analyze a model for particle transport in a stochastic ratchet with a periodic piecewise linear potential, with diffusion coefficient D, where the force is discontinuous in position and fluctuating in time via additive telegraph noise with correlation time tau. We find asymptotic formulas for the steady-state particle current J for large and small D and tau. For example, for small tau, the sharp corners in the potential lead to J=O(tau(2) exp[-(Dtau)(-1/2)])+O(tau(5/2)), in contrast to O(tau(3)) when the potential is smooth. We show that diffusion can increase or decrease J, and derive an approximate equation for the value of D that maximizes J.

Memory correlation effect on thermal ratchets

We introduce correlation between steps in a random walk model with asymmetric piecewise linear potential where either a force or the barrier height fluctuates between two states. We call the correlation between steps that we introduce in the model memory correlation to distinguish it from the usual time correlation of the fluctuations on ratchets. The model gives a qualitative idea of the effect of finite inertia on thermal ratchets. The statistical measures that we have used to characterize the particle movement are the ratchet current and the temporal autocorrelation function. We find steady-state current reversals upon varying the memory effect parameter. We find also a transition in the autocorrelation function with increasing memory correlation that indicates a change from long memory correlated to randomly correlated motion.