Single-file Model Research Articles

Exact fluctuation and long-range correlations in a single-file model under resetting.

Resetting is a renewal mechanism in which a process is intermittently repeated after a random or fixed time. This simple act of stop and repeat profoundly influences the behavior of a system as exemplified by the emergence of nonequilibrium properties and expedition of search processes. Herein we explore the ramifications of stochastic resetting in the context of a single-file system called random average process (RAP) in one dimension. In particular, we focus on the dynamics of tracer particles and analytically compute the variance, equal time correlation, autocorrelation, and unequal time correlation between the positions of different tracer particles. Our study unveils that resetting gives rise to rather different behaviors depending on whether the particles move symmetrically or asymmetrically. For the asymmetric case, the system for instance exhibits a long-range correlation which is not seen in absence of the resetting. Similarly, in contrast to the reset-free RAP, the variance shows distinct scalings for symmetric and asymmetric cases. While for the symmetric case, it decays (towards its steady value) as ∼e^{-rt}/sqrt[t], we find ∼te^{-rt} decay for the asymmetric case (r being the resetting rate). Finally, we examine the autocorrelation and unequal time correlation in the steady state and demonstrate that they obey interesting scaling forms at late times. All our analytical results are substantiated by extensive numerical simulations.

Stabilisation of stochastic single-file dynamics using port-Hamiltonian systems

This study revisits a recently proposed symmetric port-Hamiltonian single-file model in one dimension. The uniform streaming solutions are stable in the deterministic model. However, the introduction of white noise into the dynamics causes the model to exhibit divergence. In response, we add a relaxation term that draws the agents’ speed to a desired constant speed and plays the role of the input in the port-Hamiltonian framework. Our results show that this relaxation term effectively stabilises the dynamics even in the presence of stochastic noise.

On the role of adhesion in single-file dynamics

For a one-dimensional interacting system of Brownian particles with hard-core interactions (a single-file model), we study the effect of adhesion on both the collective diffusion (diffusion of the entire system with respect to its center of mass) and the tracer diffusion (diffusion of the individual tagged particles). For the case with no adhesion, all properties of these particle systems that are independent of particle labeling (symmetric in all particle coordinates and velocities) are identical to those of non-interacting particles (Lebowitz and Percus, 1967). We clarify this last fact twice. First, we derive our analytical predictions that show that the probability-density functions of single-file (ρsf) and ordinary (ρord) diffusion are identical, ρsf=ρord, predicting a nonanomalous (ordinary) behavior for the collective single-file diffusion, where the average second moment with respect to the center of mass, 〈x(t)2〉, is calculated from ρ for both diffusion processes. Second, for single-file diffusion, we show, both analytically and through large-scale simulations, that 〈x(t)2〉 grows linearly with time, confirming the nonanomalous behavior. This nonanomalous collective behavior comes in contrast to the well-known anomalous sub-diffusion behavior of the individual tagged particles (Harris, 1965). We introduce adhesion to single-file dynamics as a second inter-particle interaction rule and, interestingly, we show that adding adhesion does reduce the magnitudes of both 〈x(t)2〉 and the mean square displacement per particle Δx2; but the diffusion behavior remains intact independent of adhesion in both cases. Moreover, we study the dependence of both the collective diffusion constant D and the tracer diffusion constant DT on the adhesion coefficient α.

Single file dynamics in soft materials.

The term single file (SF) dynamics refers to the motion of an assembly of particles through a channel with cross-sections comparable to the particles' diameter. Single file diffusion (SFD) is then the diffusion of a tagged particle in a single file, i.e., under the condition that particle passing is not allowed. SFD accounts for a large variety of processes in nature, including diffusion of colloids in synthetic and natural channels, biological motors along molecular chains, electrons in proteins and liquid helium, ions through membranes, just to mention a few examples. Albeit introduced in 1965s, over the last decade the classical notion of SF dynamics has been generalised to account for a more realistic modelling of the particle properties, file geometry, particle-particle and channel-particle interactions, which paves the way to remarkable applications of the SF model, for instance, in the technology of bio-integrated nanodevices. We provide here a comprehensive review of the recent advances in the theory of SF dynamics with the purpose of spurring further experimental work.

Evidences of Efficient Investment Portfolio in Indian Capital Markets - An Analysis Based on BSE and NSE Indices

A decent budgetary portfolio is nothing more, and nothing less, than an accumulation of advantages that develop in quality and produce abundance money for the financial specialist to spend or reinvest. Markowitz (1959) is one of the pioneers of present day portfolio hypothesis. Generally, the measure of danger utilized as a part of portfolio advancement models is the fluctuation. On the other hand, option measures of danger i.e., beta (un-standardized coefficient) has been utilized by Sharpe as a part of single file model. This paper goes for applying so as to build an ideal portfolio Sharpe's single record model. For this reason the day by day shutting costs of 50 organizations recorded on the National Stock Exchange (NSE) which include the Nifty Index would be considered for the period July 2012 to June 2014. The study shows financial specialist ought to be making interest in HCL Technologies Ltd. with an extent of 77.91%, and Housing Development Finance Corporation Ltd. with an extent of 22.09%. Financial specialist is obliged to short offer Bharat Petroleum Corporation Ltd., Asian Paints Ltd., United Spirits Ltd., and Bharti Airtel Ltd., stocks to expand portfolio return. This paper would be of extensive importance and valuable to the different financial specialists in determination of stocks for their portfolios.

Everlasting effect of initial conditions on single-file diffusion

We study the dynamics of a tagged particle in an environment of point Brownian particles with hard-core interactions in an infinite one-dimensional channel (a single-file model). In particular, we examine the influence of initial conditions on the dynamics of the tagged particle. We compare two initial conditions: equal distances between particles and uniform density distribution. The effect is shown by the differences of mean-square-displacement and correlation function for the two ensembles of initial conditions. We discuss the violation of Einstein relation, and its dependence on the initial condition, and the difference between time and ensemble averaging. More specifically, using the Jepsen line, we will discuss how transport coefficients, like diffusivity, depend on the initial state. Our work shows that initial conditions determine the long time limit of the dynamics, and in this sense the system never forgets its initial state in complete contrast with thermal systems (i.e., a closed system that attains equilibrium independent of the initial state).

Langevin formulation for single-file diffusion

We introduce a stochastic equation for the microscopic motion of a tagged particle in the single-file model. This equation provides a compact representation of several of the system's properties such as fluctuation-dissipation and linear-response relations, achieved by means of a diffusion noise approach. Most importantly, the proposed Langevin equation reproduces quantitatively the three temporal regimes and the corresponding time scales: ballistic, diffusive, and subdiffusive.

Filled pore approximation: a theoretical framework for solute-solvent coupling in narrow water channels.

A phenomenological model is presented of water and solute transport that is applicable to water pores with radii less than approximately 2 A. This includes such examples as gramicidin A, the proximal tubule basolateral membrane, and the aquaporin 1 (CHIP28) water channel. The model differs from the conventional single-file model by allowing for a variation of unoccupied volume within the pores. It is shown that the accessible or free portion of the unoccupied volume can be related to the mechanical frictional coefficients and thereby to the filtration and diffusive permeabilities by the filled pore approximation. In general, the smallness of the unoccupied volume represents the compactness of the molecules within the pore and is indicative of the single-file character of the motion of water and solute moving together. When that volume is equal to a single water volume, the results are identical to the conventional single-file model. An important result is that, despite very low diffusive permeabilities, the reflection coefficient of a solute can remain at approximately 0.5 if its frictional interaction with the channel walls is comparable with its frictional interaction with neighboring water molecules. This is consistent with values previously reported for NaCl in cell membranes of proximal tubule. The model predicts a minimum effective pore radius for a water channel of 1.78 A and corresponds to a maximum filtration-to-diffusion permeability ratio that is proportional to the length of the effective pore or channel. This limiting condition corresponds to a water channel completely filled by water and may be applicable to the aquaporin 1 water channel.

A multi-substrate single-file model for ion-coupled transporters

Ion-coupled transporters are simulated by a model that differs from contemporary alternating-access schemes. Beginning with concepts derived from multi-ion pores, the model assumes that substrates (both inorganic ions and small organic molecules) hop a) between the solutions and binding sites and b) between binding sites within a single-file pore. No two substrates can simultaneously occupy the same site. Rate constants for hopping can be increased both a) when substrates in two sites attract each other into a vacant site between them and b) when substrates in adjacent sites repel each other. Hopping rate constants for charged substrates are also modified by the membrane field. For a three-site model, simulated annealing yields parameters to fit steady-state measurements of flux coupling, transport-associated currents, and charge movements for the GABA transporter GAT1. The model then accounts for some GAT1 kinetic data as well. The model also yields parameters that describe the available data for the rat 5-HT transporter and for the rabbit Na(+)-glucose transporter. The simulations show that coupled fluxes and other aspects of ion transport can be explained by a model that includes local substrate-substrate interactions but no explicit global conformational changes.

ANALYSIS OF ENERGY BROADENING IN CHARGED PARTICLE BEAMS

The anomalous energy distribution of a monomer ion or electron beam is described by a one parameter, single file model governed by shot noise emission and pairwise Coulomb interactions. The dependence of the width of the distribution and the shift in the mean energy on current, mass, and voltage is calculated. A unification of a number of other theories of energy broadening is made by variation of a free parameter that represents field-geometrical effects. A feature unique to the present model is that can be inversely related to charge.

Rate theory models for ion transport through rigid pores. II. Time-dependent analysis of the single-file model with special reference to oscillatory behaviour

This article deals with the time-dependent evolution of the single-file movement of ions through channels of both biological and artificial membranes. The single-file transport process may exhibit not only the usual relaxation behaviour but also oscillatory behaviour as a steady state is approached after an initial perturbation. A necessary condition for the occurrence of oscillations is that the system acts sufficiently far from equilibrium. The occurrence of oscillations is due to the interactions within the transport system which are taken into account by the single-file model; these are the electrostatic repulsion between the ions being transported, and the competition of the ions for the free binding sites within the pore. Information about the strength of the interactions can be obtained by measuring the damping of the transport observables (e.g. the electric current): The stronger the inter-ionic repulsion, the more apparent the oscillatory behaviour will become. Furthermore, the damping is influenced by the microscopic structure of the transport system (i.e. the energy profile of the pores). With an increasing degree of microscopicity, i.e. with a decreasing number of binding sites and an increasingly irregular pore profile, the oscillations become more damped. However, a considerable oscillatory behaviour can only be predicted for pores with both a sufficiently regular structure and a sufficiently large number of binding sites. For this class of pores, however, the measurement of the damping represents an appropriate method of gaining information which could exceed that obtainable from the. usual methods of measuring stationary quantities (e.g. stationary conductance). Moreover, our goal is to explain theoretically how the oscillatory behaviour can be interpreted in terms of the order inherent in the ionic movement, which is determined by both the external and internal forces and the microscopic properties of the system.

Rate theory models for ion transport through rigid: Pores. III. Continuum vs discrete models in single file diffusion

In studying the single file model in its discrete as well as in its continuum form the relationship between the phenomenological continuum theory of diffusion and the rate theory approach is analyzed. The single file model in its original form is discrete and represents the most general rate theory model for ion transport through rigid pores in biological membranes. In neglecting the interionic interactions which the single file model takes into account, the Nernst-Planck equation of macroscopic free diffusion can be derived from single file by means of the procedure n leads to infinity (where n is the number of binding sites within a pore) and the classical diffusion theory can thereby be integrated into the more general concept of single filing transport. Moreover, the single file model has been transformed in the limit n leads to infinity into the corresponding continuum form involving interionic interactions. The essential differences between the two derived continuum forms are: In the macroscopic diffusion model, the interionic interactions are regarded in the form of a "mean field". Thus we only get one equation of motion (Nernst-Planck equation) for the ionic concentration c(x, t) within the membrane. In the continuum version of the single file model, however, we obtain a hierarchy of Fokker-Planck equations for the probability density functions Pm(x1, . . . , xm, t) (where m is the number of ions within a pore). The interactions of the single file system are incorporated in detail into the Fokker-Planck equation as well as into the corresponding boundary conditions. As a consequence, the boundary conditions are highly complex in comparison with periodic conditions or Dirichlet conditions often used for the Nernst-Planck equation in electrophysiology. Two types of boundary conditions have been found which are principally different: The first one is to regulate the entry and exit of the ions at the pore mouth by a negative feedback mechanism, the second one describes the collisions of the ions within multiply occupied pores. In this context the question is discussed of whether the continuum version of single file has advantages over the discrete one.

Analysis of cell cycle in root meristems

We have analyzed the cell proliferation in a meristem assuming a single file model for root architecture. The meristem file appears to be built up by two clearly separated zones: the first going from the initial cell to the middle of the meristem and the second from the middle to the meristem boundary. The first half of the meristem shows an exponential age distribution for the cell population. In contrast, in the second half of the meristem, the cell kinetics of cycling cells strongly disagree with exponential kinetics and due to the compensation between the observed deviations in both halves, cell supply in the file meristem is in line with linear kinetics. However, we proposed that exponential kinetic equations offer a suitable approach to problems of cell cycle compartments and population age distributions in real meristems, where non-cycling cells cannot be identified inside the meristem, whether we consider the meristem as a whole or study a “window” inside it. Nevertheless, for more exact kinetic analysis, when estimating the proliferative fraction, the width of the “window” and its location along the axis must carefully be taken into account.

Random walk analysis of potassium fluxes associated with nerve impulses.

We present a novel recursion method for obtaining theoretical expressions for unidirectional single-file fluxes of ions through narrow membrane channels containing an arbitrary number of ion sites. The theory is applied to experimental tracer fluxes associated with nerve impulses from cephalopod giant axon membranes at various temperatures between 7 degrees and 27 degrees. The comparison between the theoretical and experimental one-way fluxes suggests that the potassium channel in nerve membrane contains three ion sites, which is consistent with the deduction by Hodgkin and Keynes that the potassium channel contains two or three sites on the basis of the ratio of tracer influx to tracer efflux. The analytical results in this paper provide a further test of the single-file model for nerve and other membrane preparations.

A Single-File Model for Potassium Transport in Squid Giant Axon: Simulation of Potassium Currents at Normal Ionic Concentrations

A physical model for potassium transport in squid giant axon is proposed. The model is designed to explain the empirical data given by the Hodgkin-Huxley model and related experiments. It is assumed that K(+) moves across the axon membrane by single-file diffusion through narrow pores. In the model a pore has three negatively charged sites that can be occupied alternatively by K(+) or by a gating particle, GP(++), coming from the external surface. GP(++) is considered to be part of the membrane rather than a diffusible component of the surrounding solutions. A high activation barrier for GP(++) is supposed at the inner membrane border so that it cannot change over to the internal surface. Therefore potassium diffusion can be blocked by GP(++) penetrating into the pores. This mechanism controls the dynamic behaviour of the model. The time-dependent probabilities of the pore states are described by a system of differential equations. The rate constants in these equations depend on the ionic concentrations, the membrane voltage, and the electrostatic interaction between ions in a single pore. Detailed computational tests for normal composition of external and internal solutions show that the model agrees remarkably well with the stationary and dynamic behaviour of the Hodgkin-Huxley model. However, the hyperpolarization delay is not reproduced. A structural modification, concerning this delay and the way in which GP(++) is attached to the membrane, is proposed, and the qualitative behavior of the model at varied external and internal concentrations is discussed.

A model for cell cycle and growth kinetics in roots

A model is proposed which accounts for the cell cycle and growth kinetics in roots growing under steady state conditions. In the model the root is conceived as a single file of cells and the three sequential zones of the root are considered. Nc cells, arranged in a single longitudinal file, make up the meristem zone. The most distal cell of this file is the initial and it has an indefinite cycle capacity, whereas all the others are derived cells with a limited programme of n cycles. The model predicts Nc = 2n. The number (Nc) of meristem cells will allow us to calculate the number of cycles that any derived cell undergoes before arresting division and triggering off its elongation. The cycle kinetics are linear and therefore the distribution of the Nc cells in the cycle compartments is directly proportional to each phase duration, and constant with time. The rate of meristem cell formation is defined by the cell flow (ϕ) as the frequency of cells that pass any point of the cycle per time unit. It also represents the frequency of cells that the meristem supplies to the elongation zone per time unit. The elongation zone is formed by Ntr cells in the process of enlargement. The model predicts thatNtr=ϕ.No.Ttr Ttr being the time that cells devote to their enlargement throughout this zone. The equation allows us to estimate this value. Changes in root length will depend only on length changes in the mature zone where G=No.ϕ.L G being the rate of growth and L the final cell size. During symplastic growth the root may be regarded as a bundle of files with identical growth rates. This leads to quantitative relationships between the growth components of the different kinds of files. Lastly, the fitness of the single file model against the actual growing root is discussed.

Relevance of the single-file model to water flow through porous cell membranes

A physical argument shows that pores of molecular width and 100 Å length connecting two water-filled vessels are empty of water molecules approximately 98% of the time, contain one molecule approximately 2% of the time, and rarely contain two or more molecules. As a result, the “single-file” model of Hodgkin and Keynes does not apply to the flow of water through porous cell membranes. The same result is obtained from the equation of state of a one-dimensional fluid.

THE SINGLE FILE MODEL FOR THE DIFFUSION OF IONS THROUGH A MEMBRANE.

The model proposed by A. L. Hodgkin and R. D. Keynes (Jour. of Physiol.,128, 61–88, 1955) for the diffusion of potassium through the nerve membrane is extended to cover an arbitrary number of species of ions with charges not necessarily the same. One type of interference is also investigated.