Single-file Diffusion Research Articles

Single-file diffusion on self-similar substrates

.We study the single file diffusion problem on a one-dimensional lattice with a self-similar distribution of hopping rates. We find that the time dependence of the mean-square displacement of both a tagged particle and the center of mass of the system present anomalous power laws modulated by logarithmic periodic oscillations. The anomalous exponent of a tagged particle is one half of the exponent of the center of mass, and always smaller than 1/4. Using heuristic arguments, the exponents and the periods of oscillation are analytically obtained and confirmed by Monte Carlo simulations.

Single-file and normal diffusion of magnetic colloids in modulated channels.

Diffusive properties of interacting magnetic dipoles confined in a parabolic narrow channel and in the presence of a periodic modulated (corrugated) potential along the unconfined direction are studied using Brownian dynamics simulations. We compare our simulation results with the analytical result for the effective diffusion coefficient of a single particle by Festa and d'Agliano [Physica A 90, 229 (1978)] and show the importance of interparticle interaction on the diffusion process. We present results for the diffusion of magnetic dipoles as a function of linear density, strength of the periodic modulation and commensurability factor.

Tracer dynamics in a single-file system with absorbing boundary

The paper addresses the single-file diffusion in the presence of an absorbing boundary. The emphasis is on an interplay between the hard-core interparticle interaction and the absorption process. The resulting dynamics exhibits several qualitatively new features. First, starting with the exact probability density function for a given particle (a tracer), we study the long-time asymptotics of its moments. Both the mean position and the mean-square displacement are controlled by dynamical exponents which depend on the initial order of the particle in the file. Second, conditioning on nonabsorption, we study the distribution of long-living particles. In the conditioned framework, the dynamical exponents are the same for all particles, however, a given particle possesses an effective diffusion coefficient which depends on its initial order. After performing the thermodynamic limit, the conditioned dynamics of the tracer is subdiffusive, the generalized diffusion coefficient D(1/2) being different from that reported for the system without absorbing boundary.

A transition state theory for calculating hopping times and diffusion in highly confined fluids

Monte Carlo simulation is used to study the dynamical crossover from single file diffusion to normal diffusion in fluids confined to narrow channels. We show that the long time diffusion coefficients for a series of systems involving hard and soft interaction potentials can be described in terms of a hopping time that measures the time it takes for a particle to escape the cage formed by its neighbors in the pore. Free energy barriers for the particle hopping process are calculated and used to show that transition state theory effectively describes the hopping time for all the systems studied over a range of pore radii. Our work suggests that the combination of hopping times and transition state theory offers a useful and general framework to describe the dynamics of highly confined, single file fluids.

Multi-functionalization of nanoporous catalytic materials to enhance reaction yield: Statistical mechanical modeling for conversion reactions with restricted diffusive transport

ABSTRACTMulti-functionalization of catalytically-active nanomaterials provides a valuable tool for enhancing reaction yield by shifting reaction equilibrium, and potentially also by adjusting reaction-diffusion kinetics. For example, multi-functionalization of mesoporous silica to make the interior pore surface hydrophobic can enhance yield in dehydration reactions. Detailed molecular-level modeling to describe the pore environment, as well as the reaction and diffusion kinetics is challenging, although we briefly discuss current strategies. Our focus, however, is on coarse-grained stochastic modeling of the overall catalytic process for highly restricted transport within narrow pores (with single-file diffusion), while accounting for a tunable interaction of the pore interior with reaction products. We show that making the pore interior unfavorable to products can significantly enhance yield due to both thermodynamic and kinetics factors.

Anisotropic dynamics of water ultraconfined in macroscopically oriented channels of single-crystal beryl: A multifrequency analysis

The properties of fluids can be significantly altered by the geometry of their confining environments. While there has been significant work on the properties of such confined fluids, the properties of fluids under ultraconfinement, environments where, at least in one plane, the dimensions of the confining environment are similar to that of the confined molecule, have not been investigated. This paper investigates the dynamic properties of water in beryl (Be(3)Al(2)Si(6)O(18)), the structure of which contains approximately 5-Å-diam channels parallel to the c axis. Three techniques, inelastic neutron scattering, quasielastic neutron scattering, and dielectric spectroscopy, have been used to quantify these properties over a dynamic range covering approximately 16 orders of magnitude. Because beryl can be obtained in large single crystals we were able to quantify directional variations, perpendicular and parallel to the channel directions, in the dynamics of the confined fluid. These are significantly anisotropic and, somewhat counterintuitively, show that vibrations parallel to the c-axis channels are significantly more hindered than those perpendicular to the channels. The effective potential for vibrations in the c direction is harder than the potential in directions perpendicular to it. There is evidence of single-file diffusion of water molecules along the channels at higher temperatures, but below 150 K this diffusion is strongly suppressed. No such suppression, however, has been observed in the channel-perpendicular direction. Inelastic neutron scattering spectra include an intramolecular stretching O-H peak at ~465 meV. As this is nearly coincident with that known for free water molecules and approximately 30 meV higher than that in liquid water or ice, this suggests that there is no hydrogen bonding constraining vibrations between the channel water and the beryl structure. However, dielectric spectroscopic measurements at higher temperatures and lower frequencies yield an activation energy for the dipole reorientation of 16.4 ± 0.14 kJ/mol, close to the energy required to break a hydrogen bond in bulk water. This may suggest the presence of some other form of bonding between the water molecules and the structure, but the resolution of the apparent contradiction between the inelastic neutron and dielectric spectroscopic results remains uncertain.

Single-file diffusion with non-thermal initial conditions

Single-file diffusion is a theoretically challenging many-body problem where the calculation of even the simplest observables, e.g. mean square displacement, for a tracer particle requires an elaborate mathematical machinery. There is therefore a need for simple approaches which provide intuitive understandings and predict qualitatively correct behaviours. Here we put forward a scaling-type method which we use to investigate the influence of non-thermal initial conditions on the dynamics of a tracer particle. With our new approach we reproduce, up to scaling, several known long and short time asymptotic results for the tracer particle mean square displacement.

Xenon in l-Alanyl-l-Valine Nanochannels: A Highly Ideal Molecular Single-File System

Xenon-129 pulsed field gradient NMR and hyperpolarized xenon-129 spin tracer exchange NMR experiments were performed on polycrystalline l-Ala-l-Val (AV) and l-Val-l-Ala (VA) nanochannels under identical conditions of temperature and Xe pressure. Displacements of up to 3 μm were measured in specimens with mean crystallite lengths of around 50–100 μm. The combination of the two NMR techniques yielded the most definitive evidence for molecular single file diffusion to date, in which the time-scaling of the mean squared displacement is proportional to the square-root of time. The PFG-NMR echo attenuation data yield single-file mobility factors of 6 ± 0.7 × 10–13 m2 s–1/2 and 4.4 ± 0.2 × 10–13 m2 s–1/2. The results establish Xe in AV nanochannels, in particular, as an ideal experimental model system for fundamental studies of single-file diffusion dynamics.

Evolution equation for tagged-particle density and correlations in single-file diffusion

We derive and study a theoretical description for single-file diffusion, i.e., diffusion in a one-dimensional lattice of particles with hard core interaction. It is well known that for this system a tagged particle has anomalous diffusion for long times. The novelty of the present approach is that it allows for the derivation of correlations between a tagged particle and other particles of the system at a given distance with empty sites in between. The behavior of the correlation gives deeper insights into the processes involved. The numerical integration of differential equations are in good agreement with Monte Carlo simulations.

Active Transport in Dense Diffusive Single-File Systems

We study a minimal model of active transport in crowded single-file environments which generalizes the emblematic model of single-file diffusion to the case when the tracer particle (TP) performs either an autonomous directed motion or is biased by an external force, while all other particles of the environment (bath) perform unbiased diffusions. We derive explicit expressions, valid in the limit of high density of bath particles, of the full distribution P((n))(X) of the TP position and of all its cumulants, for arbitrary values of the bias f and for any time n. Our analysis reveals striking features, such as the anomalous scaling [proportionality] √[n] of all cumulants, the equality of cumulants of the same parity characteristic of a Skellam distribution and a convergence to a Gaussian distribution in spite of asymmetric density profiles of bath particles. Altogether, our results provide the full statistics of the TP position and set the basis for a refined analysis of real trajectories of active particles in crowded single-file environments.

Single-File Diffusion of Flagellin in Flagellar Filaments

A bacterial flagellar filament is a cylindrical crystal of a protein known as flagellin. Flagellin subunits travel from the cytoplasm through a 2 nm axial pore and polymerize at the filament’s distal end. They are supplied by a pump in the cell membrane powered by a proton-motive force. In a recent experiment, it was observed that growth proceeded at a rate of approximately one subunit every 2 s. Here, we asked whether transport of subunits through the pore at this rate could be effected by single-file diffusion, which we simulated by a random walk on a one-dimensional lattice. Assuming that the subunits are α-helical, the answer is yes, by a comfortable margin.

Mapping the Native Conformational Ensemble of Proteins from a Combination of Simulations and Experiments: New Insight into the src-SH3 Domain.

The biological function of a protein is strongly tied to the ensemble of three-dimensional conformations populated at physiological temperature, and dynamically transforming into each other. Experimental techniques such as nuclear magnetic resonance spectroscopy (NMR) provide a wealth of structural and dynamical information, which, in combination with an accurate atomic-level computational modeling, can disclose the details of protein behavior. We here propose a fast and efficient protocol employing molecular dynamics (MD) simulations and NMR chemical shifts, which allows one to reconstruct the detailed conformational ensemble of small globular proteins. In the case of the well-studied src-SH3 domain, we are able to obtain new important insight including the existence of a helical state in the RT loop and a pathway for single-file water diffusion in and out of the core.

Molecular dynamics simulation of single-walled silicon carbide nanotubes immersed in water

The structure and dynamics of water confined in single-walled silicon carbon nanotubes (SWSiCNTs) are investigated using molecular dynamics (MD) simulations. The density of water inside SWSiCNTs is reported, and an equation is suggested to predict the density of water inside SWSiCNTs. Interestingly, the water diffusion coefficients (D) here are larger compared with those in SWCNTs and single-walled boron-nitride nanotubes (SWBNNTs). Furthermore, water inside zigzag SWCNTs has a lower diffusion coefficient than water inside armchair SWCNTs. A thorough analysis of the density profiles, hydrogen bonding, and water molecule orientation inside SWSiCNTs is presented to explore the mechanism behind the diffusive behavior of water observed here. It is shown here, by mean square displacement (MSD) analysis, that water molecules inside SWSiCNTs diffuse with a ballistic motion mechanism for up to 500ps. Additionally it is confirmed here for the first time that water molecules confined in the SWSiCNTs with diameters of less than 10Å obey the single-file diffusion mechanism at time scales in excess of 500ps. The orientation of water molecules inside SWSiCNTs could be a good explanation for the difference between the diffusion coefficient in (6,6) and (10,0) SWSiCNTs. Finally, a PMF analysis explains the difficulty of water entrance into SWSiCNTs and also the different water self-diffusion inside armchair and zigzag SWSiCNTs. These results are motivating reasons to use SWSiCNTs in nanoscale biochannels, for instance, in drug-delivery applications.

Diffusion of Finite-Size Particles in Confined Geometries

The diffusion of finite-size hard-core interacting particles in two- or three-dimensional confined domains is considered in the limit that the confinement dimensions become comparable to the particle's dimensions. The result is a nonlinear diffusion equation for the one-particle probability density function, with an overall collective diffusion that depends on both the excluded-volume and the narrow confinement. By including both these effects, the equation is able to interpolate between severe confinement (for example, single-file diffusion) and unconfined diffusion. Numerical solutions of both the effective nonlinear diffusion equation and the stochastic particle system are presented and compared. As an application, the case of diffusion under a ratchet potential is considered, and the change in transport properties due to excluded-volume and confinement effects is examined.

Single-file diffusion in an interval: First passage properties

We investigate the long-time behavior of the survival probability of a tagged particle in a single-file diffusion in a finite interval. The boundary conditions are of two types: (1) one boundary is absorbing the second is reflecting and (2) both boundaries are absorbing. For each type of the boundary conditions we consider two types of initial conditions: (a) initial number of particles N is given and (b) initial concentration of particles is given (N is random). In all four cases the tagged-particle survival probability exhibits different asymptotic behavior. When the both boundaries are absorbing we also consider a case of a random interval length (single-file diffusion on a line with randomly distributed traps). In the latter setting, the initial concentration of particles has the same effect on the asymptotic decay of the survival probability as the concentration of traps.

Publisher's Note: Single-file diffusion and kinetics of template-assisted assembly of colloids [Phys. Rev. E85, 020402(R) (2012)

Publisher's Note: Single-file diffusion and kinetics of template-assisted assembly of colloids [Phys. Rev. E<b>85</b>, 020402(R) (2012)

Controlling reactivity of nanoporous catalyst materials by tuning reaction product-pore interior interactions: Statistical mechanical modeling

Statistical mechanical modeling is performed of a catalytic conversion reaction within a functionalized nanoporous material to assess the effect of varying the reaction product-pore interior interaction from attractive to repulsive. A strong enhancement in reactivity is observed not just due to the shift in reaction equilibrium towards completion but also due to enhanced transport within the pore resulting from reduced loading. The latter effect is strongest for highly restricted transport (single-file diffusion), and applies even for irreversible reactions. The analysis is performed utilizing a generalized hydrodynamic formulation of the reaction-diffusion equations which can reliably capture the complex interplay between reaction and restricted transport.

Signatures of normal and anomalous diffusion in nanotube systems by NMR

This paper reviews results of our recent studies of diffusion of gaseous sorbates in nanotube systems. 13C pulsed field gradient (PFG) NMR at 17.6T was used to probe diffusion of tetrafluoromethane in aluminosilicate nanotubes at several sorbate loadings. The results of PFG NMR measurements were compared with the corresponding data obtained by molecular dynamics simulations. The diffusion in this system was found to follow the laws of normal (i.e. Fickian) transport. At the same time, experimental data pointing at single-file diffusion were recorded for xenon gas confined inside dipeptide l-alanyl l-valine (AV) nanotubes. These data were obtained by means of high field 129Xe PFG NMR. The PFG NMR data are compared with recently reported hyperpolarized tracer exchange 129Xe NMR in the same material.

Stochastic models of intracellular transport

The interior of a living cell is a crowded, heterogenuous, fluctuating environment. Hence, a major challenge in modeling intracellular transport is to analyze stochastic processes within complex environments. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion and motor-driven active transport. Diffusive transport can be formulated in terms of the motion of an overdamped Brownian particle. On the other hand, active transport requires chemical energy, usually in the form of adenosine triphosphate hydrolysis, and can be direction specific, allowing biomolecules to be transported long distances; this is particularly important in neurons due to their complex geometry. In this review a wide range of analytical methods and models of intracellular transport is presented. In the case of diffusive transport, narrow escape problems, diffusion to a small target, confined and single-file diffusion, homogenization theory, and fractional diffusion are considered. In the case of active transport, Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasi-steady-state reduction methods, and mean-field approximations are considered. Applications include receptor trafficking, axonal transport, membrane diffusion, nuclear transport, protein-DNA interactions, virus trafficking, and the self-organization of subcellular structures.

Two Distinct Transport Mechanisms in AmtB and RhCG Proteins

The Amt/Rh family of membrane proteins facilitates the diffusion of ammonium across cellular membranes. Functional data show that Amt proteins, notably found in plants, transport ammonium ion (NH4+) while human rhesus (Rh) proteins transport ammonia (NH3). Comparison between the X-ray structures of the prokaryotic AmtB and the human RhCG reveals important differences at the level of their pore. Despite these important functional and structural differences between Amt and Rh proteins, studies of the AmtB transporter have led to the conclusion that proteins of both sub-families work according to the same mechanism and all transport ammonia. We performed molecular dynamics simulations of the AmtB and RhCG proteins under different conditions. Our free energy calculations suggest that the probability of finding NH3 molecules in the pore of AmtB is negligible in comparison to water and refute a single file diffusion of ammonia in AmtB. The pore lumen of RhCG is found to be more hydrophobic due to the presence of a phenylalanine conserved among Rh proteins. Simulations of RhCG also reveal that one of the signature histidine residues is occasionally exposed to the extra-cellular bulk, which is never observed in AmtB. These quite different hydration patterns are consistent with the idea that permeation in Amt and Rh proteins are not functionally equivalent and take place according to two distinct mechanisms. In both cases, after binding to the histidine dyad, ammonium deprotonates and diffuse down the pore as ammonia. Hydration of the pore in Amt proteins allows proton diffusion, which results in electrogenic transport. Such mechanism cannot take place in Rh proteins due to lack of pore hydration. Our QM/MM simulations rather show that the excess proton is released back to the extracellular bulk through a network of H-bonds, resulting in net NH3 transport.