Short-time Diffusion Research Articles

Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions

We report the results of single tracer particle tracking by optical tweezers and video microscopy in micellar solutions. From careful analysis in terms of different stochastic models, we show that the polystyrene tracer beads of size 0.52–2.5 μm after short-time normal diffusion turn over to perform anomalous diffusion of the form 〈r2(t)〉 ≃ tα with α ≈ 0.3. This free anomalous diffusion is ergodic and consistent with a description in terms of the generalized Langevin equation with a power-law memory kernel. With optical tweezers tracking, we unveil a power-law relaxation over several decades in time to the thermal plateau value under the confinement of the harmonic tweezer potential, as predicted previously (Phys. Rev. E 85 021147 (2012)). After the subdiffusive motion in the millisecond range, the motion becomes faster and turns either back to normal Brownian diffusion or to even faster superdiffusion, depending on the size of the tracer beads.

Phase Separation in a Lipid/Cholesterol System: Comparison of Coarse-Grained and United-Atom Simulations

Ternary mixtures of saturated and unsaturated phospholipids and cholesterol constitute a well-known model system to study raft formation in membranes. This phenomenon is, e.g., observed in cell membranes. Here, coarse-grained (CG) and microscopic united-atom (UA) simulations are performed to investigate the phase separation of a membrane bilayer for the ternary system of saturated 16:0 (DPPC) and unsaturated 18:2 (DUPC) phospholipids and cholesterol (CHOL). The results of a 9 μs UA simulation demonstrate the onset of phase separation and can be compared with properties of the corresponding CG system. While sharing the structural features of phase separation in the CG model, the onset of demixing for the UA model is 40 times slower. This factor can be rationalized by the different short-time diffusion constants. Various system properties such as order parameters and lipid-CHOL and CHOL-CHOL interactions are analyzed and compared between the UA and CG models. From the structural perspective, the phase separation process appears to be rather similar for both models, which illustrates once more the power of using CG approaches.

Investigation of field and diffusion time dependence of the diffusion‐weighted signal at ultrahigh magnetic fields

Over the last decade, there has been a significant increase in the number of high-magnetic-field MRI magnets. However, the exact effect of a high magnetic field strength (B0 ) on diffusion-weighted MR signals is not yet fully understood. The goal of this study was to investigate the influence of different high magnetic field strengths (9.4 T and 14.1 T) and diffusion times (9, 11, 13, 15, 17 and 24 ms) on the diffusion-weighted signal in rat brain white matter. At a short diffusion time (9 ms), fractional anisotropy values were found to be lower at 14.1 T than at 9.4 T, but this difference disappeared at longer diffusion times. A simple two-pool model was used to explain these findings. The model describes the white matter as a first hindered compartment (often associated with the extra-axonal space), characterized by a faster orthogonal diffusion and a lower fractional anisotropy, and a second restricted compartment (often associated with the intra-axonal space), characterized by a slower orthogonal diffusion (i.e. orthogonal to the axon direction) and a higher fractional anisotropy. Apparent T2 relaxation time measurements of the hindered and restricted pools were performed. The shortening of the pseudo-T2 value from the restricted compartment with B0 is likely to be more pronounced than the apparent T2 changes in the hindered compartment. This study suggests that the observed differences in diffusion tensor imaging parameters between the two magnetic field strengths at short diffusion time may be related to differences in the apparent T2 values between the pools.

Polymer-Mediated Clustering of Charged Anisotropic Colloids

Formation of stable, dense nanoparticle clusters is interesting due to both the underlying physics and use of nanoclusters in applications such as digital printing, imaging and biosensing, and energy storage. Here, we explore formation of nanoparticle clusters in dispersions of the model disk-shaped colloid Laponite. Under basic conditions, the model disk-shaped colloid Laponite forms a repulsive glass in water due to strong electrostatic interactions. Addition of a nonadsorbing polymer, the sodium salt of poly(acrylic acid) (PAA), induces a depletion attraction between particles. Through dynamic light scattering (DLS) and rheology, we see that the polymer initially causes a transition from the glassy phase to an ergodic fluid. Samples at higher particle concentration age to a weak nonergodic state, while samples at lower Laponite remain as fluids. As the strength of attraction between particles is increased, we find an increase in the fast relaxation time measured via dynamic light scattering (e.g., slowing of the short-time diffusion of a single particle). While this may in part be attributed to an increase in the ionic strength, the aging behavior and glass-fluid transition we observe appear to be unique to the presence of polymer, suggesting that depletion plays an important role. DLS data on the fluid samples were consistent with two widely spaced diffusive relaxation modes, corresponding to motion of single particles and motion of large clusters, although other slow dynamic processes may be present. On the basis of the estimated volume fraction and depletion attraction, we believe the Laponite-PAA suspensions to be either fluids of stable clusters or glasses of clusters, although it is possible that the nonergodic state we observe is instead a gel of clusters. Additionally, the cluster size was found to be stable for at least 120 days and was directly related to the polymer concentration. This may serve as an important means of tuning cluster size in products and processes based on dense nanoparticle assemblies.

Modified FRAP Methodology with Millisecond Time Resolution for the Investigation of Heterogeneous RNA Polymerase II Diffusion Dynamics

We have developed a laser scanning fluorescence recovery after photobleaching (FRAP) methodology capable of investigating protein diffusion and binding dynamics over several decades in time, from the millisecond through minute timescale. A several micron diameter bleach spot is established within 800 us, followed by fluorescence monitoring by a line scan through the bleach region. This method of FRAP permits the recovery to be captured with 1.6 ms time resolution, enabling observation of short time diffusion and long time binding kinetics. The rapid time resolution, precisely defined bleach geometry, and well-characterized bleach depth enable this method to be made highly quantitative through the application of a diffusion-anomalous subdiffusion model. This modeling can provide insight into the local environment through which biological molecules move. We have applied this novel implementation to investigate a fluorescently labeled subunit of RNA Polymerase II (Rpb3-GFP) in the polytene nuclei of Drosophila salivary glands. This unique chromosomal architecture creates segregated nucleoplasmic and chromatin regions. This has allowed for discriminating observations of protein dynamics while diffusing in the presence or absence of DNA obstructions and binding sites, permitting observation of three distinct regimes of protein diffusion - nucleoplasmic (absence of binding/obstruction), intrachromatin (binding/obstruction), and active transcription regions. We have observed that Rpb3-GFP diffuses anomalously in the nucleoplasm, in contrast to our studies of GFP expressed in polytene cells, which moves by Brownian diffusion through the nucleoplasm. Given the comparable size between the proteins and absence of possible chromatin binding sites, this indicates that Rpb3-GFP is engaged with other RNAP II subunits, giving rise to a distribution of diffusion coefficients, manifest as anomalous subdiffusion. We also see highly anomalous behavior for Rbp3-GFP moving through chromatin regions coupled with reduced diffusion coefficients.

Minimal model for short-time diffusion in periodic potentials

We investigate the dynamics of a single, overdamped colloidal particle, which is driven by a constant force through a one-dimensional periodic potential. We focus on systems with large barrier heights where the lowest-order cumulants of the density field, that is, average position and the mean-squared displacement, show nontrivial (nondiffusive) short-time behavior characterized by the appearance of plateaus. We demonstrate that this "cage-like" dynamics can be well described by a discretized master equation model involving two states (related to two positions) within each potential valley. Nontrivial predictions of our approach include analytic expressions for the plateau heights and an estimate of the "de-caging time" obtained from the study of deviations from Gaussian behavior. The simplicity of our approach means that it offers a minimal model to describe the short-time behavior of systems with hindered dynamics.

The influence of lung airways branching structure and diffusion time on measurements and models of short-range 3He gas MR diffusion

Hyperpolarized 3He diffusion experiments have been shown to be sensitive to changes in acinar structure due to emphysematous lung disease. Extracting quantitative information about lung microstructure from the diffusion signal is complicated due its dependence on a number of factors including diffusion time and the complex branching acinar geometry. A theoretical model (cylinder model) has been proposed as a means of estimating acinar airway dimensions from measured diffusivities. This model assumes that the effects of acinar branching geometry and finite airway length upon 3He diffusion behaviour are negligible. In this work, we use finite element simulations of diffusion in a model of branching alveolar ducts to investigate in detail the effects of acinar branching structure and finite airway length on short-range 3He diffusion measurements. The results show that branching effects have a significant influence upon 3He diffusivity, even at short diffusion times. The expressions of the cylinder model theory do not account for significant dependences upon diffusion time, branching geometry and airway length, as a consequence of the oversimplified geometrical model used. The effect of diffusion time on 3He ADC was also investigated through experiments with healthy human volunteers. The results demonstrate that the cylinder model can produce inaccurate estimates of the airway dimensions as a consequence of incompletely accounting for the diffusion-time dependence in the model equations and confirmed the predicted limitations of the cylinder model for reliable lung morphometry measurements. The results and models presented in this work may help in the development of a more realistic theoretical framework for ‘in vivo lung morphometry’ using 3He diffusion MR.

Microfluidic mobility of single (DNA-grafted) colloids in dilute DNA suspensions

By employing single particle tracking on the basis of fast video microscopy, we directly measure the equilibrium short and long time diffusion coefficients of blank and DNA-grafted (molecular weight of 250, 1000 and 4000 base pairs (bp)) tracer colloids in dilute λ-DNA solutions. We compare these results to recently reported microfluidic mobilities under non-equilibrium steady state flow and to the predictions of quantitative theoretical models. A pronounced discrepancy between the measured equilibrium and non-equilibrium mobilities is observed for colloids grafted with 4000 bp DNA molecules, revealing the inapplicability of the Stokes–Einstein relation under these conditions. We demonstrate that this deviation may be interpreted as a strong micro-shear-thinning effect that is caused by the formation of entanglements between the grafted DNA brush and λ-DNA coils from the suspension.

Effect of sweep and ejection events on particle dispersion in wall bounded turbulent flows

This paper studies the sweep and ejection events in a channel flow with Reτ =80 by using Direct Numerical Simulation (DNS). The effects of ejection and sweep events on the transport of fluid particles are analyzed separately through a quadrant technique. By analyzing trajectories of the particles released at different wall-normal locations, it is found that the particles from the ejection events mainly move upward while the particles from the sweep events move downward of the flow during short and intermediate diffusion time durations. Numerical results show that the effects of the ejection and sweep events on the mean displacement and the mean square dispersion remain for a long time, one-order of magnitude larger than the streamwise Lagrangian integral scales.

Structure and short-time dynamics in concentrated suspensions of charged colloids

We report a comprehensive joint experimental-theoretical study of the equilibrium pair-structure and short-time diffusion in aqueous suspensions of highly charged poly-acrylate (PA) spheres in the colloidal fluid phase. Low-polydispersity PA sphere systems with two different hard-core radii, R(0) = 542 and 1117 Å, are explored over a wide range of concentrations and salinities using static and dynamic light scattering (DLS), small angle x-ray scattering, and x-ray photon correlation spectroscopy (XPCS). The measured static and dynamic scattering functions are analyzed using state-of-the-art theoretical methods. For all samples, the measured static structure factor, S(Q), is in good agreement with results by an analytical integral equation method for particles interacting by a repulsive screened Coulomb plus hard-core pair potential. In our DLS and XPCS measurements, we have determined the short-time diffusion function D(Q) = D(0) H(Q)∕S(Q), comprising the free diffusion coefficient D(0) and the hydrodynamic function H(Q). The latter is calculated analytically using a self-part corrected version of the δγ-scheme by Beenakker and Mazur which accounts approximately for many-body hydrodynamic interactions (HIs). Except for low-salinity systems at the highest investigated volume fraction φ ≈ 0.32, the theoretical predictions for H(Q) are in excellent agreement with the experimental data. In particular, the increase in the collective diffusion coefficient D(c) = D(Q → 0), and the decrease of the self-diffusion coefficient, D(s) = D(Q → ∞), with increasing φ is well described. In accord with the theoretical prediction, the peak value, H(Q(m)), of H(Q) relates to the nearest neighbor cage size ∼2π∕Q(m), for which concentration scaling relations are discussed. The peak values H(Q(m)) are globally bound from below by the corresponding neutral hard-spheres peak values, and from above by the limiting peak values for low-salinity charge-stabilized systems. HIs usually slow short-time diffusion on colloidal length scales, except for the cage diffusion coefficient, D(cge) = D(Q(m)), in dilute low-salinity systems where a speed up of the system dynamics and corresponding peak values of H(Q(m)) > 1 are observed experimentally and theoretically.

Anomalous Diffusion of Brain Metabolites Evidenced by Diffusion-Weighted Magnetic Resonance Spectroscopy in Vivo

Translational displacement of molecules within cells is a key process in cellular biology. Molecular motion potentially depends on many factors, including active transport, cytosol viscosity and molecular crowding, tortuosity resulting from cytoskeleton and organelles, and restriction barriers. However, the relative contribution of these factors to molecular motion in the cytoplasm remains poorly understood. In this work, we designed an original diffusion-weighted magnetic resonance spectroscopy strategy to probe molecular motion at subcellular scales in vivo. This led to the first observation of anomalous diffusion, that is, dependence of the apparent diffusion coefficient (ADC) on the diffusion time, for endogenous intracellular metabolites in the brain. The observed increase of the ADC at short diffusion time yields evidence that metabolite motion is characteristic of hindered random diffusion rather than active transport, for time scales up to the dozen milliseconds. Armed with this knowledge, data modeling based on geometrically constrained diffusion was performed. Results suggest that metabolite diffusion occurs in a low-viscosity cytosol hindered by ∼2-μm structures, which is consistent with known intracellular organization.

Single-molecule study of molecular mobility in the cytoplasm ofEscherichia coli

The cytoplasm of bacterial cells is filled with individual molecules and molecular complexes that rely on diffusion to bring them together for interaction. The mobility of molecules in the cytoplasm has been characterized by several techniques mainly using fluorescent probes and ensemble methods. In order to probe the microenvrionment inside the cytoplasm as viewed by an individual molecule, we have studied single green fluorescent proteins (GFPs) diffusing in the cytoplasm of Escherichia coli cells at observation at rates ranging from 60 to 1000 Hz. Over long times the diffusion shows confinement due to the geometry of the cells themselves. A simulation in model cells using the actual distribution of cell sizes found in the experiments describes accurately the experimental results as well as reveals a short time diffusion coefficient that agrees well with that determined by ensemble methods. Higher short time diffusion coefficients can be obtained by filling the simulated cell with small spheres modeling cytoplasmic molecules and, depending on the density of particles included in the modeled cytoplasm, can approach the diffusion coefficient of GFPs found in water. Thus, single-molecule tracking combined with analysis using simple simulation of Brownian motion is able to reveal the main contributors to the GFP mobility in the cytoplasm of E. coli.

Dynamic Monte Carlo simulations of anisotropic colloids

We put forward a simple procedure for extracting dynamical information from Monte Carlo simulations, by appropriate matching of the short-time diffusion tensor with its infinite-dilution limit counterpart, which is supposed to be known. This approach--discarding hydrodynamics interactions--first allows us to improve the efficiency of previous dynamic Monte Carlo algorithms for spherical Brownian particles. In the second step, we address the case of anisotropic colloids with orientational degrees of freedom. As an illustration, we present a detailed study of the dynamics of thin platelets, with emphasis on long-time diffusion and orientational correlations.

Structure and Dynamics of Loosely Cross-Linked Ionic Microgel Dispersions in the Fluid Regime

We report a comprehensive experimental-theoretical study of the temperature- and concentration-dependent swelling behavior of weakly cross-linked PNiPAm ionic microgel particles in the deionized fluid phase. The particles swell reversibly when the dispersion is cooled from the collapsed state to lower temperatures. While the collapsed state shows no dependence on the microgel number density, the swelling at lower T is more pronounced at lower concentrations. The static pair correlations and short-time diffusion functions, and the concentration and temperature dependence of the microgel radius and effective charge, are studied using static and dynamic light scattering in combination with state-of-the-art analytical theoretical schemes based on a Yukawa-type effective pair potential and a core-shell model. We show that only such a combined, simultaneous fit of static and dynamic scattering functions allows for an unambiguous determination of the microgel radius and effective charge.

Characterizing Tumor Response to Chemotherapy at Various Length Scales Using Temporal Diffusion Spectroscopy

Measurements of apparent diffusion coefficient (ADC) using magnetic resonance imaging (MRI) have been suggested as potential imaging biomarkers for monitoring tumor response to treatment. However, conventional pulsed-gradient spin echo (PGSE) methods incorporate relatively long diffusion times, and are usually sensitive to changes in cell density and necrosis. Diffusion temporal spectroscopy using the oscillating gradient spin echo (OGSE) sequence is capable of probing short length scales, and may detect significant intracellular microstructural changes independent of gross cell density changes following anti-cancer treatment. To test this hypothesis, SW620 xenografts were treated by barasertib (AZD1152), a selective inhibitor of Aurora B kinase which causes SW620 cancer cells to develop polyploidy and increase in size following treatment, ultimately leading to cell death through apoptosis. Following treatment, the ADC values obtained by both the PGSE and low frequency OGSE methods increased. However, the ADC values at high gradient frequency (i.e. short diffusion times) were significantly lower in treated tumors, consistent with increased intracellular restrictions/hindrances. This suggests that ADC values at long diffusion times are dominated by tumor microstructure at long length scales, and may not convey unambiguous information of subcellular space. While the diffusion temporal spectroscopy provides more comprehensive means to probe tumor microstructure at various length scales. This work is the first study to probe intracellular microstructural variations due to polyploidy following treatment using diffusion MRI in vivo. It is also the first observation of post-treatment ADC changes occurring in opposite directions at short and long diffusion times. The current study suggests that temporal diffusion spectroscopy potentially provides pharmacodynamic biomarkers of tumor early response which distinguish microstructural variations following treatment at both the subcellular and supracellular length scales.

Oscillating and pulsed gradient diffusion magnetic resonance microscopy over an extendedb‐value range: Implications for the characterization of tissue microstructure

Oscillating gradient spin-echo (OGSE) pulse sequences have been proposed for acquiring diffusion data with very short diffusion times, which probe tissue structure at the subcellular scale. OGSE sequences are an alternative to pulsed gradient spin echo measurements, which typically probe longer diffusion times due to gradient limitations. In this investigation, a high-strength (6600 G/cm) gradient designed for small-sample microscopy was used to acquire OGSE and pulsed gradient spin echo data in a rat hippocampal specimen at microscopic resolution. Measurements covered a broad range of diffusion times (TDeff = 1.2-15.0 ms), frequencies (ω = 67-1000 Hz), and b-values (b = 0-3.2 ms/μm2). Variations in apparent diffusion coefficient with frequency and diffusion time provided microstructural information at a scale much smaller than the imaging resolution. For a more direct comparison of the techniques, OGSE and pulsed gradient spin echo data were acquired with similar effective diffusion times. Measurements with similar TDeff were consistent at low b-value (b < 1 ms/μm(2) ), but diverged at higher b-values. Experimental observations suggest that the effective diffusion time can be helpful in the interpretation of low b-value OGSE data. However, caution is required at higher b, where enhanced sensitivity to restriction and exchange render the effective diffusion time an unsuitable representation. Oscillating and pulsed gradient diffusion techniques offer unique, complementary information. In combination, the two methods provide a powerful tool for characterizing complex diffusion within biological tissues.

Sorption and Desorption of PVA-Pyrene Chains in and out of Agarose Gel

In situ steady-state fluorescence (SSF) measurement technique was applied to investigation of pyrene labeled Poly(vinyl alcohol) (PVA-Py) molecules diffusion in and out of agarose gels. Gel samples with four different concentration of agarose were prepared. PVA-Py was synthesized by "click" chemistry method and dissolved in water to use in diffusion experiments. The results were analyzed by using Fickian type diffusion model, and it was found that sorption and desorption processes of PVA-Py molecules in and out of agarose gel have two distinct regions for short and long diffusion times. Sorption and desorption coefficients were measured and it was seen that the diffusion rates were much larger at short times and at lower agarose concentrations.

Diffusion-induced stress in inhomogeneous materials: concentration-dependent elastic modulus

By incorporating the contribution of solute atoms to the Helmholtz free energy of solid solution, a linear relation is derived between Young’s modulus and the concentration of solute atoms. The solute atoms can either increase or decrease Young’s modulus of solid solution, depending on the first-order derivative of the Helmholtz free energy with respect to the concentration of solute atoms. Using this relation, a closed-form solution of the chemical stress in an elastic plate is obtained when the diffusion behavior in the plate can be described by the classical Fick’s second law with convection boundary condition on one surface and zero flux on the other surface. The plate experiences tensile stress after short diffusion time due to asymmetrical diffusion, which will likely cause surface microcracking. The results show that the effect of the concentration dependence of Young’s modulus on the evolution of chemical stress in elastic plates is negligible if the change of Young’s modulus due to the diffusive motion of solute atomsis is not compatible in magnitude with Young’s modulus of the pure material. Also, a new diffusion equation is developed for strictly regular binary solid solution. The effective diffusivity is a nonlinear function of the concentration of solute atoms.

Photoanodes Consisting of Mesoporous Anatase TiO2 Beads with Various Sizes for High-Efficiency Flexible Dye-Sensitized Solar Cells

The synthesis and characterization of TiO2 beads and the use of them in the photoanodes of all-plastic flexible dye-sensitized solar cells (FDSCs) are reported. Pure anatase TiO2 beads having different sizes and characteristics were first made using a novel two-step chemical method under different conditions. Photoanodes consisting of the beads as scattering layers were then fabricated. The use of beads largely enhances the dye loading and gives highly effective light scattering, leading to improved light absorbance. The resulting cells were evaluated for the electron diffusion time, electron lifetime, charge collection efficiency, incident photon-to-electron conversion efficiency, electron injection efficiency, and IV characteristics. The pure anatase TiO2 beads, having low oxygen vacancy concentrations and directional attached grains, lead to more photoelectrons and enhance the electron diffusion, giving very short diffusion times. We have demonstrated for the first time that the use of beads, having diameters ranging from 250 to 750 nm, enhances the light-to-electricity conversion efficiency of FDSCs having plastic substrates by as much as 28%. The cell conversion efficiency is also enhanced from 4.3 to 5.5%.

Dynamics in dense hard-sphere colloidal suspensions

The dynamic behavior of a hard-sphere colloidal suspension was studied by x-ray photon correlation spectroscopy and small-angle x-ray scattering over a wide range of particle volume fractions. The short-time mobility of the particles was found to be smaller than that of free particles even at relatively low concentrations, showing the importance of indirect hydrodynamic interactions. Hydrodynamic functions were derived from the data, and for moderate particle volume fractions (Φ≤ 0.40) there is good agreement with earlier many-body theory calculations by Beenakker and Mazur [Physica A 120, 349 (1984)]. Important discrepancies appear at higher concentrations, above Φ≈ 0.40, where the hydrodynamic effects are overestimated by the Beenakker-Mazur theory, but predicted accurately by an accelerated Stokesian dynamics algorithm developed by Banchio and Brady [J. Chem. Phys. 118, 10323 (2003)]. For the relaxation rates, good agreement was also found between the experimental data and a scaling form predicted by the mode coupling theory. In the high concentration range, with the fluid suspensions approaching the glass transition, the long-time diffusion coefficient was compared with the short-time collective diffusion coefficient to verify a scaling relation previously proposed by Segrè and Pusey [Phys. Rev. Lett. 77, 771 (1996)]. We discuss our results in view of previous experimental attempts to validate this scaling law [L. Lurio et al., Phys. Rev. Lett. 84, 785 (2000)].