Time-dependent Mean Square Research Articles

Side Chain Dynamics of Poly(norbornene)-g-Poly(propylene oxide) Bottlebrush Polymers.

The segmental dynamics of the side chains of poly(norbornene)-g-poly(propylene oxide) (PNB-g-PPO) bottlebrush polymer in comparison to PPO is studied by quasi-elastic neutron scattering. Having experimental time and length scale information simultaneously allows to extract spatial information in addition to relaxation time. Tethering one end of the PPO side chain onto a stiff PNB backbone slows down the segmental relaxation, over the length and time scales investigated. The power law dependence of the relaxation time on the momentum transfer, Q, indicates a more heterogeneous relaxation pattern for the bottlebrush polymer, whereas the linear PPO has less deviations from a homogenous relaxation. Similar conclusions can be drawn from the time dependent mean square displacement, 〈r2 (t)〉, and the non-Gaussian parameter, α2 (t). Herein, the bottlebrush polymer shows a more restricted dynamics, whereas the linear PPO reaches 〈r2 (t)〉∝t0.5 at the highest temperature. The deviations from Gaussian behavior are evident at the α2 (t). Both samples show a decaying α2 (t). The non-Gaussian parameter supports the results from the power law dependence of the relaxation times, with lower α2 (t) values for PPO compared to those for PNB-g-PPO, pointing to less deviations from Gaussian behavior.

Computational modeling of charge hopping dynamics along a disordered one-dimensional wire with energy gradients in quantum environments.

This computational study investigates the effects of energy gradients on charge hopping dynamics along a one-dimensional chain of discrete sites coupled to quantum bath, which is modeled at the level of Pauli master equation (PME). This study also assesses the performance of different approximations for the hopping rates. Three different methods for solving the PME, a fourth order Runge-Kutta method, numerical diagonalization of the rate matrix followed by analytic propagation, and kinetic Monte Carlo simulation method, are tested and confirmed to produce virtually identical values of time dependent mean square displacement, diffusion constant, and mobility. Five different rate expressions, exact numerical evaluation of Fermi's Golden Rule (FGR) rate, stationary phase interpolation (SPI) approximation, semiclassical approximation, classical Marcus rate, and Miller-Abrahams rate, are tested to help understand the effects of approximations in representing quantum environments in the presence of energy gradients. The results based on direct numerical evaluation of FGR rate exhibit transition from diffusive to non-diffusive behavior with the increase in the gradient and show that the charge transport in the quantum bath is more sensitive to the magnitude of the gradient and the disorder than in the classical bath. Among all the four approximations for the hopping rates, the SPI approximation is confirmed to work best overall. A comparison of two different methods to calculate the mobility identifies drift motion of the population distribution as the major source of non-diffusive behavior and provides more reliable information on the contribution of quantum bath.

Asymptotic neutron scattering laws for anomalously diffusing quantum particles.

The paper deals with a model-free approach to the analysis of quasielastic neutron scattering intensities from anomalously diffusing quantum particles. All quantities are inferred from the asymptotic form of their time-dependent mean square displacements which grow ∝t(α), with 0 ≤ α < 2. Confined diffusion (α = 0) is here explicitly included. We discuss in particular the intermediate scattering function for long times and the Fourier spectrum of the velocity autocorrelation function for small frequencies. Quantum effects enter in both cases through the general symmetry properties of quantum time correlation functions. It is shown that the fractional diffusion constant can be expressed by a Green-Kubo type relation involving the real part of the velocity autocorrelation function. The theory is exact in the diffusive regime and at moderate momentum transfers.

Communication: Probing anomalous diffusion in frequency space.

Anomalous diffusion processes are usually detected by analyzing the time-dependent mean square displacement of the diffusing particles. The latter evolves asymptotically as W(t) ∼ 2Dαt(α), where Dα is the fractional diffusion constant and 0 < α < 2. In this article we show that both Dα and α can also be extracted from the low-frequency Fourier spectrum of the corresponding velocity autocorrelation function. This offers a simple method for the interpretation of quasielastic neutron scattering spectra from complex (bio)molecular systems, in which subdiffusive transport is frequently encountered. The approach is illustrated and validated by analyzing molecular dynamics simulations of molecular diffusion in a lipid POPC bilayer.

Ion dynamics in glassy ionic conductors: Scaling of mean square displacement of mobile ions

In this letter, we present for the first time a new scaling for the mean square displacement of mobile ions in glasses. We have obtained time-dependent mean square displacement of mobile ions by mapping the frequency-dependent conductivity into time domain. We have obtained a universal scaling for the mean square displacement of mobile ions for silver phosphate glasses. The present scaling for the mean square displacement is consistent with the scaling of the conductivity spectra and is supported by the structural features.

In Vivo Studies of Active Processes in the Escherichia Coli Nucleoid

The cell is the site of actively motor-driven processes which drive the intracellular environment far from thermodynamic equilibrium. The dynamics of biological macromolecules such as DNA in such an environment are complex and subject to a multitude of constraints and forces. Inspired by our in vitro studies of DNA looping with optical tweezers that showed that additional non-thermal fluctuations in the DNA can substantially enhance the formation of regulatory DNA-protein complexes, we study the conformational fluctuations of chromosomal DNA in vivo in Escherichia coli by Fluorescence Correlation Spectroscopy (FCS).Conformational fluctuations of the DNA move the bound fluorophores stochastically into the diffraction-limited excitation volume of a focused laser beam in a confocal microscope. From the time correlation functions of the measured fluorescence intensity, we quantify the fluctuations of the DNA as measured by its time-dependent mean square displacement, and the viscoelastic moduli of the nucleoid. These quantities in live cells significantly differ from the ATP-depleted dead cells on longer time scales, indicating that the fluctuations on longer time scale may be driven by active processes involving molecular motors that generate forces by ATP hydrolysis. On shorter time scales, we see little difference between live and dead cells, suggesting that the processes on corresponding short length scales rely primarily on thermally-driven diffusive mechanisms. We also note that the rheological properties of E. coli nucleoid significantly change when the ATP hydrolysis in cells is inhibited.

Communication: Translational Brownian motion for particles of arbitrary shape

A single Brownian particle of arbitrary shape is considered. The time-dependent translational mean square displacement W(t) of a reference point at this particle is evaluated from the Smoluchowski equation. It is shown that at times larger than the characteristic time scale of the rotational Brownian relaxation, the slope of W(t) becomes independent of the choice of a reference point. Moreover, it is proved that in the long-time limit, the slope of W(t) is determined uniquely by the trace of the translational-translational mobility matrix μ(tt) evaluated with respect to the hydrodynamic center of mobility. The result is applicable to dynamic light scattering measurements, which indeed are performed in the long-time limit.

Some Anomalies in the Self-Diffusion of Water in Disordered Carbons

The unusual dynamics of water under confinement is important to a number of conventional and emerging industrial processes and to life in general; nevertheless, our understanding of this critical area is still in an early stage. Nanoporous carbons have provided a key material through which to investigate this dynamics; however, existing simulations have been based on the use of idealized slit pore carbons or carbon nanotubes with smooth energy landscapes and fail to capture the influence of structural disorder inherent to real carbons. We show here that the irregular structure of such carbons critically influences the dynamics and the mode of diffusion (single file, subdiffusion or Fickian). Our molecular dynamics simulations, using a realistic hydrophobic carbon model based on hybrid reverse Monte Carlo simulation of the structure of an activated carbon fiber, show the existence of a single-file diffusion mode between the ballistic and Fickian modes in the narrowest pore regions of this material, not seen in simulations using model 2-D slit pores. A rich variety of behavior is found in this subdiffusion regime, with the fits of time-dependent mean square displacement to the power law of time (∝tα), revealing that the exponent α varies significantly with temperature, especially at low temperatures (273–350 K). It reaches a minimum value of 0.5 at 298 K corresponding to the single-file diffusion regime and approaches unity at 610 K, corresponding to the Fickian mode. It is demonstrated that confinement effects lead to the experimentally observed non-Arrhenius behavior of the water dynamics at low temperatures (<350 K), for both the realistic carbon model and an idealized 2-D slit pore model, due to the transport of the adsorbed water as a large cluster in a water monolayer.

Atomic-scale structural evolution from disorder to order in an amorphous metal

In this paper, we performed molecular dynamics simulations to study the atomic-scale structural evolution from disorder to order during the isothermal annealing of an amorphous Ni. Three plateaus in the time dependent potential energy and mean square displacement (MSD) curves were observed, indicating that the atomic ordering process from amorphous to nanocrystalline Ni undergoes three distinct stages. The structural analyses reveal that the atomic structural evolution is associated with these three stages: Disordered atoms adjust their relative positions to form a one-dimensional (1D) periodic structure at the first stage, then form a 2D periodic structure at the second stage, and finally form a 3D periodic nanocrystal. Further analyses of potential energy and MSD difference and dynamics demonstrate that the structural change from the 2D to 3D structure is more difficult than that from the 1D to 2D structure, because both the 1D and 2D quasi-ordered structures belong to transition states and have similar...

Implementation of a Protein Reduced Point Charge Model toward Molecular Dynamics Applications

A reduced point charge model was developed in a previous work from the study of extrema in smoothed charge density distribution functions generated from the Amber99 molecular electrostatic potential. In the present work, such a point charge distribution is coupled with the Amber99 force field and implemented in the program TINKER to allow molecular dynamics (MD) simulations of proteins. First applications to two polypeptides that involve α-helix and β-sheet motifs are analyzed and compared to all-atom MD simulations. Two types of coarse-grained (CG)-based trajectories are generated using, on one hand, harmonic bond stretching terms and, on the other hand, distance restraints. Results show that the use of the unrestrained CG conditions are sufficient to preserve most of the secondary structure characteristics but restraints lead to a better agreement between CG and all-atom simulation results such as rmsd, dipole moment, and time-dependent mean square deviation functions.

Particle Tracking Microrheology of Lyotropic Liquid Crystals

We present comprehensive results on the microrheological study of lyotropic liquid crystalline phases of various space groups constituted by water-monoglyceride (Dimodan) mixtures. In order to explore the viscoelastic properties of these systems, we use particle tracking of probe colloidal particles suitably dispersed in the liquid crystals and monitored by diffusing wave spectroscopy. The identification of the various liquid crystalline phases was separately carried out by small-angle X-ray scattering. The restricted motion of the particles was monitored and identified by the decay time of intensity autocorrelation function and the corresponding time-dependent mean square displacement (MSD), which revealed space group-dependent behavior. The characteristic time extracted by the intersection of the slopes of the MSD at short and long time scales, provided a characteristic time which could be directly compared with the relaxation time obtained by microrheology. Further direct comparison of microrheology and bulk rheology measurements was gained via the Laplace transform of the generalized time-dependent MSD, yielding the microrheology storage and loss moduli, G'(ω) and G''(ω), in the frequency domain ω. The general picture emerging from the microrheology data is that all liquid crystals exhibit viscoelastic properties in line with results from bulk rheology and the transition regime (elastic to viscous) differs according to the specific liquid crystal considered. In the case of the lamellar phase, a plastic fluid is measured by bulk rheology, while microrheology indicates viscoelastic behavior. Although we generally find good qualitative agreement between the two techniques, all liquid crystalline systems are found to relax faster when studied with microrheology. The most plausible explanation for this difference is due to the different length scales probed by the two techniques: that is, microscopical relaxation on these structured fluids, is likely to occur at shorter time scales which are more suitably probed by microrheology, whereas bulk, macroscopic relaxations occurring at longer time scales can only be probed by bulk rheology.

Proteins as fractals: Role of the hydrodynamic interaction

Exploiting the fractal nature of folded proteins, we study the effect of the hydrodynamic interaction between amino acids using a Zimm-type model. We compute the time-dependent mean square displacement of an amino acid and the time-dependent autocorrelation function of the distance between two amino acids, and we show that these dynamic quantities evolve anomalously, similar to the Rouse-type behavior, yet with modified dynamic exponents. Good agreement is found with recent neutron spin-echo studies of myoglobin and hemoglobin.

Dynamics of dilute colloidal suspensions in modulated potentials

The dynamics of dilute suspensions of charged polystyrene spheres have been studied in a sinusoidal potential. We investigated experimentally and theoretically the effect of the wavelength and amplitude of this potential as well as the particle size on the mean square displacement, the distribution of displacements and the non-Gaussian parameter. These properties scale with the radius of the particles and the magnitude of the wavevector of the potential as expected from a dimensional analysis. In contrast, they show a non-trivial dependence on the amplitude of the potential, which determines the barrier height encountered in long-distance motions and also the intermediate-time dynamics in the potential well. The periodic potential leads to particle dynamics which resemble the self dynamics of various systems approaching their glass transition. In particular, we found that the time-dependent mean square displacements determined in our system are surprisingly close to the ones in a quasi two-dimensional colloidal supercooled liquid. In this case, the role of the colloid volume fraction is played by the amplitude of the potential. The similarity of the mean square displacements is particularly striking since an individual colloidal particle in a periodic potential represents a considerably simpler situation than a highly concentrated multi-particle system.

Thermal behaviour of hydrated lysozyme in the presence of sucrose and trehalose by EINS

The main aim of the present work is to investigate the dynamical effects of the addition of disaccharides to hydrated lysozyme. The Self-Distribution-Function procedure is at first applied on Elastic Incoherent Neutron Scattering data obtained by the IN13 spectrometer on trehalose/H 2O and sucrose/H 2O where it highlights a different Q-dependence for the two disaccharides. Then a quantitative analysis of the Mean Square Displacement of lysozyme/trehalose/H 2O and of lysozyme/sucrose/H 2O from Elastic Incoherent Neutron Scattering data obtained by the IN10 spectrometer is presented. It is shown how the resolution function gives rise to a time integration of a given time-dependent Mean Square Displacement function. Furthermore the analysis shows that the protein dynamical transition, registered at a temperature value of about T = 200–220 K in the H 2O hydrated lysozyme sample, is inhibited by the addition of the disaccharides.

Analysis of Microscopic Parameters of Single-Particle Trajectories in Neurons

We performed a comparative study of the statistical uncertainties that arise when calculating the velocity and diffusion coefficients from single-particle trajectories. We show that a method where particle mean displacement is used to calculate velocity and mean square fluctuation is used to calculate diffusion coefficient offers greater accuracy than analysis of time-dependent mean square displacement. Our assessment of the performance of the two analysis strategies is conducted in two ways. First, we apply each of the methods to simulated trajectories where each parameter term is known. Second, we analyze the motion of previously uncharacterized EphB2 receptors in the membrane of hippocampal neurons. We find that EphB2 receptors display different types of motion mode and transition between these modes. We present our data as a distribution of microscopic diffusion coefficients for each particle trajectory, which we refer to as partial distributions. Partial distributions are summed to form a cumulative distribution of diffusion coefficients for EphB2 receptors in hippocampal neurons. The structure and interpretation of the EphB2 cumulative distribution are discussed.

Anomalies in the vibrational dynamics of proteins are a consequence of fractal-like structure

Proteins have been shown to exhibit strange/anomalous dynamics displaying non-Debye density of vibrational states, anomalous spread of vibrational energy, large conformational changes, nonexponential decay of correlations, and nonexponential unfolding times. The anomalous behavior may, in principle, stem from various factors affecting the energy landscape under which a protein vibrates. Investigating the origins of such unconventional dynamics, we focus on the structure-dynamics interplay and introduce a stochastic approach to the vibrational dynamics of proteins. We use diffusion, a method sensitive to the structural features of the protein fold and them alone, in order to probe protein structure. Conducting a large-scale study of diffusion on over 500 Protein Data Bank structures we find it to be anomalous, an indication of a fractal-like structure. Taking advantage of known and newly derived relations between vibrational dynamics and diffusion, we demonstrate the equivalence of our findings to the existence of structurally originated anomalies in the vibrational dynamics of proteins. We conclude that these anomalies are a direct result of the fractal-like structure of proteins. The duality between diffusion and vibrational dynamics allows us to make, on a single-molecule level, experimentally testable predictions. The time dependent vibrational mean square displacement of an amino acid is predicted to be subdiffusive. The thermal variance in the instantaneous distance between amino acids is shown to grow as a power law of the equilibrium distance. Mean first passage time analysis is offered as a practical tool that may aid in the identification of amino acid pairs involved in large conformational changes.

Direct Correlation between Nonrandom Ion Hopping and Network Structure in Ion-Conducting Borophosphate Glasses

We present temperature-dependent conductivity spectra of sodium borophosphate glasses with a varying borate/phosphate ratio but a constant sodium oxide content which can be mapped into time-dependent mean square displacements. For the first time, we show that characteristic lengths of ion transport derived thereof are directly linked to features of network structure, viz., the number of boron oxide tetrahedra. Our results also shed light on the mixed network former effect.

Experimental confirmation of oscillating properties of the complex conductivity: Dielectric study of polymerization/vitrification reaction

Clear evidence of the existence of fractional kinetics containing the complex power-law exponents were obtained by conductivity measurements of polymerization reaction of polyvinylpyrrolidone (PVP) performed inside a dielectric cell. We established the relationship between the Fourier image R( jω) of the complex memory function K( t) and the time-dependent mean square displacement 〈 r 2( t)〉. This relationship helps to understand the origin of the different power-law exponents appearing in the real part of complex conductivity Re[ σ( ω)] and find a physical/geometrical meaning of the power-law exponents that can form the complex-conjugated values. The complex-conjugated values of the power-law exponents leading to oscillating behavior of conductivity follows from the fractional kinetics suggested by one of the authors (R.R.N.). The relationships [ R( jω)⇔Re[ σ( ω)]⇔〈 r 2( t)〉] are becoming very efficient in classification of different types of collective motions belonging to light and heavy carriers involved in the relaxation/transfer process. The conductivity data obtained for Re[ σ( ω)] during the whole polymerization process of the PVP at different temperatures (80, 90, 100 °C) are very well described by the fitting function that follows from the suggested theory. Original fitting procedure based on the application of the eigen-coordinates (ECs) method helps to provide a reliable fitting procedure in two stages and use the well-developed and statistically stable linear least square method (LLSM) for obtaining the correct values of the fitting parameters that describe the behavior of Re[ σ( ω, T r)] in the available frequency range for the current time of the chemical reaction T r measured during the whole process of polymerization. The suggested theory gives a unique possibility to classify the basic types of motions that take place during the whole polymerization process.

Rotational diffusion microrheology.

Examining the rotational diffusion of a microparticle suspended in a soft material opens up exciting new opportunities for locally probing the frequency-dependent linear viscoelastic shear modulus, G*(omega). We study the one-dimensional rotational diffusion of a wax microdisk in an aqueous polymer entanglement network using light streak tracking. By measuring the disk's time-dependent mean square angular displacement, <Delta theta(2)(t)>, we predict the polymer solution's G*(omega) using a rotational generalized Stokes-Einstein relation. The good agreement of the predicted modulus with mechanical measurements confirms this new microrheological approach.

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes-Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, , of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G*(ω), in the Fourier frequency domain, and the stress relaxation modulus, Gr(t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing as a local power law. If the logarithmic slope of can be accurately determined, these estimates generally perform well at the frequency extremes.