Average Mean Square Displacement Research Articles

Effect of loops on the mean-square displacement of Rouse-model chromatin.

Chromatin polymer dynamics are commonly described using the classical Rouse model. The subsequent discovery, however, of intermediate-scale chromatin organization known as topologically associating domains (TADs) in experimental Hi-C contact maps for chromosomes across the tree of life, together with the success of loop extrusion factor (LEF) model in explaining TAD formation, motivates efforts to understand the effect of loops and loop extrusion on chromatin dynamics. This paper seeks to fulfill this need by combining LEF-model simulations with extended Rouse-model polymer simulations to investigate the dynamics of chromatin with loops and dynamic loop extrusion. We show that loops significantly suppress the averaged mean-square displacement (MSD) of a gene locus, consistent with recent experiments that track fluorescently labeled chromatin loci. We also find that loops reduce the MSD's stretching exponent from the classical Rouse-model value of 1/2 to a loop-density-dependent value in the 0.45-0.40 range. Remarkably, stretching exponent values in this range have also been observed in recent experiments [Weber et al., Phys. Rev. Lett. 104, 238102 (2010)0031-900710.1103/PhysRevLett.104.238102; Bailey et al., Mol. Biol. Cell 34, ar78 (2023)1059-152410.1091/mbc.E23-04-0119]. We also show that the dynamics of loop extrusion itself negligibly affects chromatin mobility. By studying static "rosette" loop configurations, we also demonstrate that chromatin MSDs and stretching exponents depend on the location of the locus in question relative to the position of the loops and on the local friction environment.

Dynamic mechanical contact behaviors and sintering mechanism of Al nanoparticles subjected to high-speed impact

In this work, dynamic mechanical contact behaviors and sintering mechanism of Al nanoparticles subjected to high-speed impact have been investigated using molecular dynamic simulations by analyzing radial distribution function (RDF), atomic average displacement, mean square displacement (MSD), the sintering neck radius, shrinkage, nanoparticle temperature and the proportion of amorphous atoms. The results shows that the long-ranged ordered structure of Al nanoparticles gradually transforms into short-range ordered structure and finally amorphous structure during the head-on impact. The atomic average displacement, MSD, radius ratio (x/R) and shrinkage at different initial impact speeds first experience a rapid increase to a maximum, followed by a slow decrease and subsequent a recovery to an almost constant. The displacement of surface atoms is always larger than the internal atoms during the head-on impact. There exists one critical impact velocity (e.g., 2800 m/s for given 4 nm in diameter) beyond which the ordered aluminum atoms completely transform into amorphous structure. The contact force, contact radius and contact stress obtained under a given low impact velocity and small strain are quite consistent with the predictions of the continuous Hertz model, but as the impact velocity and normal displacement further increase, the predictions of Hertz model underestimate these contact parameters. The influence of nanoparticle size on the relationship of contact radius and number of contacting atoms is inappreciable. The nanoparticles undergo elastic deformation, and then plastic deformation, followed by mechanical failure and shock-induced melting process into a whole. This work could strengthen our understanding of the industrial processes with applications in energetic nanomaterials, rocket propellant, metal sintering, etc.

Discriminating Gaussian processes via quadratic form statistics.

Gaussian processes are powerful tools for modeling and predicting various numerical data. Hence, checking their quality of fit becomes a vital issue. In this article, we introduce a testing methodology for general Gaussian processes based on a quadratic form statistic. We illustrate the methodology on three statistical tests recently introduced in the literature, which are based on the sample autocovariance function, time average mean-squared displacement, and detrended moving average statistics. We compare the usefulness of the tests by taking into consideration three very important Gaussian processes: the fractional Brownian motion, which is self-similar with stationary increments (SSSIs), scaled Brownian motion, which is self-similar with independent increments (SSIIs), and the Ornstein-Uhlenbeck (OU) process, which is stationary. We show that the considered statistics' ability to distinguish between these Gaussian processes is high, and we identify the best performing tests for different scenarios. We also find that there is no omnibus quadratic form test; however, the detrended moving average test seems to be the first choice in distinguishing between same processes with different parameters. We also show that the detrended moving average method outperforms the Cholesky method. Based on the previous findings, we introduce a novel procedure of discriminating between Gaussian SSSI, SSII, and stationary processes. Finally, we illustrate the proposed procedure by applying it to real-world data, namely, the daily EURUSD currency exchange rates, and show that the data can be modeled by the OU process.

Testing of Multifractional Brownian Motion.

Fractional Brownian motion (FBM) is a generalization of the classical Brownian motion. Most of its statistical properties are characterized by the self-similarity (Hurst) index . In nature one often observes changes in the dynamics of a system over time. For example, this is true in single-particle tracking experiments where a transient behavior is revealed. The stationarity of increments of FBM restricts substantially its applicability to model such phenomena. Several generalizations of FBM have been proposed in the literature. One of these is called multifractional Brownian motion (MFBM) where the Hurst index becomes a function of time. In this paper, we introduce a rigorous statistical test on MFBM based on its covariance function. We consider three examples of the functions of the Hurst parameter: linear, logistic, and periodic. We study the power of the test for alternatives being MFBMs with different linear, logistic, and periodic Hurst exponent functions by utilizing Monte Carlo simulations. We also analyze mean-squared displacement (MSD) for the three cases of MFBM by comparing the ensemble average MSD and ensemble average time average MSD, which is related to the notion of ergodicity breaking. We believe that the presented results will be helpful in the analysis of various anomalous diffusion phenomena.

Construction of one-dimensional random walk lattices using DNA algorithmic self-assembly

Algorithmic DNA lattices are constructed using pre-defined rules such as COPY, NOT, and XOR, where patterns are predicted based on initial values. However, the experimental implementation of an unpredictable random walk pattern (which is the implementation of a random rule, i.e., equally probable to move toward either the left or right in 1D systems) in DNA has not been reported yet. Here, we construct DNA lattices with DNA rule tiles implemented using the random rule. Patterns are visualized by atomic force microscopy. Finally, we discussed the average displacement, mean-square displacement, and number of displacement occurrences of experimental as well as simulated 1D random walk. The encoded information in sticky ends of DNA rule tiles demonstrates the feasibility of universal computation through DNA algorithmic self-assembly, which could be extremely beneficial in future computations.

Subdiffusion and ergodicity breaking in heterogeneous environments subject to Lévy noise

The anomalous diffusion and ergodicity breaking of Brownian particles in heterogeneous environments (space-dependent diffusion coefficient D(x)=D0|x|α) subject to Lévy noise are investigated, by both the analytically theory and the numerically simulating schemes. Within the tailored parameter, the competition of the power-law exponent α and the Lévy index μ ensures the long tail of probability density function with the form P(x)∼1|x|μ∕p+1 with p=22−α, falling rapidly when μ∕p>2. Meanwhile, we observe the subdiffusive motion that is characterized by the scale of ensemble averaged mean-square displacement (MSD), 〈x2(t)〉∝tκ with κ=2pμ<1. Furthermore, we also observe the weak ergodicity breaking which means the inequivalence between individual time averaged MSD and the corresponding ensemble averaged MSD. We quantify the weak ergodicity breaking behavior in terms of the dimensionless parameters EB and EB. For 1≤μ<2 (except for μ=2), the ergodicity breaking enhances as the α increases, but weakens as μ or D0. Overall, the EB follows the from, EB∝(Δ∕T)(12−2pμ), in the large time. Additionally, the anomalous diffusion above might be particularly relevant for ultracold atoms diffusion in heterogeneous media.

Determination of Dynamical Heterogeneity from Dynamic Neutron Scattering of Proteins

Motional displacements of hydrogen (H) in proteins can be measured using incoherent neutron-scattering methods. These displacements can also be calculated numerically using data from molecular dynamics simulations. An enormous amount of data on the average mean-square motional displacement (MSD) of H as a function of protein temperature, hydration, and other conditions has been collected. H resides in a wide spectrum of sites in a protein. Some H are tightly bound to molecular chains, and the H motion is dictated by that of the chain. Other H are quite independent. As a result, there is a distribution of motions and MSDs of H within a protein that is denoted dynamical heterogeneity. The goal of this paper is to incorporate a distribution of MSDs into models of the H incoherent intermediate scattering function, I(Q,t), that is calculated and observed. The aim is to contribute information on the distribution as well as on the average MSD from comparison of the models with simulations and experiment. For example, we find that simulations of I(Q,t) in lysozyme are well reproduced if the distribution of MSDs is bimodal with two broad peaks rather than a single broad peak.

Molecular Dynamics Study on the Effect of Single Grain Boundary on the Mechanical Behavior of α-Quartz

Uniaxial compression test was performed on twinned crystal (TC) models and monocrystal (MC) α-quartz to study the effect of the single grain boundary on the strength and fracture of α-quartz. Based on the stress and strain curves, for any given strain rate, MC has a higher stress than any other TC. Based on the bond length changes, bond breaks are found in the TC but no bond break is found in the MC. Thus, the grain boundary causes stress concentration and leads to bond breakage. Larger transformation of bonds absorbs more energy, which results in lower bonding strength. Moreover, the fracture does not necessarily start from the grain boundary but also depends on the initially large bond length. The averaged mean square displacement (AMSD) is also calculated. For the results of MC, only one set of peaks corresponding to the maximum stress appears on the AMSD and strain curves. For TC, two sets of peaks indicate the largest stretching of bond length and the start of fractures, separately. These characteristics illustrate that the vibration of atoms will intensify when the structure reaches the limit of compressive strength.

Thermal vibrations in the metallic glass Cu64Zr36

Neutrons with 14.7 and 34 meV energy were used to determine the elastic and inelastic part of the structure factor for the metallic glass Cu64Zr36 at 250 K. Based on the temperature dependence of the elastic scattering between 150 K and RT, an average mean-square displacement at 250 K is obtained. The experimental scattering-vector dependence of inelastic scattering in reference to elastic scattering is found to be well described by the Debye model. Both results are supported by molecular dynamics simulations. A procedure is presented to separate the elastic part also in total x-ray scattering. This allows the smearing of structural information due to thermal vibrations to be eliminated.

Ultrafast electron diffraction from non-equilibrium phonons in femtosecond laser heated Au films

We use ultrafast electron diffraction to detect the temporal evolution of non-equilibrium phonons in femtosecond laser-excited ultrathin single-crystalline gold films. From the time-dependence of the Debye-Waller factor, we extract a 4.7 ps time-constant for the increase in mean-square atomic displacements. The observed increase in the diffuse scattering intensity demonstrates that the energy transfer from laser-heated electrons to phonon modes near the X and K points in the Au fcc Brillouin zone proceeds with timescales of 2.3 and 2.9 ps, respectively, faster than the Debye-Waller average mean-square displacement.

Self-subdiffusion in solutions of star-shaped crowders: non-monotonic effects of inter-particle interactions

We examine by extensive computer simulations the self-diffusion of anisotropic star-like particles in crowded two-dimensional solutions. We investigate the implications of the area coverage fraction ϕ of the crowders and the crowder–crowder adhesion properties on the regime of transient anomalous diffusion. We systematically compute the mean squared displacement (MSD) of the particles, their time averaged MSD, and the effective diffusion coefficient. The diffusion is ergodic in the limit of long traces, such that the mean time averaged MSD converges towards the ensemble averaged MSD, and features a small residual amplitude spread of the time averaged MSD from individual trajectories. At intermediate time scales, we quantify the anomalous diffusion in the system. Also, we show that the translational—but not rotational—diffusivity of the particles D is a nonmonotonic function of the attraction strength between them. Both diffusion coefficients decrease as the power law with the area fraction ϕ occupied by the crowders and the critical value Our results might be applicable to rationalising the experimental observations of non-Brownian diffusion for a number of standard macromolecular crowders used in vitro to mimic the cytoplasmic conditions of living cells.

Motional displacements in proteins: The origin of wave-vector-dependent values.

The average mean-square displacement, 〈r(2)〉, of H atoms in a protein is frequently determined using incoherent neutron-scattering experiments. 〈r(2)〉 is obtained from the observed elastic incoherent dynamic structure factor, S(i)(Q,ω=0), assuming the form S(i)(Q,ω=0) =exp(-Q(2)〈r(2)〉/3). This is often referred to as the Gaussian approximation (GA) to S(i)(Q,ω=0). 〈r(2)〉 obtained in this way depends on the value of the wave vector, Q considered. Equivalently, the observed S(i)(Q,ω=0) deviates from the GA. We investigate the origin of the Q dependence of 〈r(2)〉 by evaluating the scattering functions in different approximations using molecular dynamics (MD) simulation of the protein lysozyme. We find that keeping only the Gaussian term in a cumulant expansion of S(Q,ω) is an accurate approximation and is not the origin of the Q dependence of 〈r(2)〉. This is demonstrated by showing that the term beyond the Gaussian is negligible and that the GA is valid for an individual atom in the protein. Rather, the Q dependence (deviation from the GA) arises from the dynamical heterogeneity of the H in the protein. Specifically it arises from representing, in the analysis of data, this diverse dynamics by a single average scattering center that has a single, average 〈r(2)〉. The observed Q dependence of 〈r(2)〉 can be used to provide information on the dynamical heterogeneity in proteins.

Hydration control of the mechanical and dynamical properties of cellulose.

The mechanical and dynamical properties of cellulose, the most abundant biomolecule on earth, are essential for its function in plant cell walls and advanced biomaterials. Cellulose is almost always found in a hydrated state, and it is therefore important to understand how hydration influences its dynamics and mechanics. Here, the nanosecond-time scale dynamics of cellulose is characterized using dynamic neutron scattering experiments and molecular dynamics (MD) simulation. The experiments reveal that hydrated samples exhibit a higher average mean-square displacement above ∼240 K. The MD simulation reveals that the fluctuations of the surface hydroxymethyl atoms determine the experimental temperature and hydration dependence. The increase in the conformational disorder of the surface hydroxymethyl groups with temperature follows the cellulose persistence length, suggesting a coupling between structural and mechanical properties of the biopolymer. In the MD simulation, 20% hydrated cellulose is more rigid than the dry form, due to more closely packed cellulose chains and water molecules bridging cellulose monomers with hydrogen bonds. This finding may have implications for understanding the origin of strength and rigidity of secondary plant cell walls. The detailed characterization obtained here describes how hydration-dependent increased fluctuations and hydroxymethyl disorder at the cellulose surface lead to enhancement of the rigidity of this important biomolecule.

Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials: extracting information from transients.

A Langevin process describing diffusion in a periodic potential landscape has a time-dependent diffusion constant, which means that its average mean-squared displacement (MSD) only becomes linear at late times. The long-time, or effective diffusion, constant can be estimated from the slope of a linear fit of the MSD at late times. Due to the crossover between a short time microscopic diffusion constant, which is independent of the potential, to the effective late-time diffusion constant, a linear fit of the MSD will not in general pass through the origin and will have a nonzero constant term. Here we address how to compute the constant term and provide explicit results for Brownian particles in one dimension in periodic potentials. We show that the constant is always positive and that at low temperatures it depends on the curvature of the minimum of the potential. For comparison we also consider the same question for the simpler problem of a symmetric continuous time random walk in discrete space. Here the constant can be positive or negative and can be used to determine the variance of the hopping time distribution.

Random time averaged diffusivities for Lévy walks

We investigate a L\'evy-Walk alternating between velocities $\pm v_0$ with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at large times is $< x^2 > \propto t^2$, the latter to enhanced diffusion with $< x^2 > \propto t^\nu$, $1<\nu<2$. The correlation function and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSD from a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function. In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yet very slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averaged MSDs.

Derivation of Mean-Square Displacements for Protein Dynamics from Elastic Incoherent Neutron Scattering

The derivation of mean-square displacements from elastic incoherent neutron scattering (EINS) of proteins is examined, with the aid of experiments on camphor-bound cytochrome P450cam and complementary molecular dynamics simulations. It is shown that a q(4) correction to the elastic incoherent structure factor (where q is the scattering vector) can be simply used to reliably estimate from the experiment both the average mean-square atomic displacement, <Δr(2)> of the nonexchanged hydrogen atoms in the protein and its variance, σ(2). The molecular dynamics simulation results are in broad agreement with the experimentally derived <Δr(2)> and σ(2) derived from EINS on instruments at two different energy resolutions, corresponding to dynamics on the ∼100 ps and ∼1 ns time scales. Significant dynamical heterogeneity is found to arise from methyl-group rotations. The easy-to-apply q(4) correction extends the information extracted from elastic incoherent neutron scattering experiments and should be of wide applicability.

Temperature Dependent Dynamics of Cytochrome P450cam from Elastic Incoherent Neutron Scattering

The derivation of mean-square displacements from elastic incoherent neutron scattering (EINS) of proteins is examined, with the aid of experiments on camphor-bound cytochrome P450cam and complementary molecular dynamics simulations. It is shown that a q4 correction to the elastic incoherent structure factor (where q is the scattering vector) can be simply used to reliably estimate from the experiment both the average mean-square atomic displacement (MSD) of the non-exchanged hydrogen atoms in the protein and its variance. The molecular dynamics simulation results are in broad agreement with the experimentally-derived MSD and its variance derived from EINS on instruments at two different energy resolutions, corresponding to dynamics on the ∼100 ps and ∼1 ns timescales. Significant dynamical heterogeneity is found to arise from methyl-group rotations. The easy-to-apply q4 correction extends the information extracted from elastic incoherent neutron scattering experiments and should be of wide applicability.

Colloidal rotation near the colloidal glass transition

We compare, using single-particle optical imaging, trajectories of rotation and translation for micron-sized spheres in index-matched colloidal suspensions near their glass transition. Rotational trajectories, while they show intermittent caged behavior associated with supercooled and glassy behavior, explore a sufficiently wider phase space such that in the averaged mean-square angular displacement there appears no plateau regime, but instead sub-Fickian angular diffusion that follows an apparent power law in time. We infer translation and rotation time constants, the former being the time to diffuse a particle diameter and the latter being the time to rotate a full revolution. Correlation between time constants increases with increasing volume fraction, but unlike the case for molecular glasses, the rotation time constant slows more weakly than the translation time.

Fractional Lévy stable motion can model subdiffusive dynamics

We show in this paper that the sample (time average) mean-squared displacement (MSD) of the fractional Lévy α -stable motion behaves very differently from the corresponding ensemble average (second moment). While the ensemble average MSD diverges for α<2 , the sample MSD may exhibit either subdiffusion, normal diffusion, or superdiffusion. Thus, H -self-similar Lévy stable processes can model either a subdiffusive, diffusive or superdiffusive dynamics in the sense of sample MSD. We show that the character of the process is controlled by a sign of the memory parameter d=H-1/α . We also introduce a sample p -variation dynamics test which allows to distinguish between two models of subdiffusive dynamics. Finally, we illustrate a subdiffusive behavior of the fractional Lévy stable motion on biological data describing the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells, but it may concern many other fields of contemporary experimental physics.

Unforced polymer translocation compared to the forced case

We present results for unforced polymer translocation from simulations using Langevin dynamics in two dimensions (2D) to four dimensions and stochastic rotation dynamics supporting hydrodynamic modes in three dimensions (3D). We compare our results to forced translocation and a simplified model where the polymer escapes from an infinite pore. The simple model shows that the scaling behavior of unforced translocation is independent of the dimension of the side to which the polymer is translocating. We find that, unlike its forced counterpart, unforced translocation dynamics is insensitive to pore design. Hydrodynamics is seen to markedly speed up the unforced translocation process but not to affect the scaling relations. Average mean-squared displacement shows scaling with average transition time in unforced but not in forced translocation. The waiting-time distribution in unforced translocation follows closely Poissonian distribution. Our measured transfer probabilities align well with those obtained from an equilibrium theory in 3D, but somewhat worse in 2D, where a polymer's relaxation toward equilibrium with respect to its translocation time is slower. Consequently, in stark contrast to forced translocation, unforced translocation is seen to remain close to equilibrium and shows clear universality.