Subdiffusive Motion Research Articles

Generalized Langevin subdiffusion in channels: The bath always wins.

We consider subdiffusive motion, modeled by the generalized Langevin equationin an equilibrium setting, of tracer particles in channels of indefinite length in the x direction: the channels of varying width and the channels with sinusoidally meandering midline. The subdiffusion in the x direction is not affected by constraints put by the channel. This is especially astonishing for meandering channels whose centerline might be quite long. The same behavior is seen in a holonomic model of a bead on a sinusoidal and meandering wire, where some analytic insights are possible.

Piggybacking functionalized DNA nanostructures into live-cell nuclei.

DNA origami nanostructures (DOs) are promising tools for applications including drug delivery, biosensing, detecting biomolecules, and probing chromatin substructures. Targeting these nanodevices to mammalian cell nuclei could provide impactful approaches for probing, visualizing, and controlling biomolecular processes within live cells. We present an approach to deliver DOs into live-cell nuclei. We show that these DOs do not undergo detectable structural degradation in cell culture media or cell extracts for 24 hours. To deliver DOs into the nuclei of human U2OS cells, we conjugated 30-nanometer DO nanorods with an antibody raised against a nuclear factor, specifically the largest subunit of RNA polymerase II (Pol II). We find that DOs remain structurally intact in cells for 24 hours, including inside the nucleus. We demonstrate that electroporated anti-Pol II antibody-conjugated DOs are piggybacked into nuclei and exhibit subdiffusive motion inside the nucleus. Our results establish interfacing DOs with a nuclear factor as an effective method to deliver nanodevices into live-cell nuclei.

Distinct mobility patterns of BRCA2 molecules at DNA damage sites.

BRCA2 is an essential tumor suppressor protein involved in promoting faithful repair of DNA lesions. The activity of BRCA2 needs to be tuned precisely to be active when and where it is needed. Here, we quantified the spatio-temporal dynamics of BRCA2 in living cells using aberration-corrected multifocal microscopy (acMFM). Using multicolor imaging to identify DNA damage sites, we were able to quantify its dynamic motion patterns in the nucleus and at DNA damage sites. While a large fraction of BRCA2 molecules localized near DNA damage sites appear immobile, an additional fraction of molecules exhibits subdiffusive motion, providing a potential mechanism to retain an increased number of molecules at DNA lesions. Super-resolution microscopy revealed inhomogeneous localization of BRCA2 relative to other DNA repair factors at sites of DNA damage. This suggests the presence of multiple nanoscale compartments in the chromatin surrounding the DNA lesion, which could play an important role in the contribution of BRCA2 to the regulation of the repair process.

Elongasome core proteins and class A PBP1a display zonal, processive movement at the midcell of Streptococcus pneumoniae

Ovoid-shaped bacteria, such as Streptococcus pneumoniae (pneumococcus), have two spatially separated peptidoglycan (PG) synthase nanomachines that locate zonally to the midcell of dividing cells. The septal PG synthase bPBP2x:FtsW closes the septum of dividing pneumococcal cells, whereas the elongasome located on the outer edge of the septal annulus synthesizes peripheral PG outward. We showed previously by sm-TIRFm that the septal PG synthase moves circumferentially at midcell, driven by PG synthesis and not by FtsZ treadmilling. The pneumococcal elongasome consists of the PG synthase bPBP2b:RodA, regulators MreC, MreD, and RodZ, but not MreB, and genetically associated proteins Class A aPBP1a and muramidase MpgA. Given its zonal location separate from FtsZ, it was of considerable interest to determine the dynamics of proteins in the pneumococcal elongasome. We found that bPBP2b, RodA, and MreC move circumferentially with the same velocities and durations at midcell, driven by PG synthesis. However, outside of the midcell zone, the majority of these elongasome proteins move diffusively over the entire surface of cells. Depletion of MreC resulted in loss of circumferential movement of bPBP2b, and bPBP2b and RodA require each other for localization and circumferential movement. Notably, a fraction of aPBP1a molecules also moved circumferentially at midcell with velocities similar to those of components of the core elongasome, but for shorter durations. Other aPBP1a molecules were static at midcell or diffusing over cell bodies. Last, MpgA displayed nonprocessive, subdiffusive motion that was largely confined to the midcell region and less frequently detected over the cell body.

Modelling Heterogeneous Anomalous Dynamics of Radiation-Induced Double-Strand Breaks in DNA during Non-Homologous End-Joining Pathway.

The process of end-joining during nonhomologous repair of DNA double-strand breaks (DSBs) after radiation damage is considered. Experimental evidence has revealed that the dynamics of DSB ends exhibit subdiffusive motion rather than simple diffusion with rare directional movement. Traditional models often overlook the rare long-range directed motion. To address this limitation, we present a heterogeneous anomalous diffusion model consisting of subdiffusive fractional Brownian motion interchanged with short periods of long-range movement. Our model sheds light on the underlying mechanisms of heterogeneous diffusion in DSB repair and could be used to quantify the DSB dynamics on a time scale inaccessible to single particle tracking analysis. The model predicts that the long-range movement of DSB ends is responsible for the misrepair of DSBs in the form of dicentric chromosome lesions.

Revealing the Existence of Long-Range Liquid-Liquid Interfacial Potential in Phase-Transfer Processes.

By employing fluorescence wide-field microscopy and a nanoparticle-based phase transfer catalyst (PTC), consisting of a fluorescent silica nanoparticle functionalized with trioctylpropylammonium bromide, we demonstrate that in the presence of NaOH, single nanoparticles display subdiffusive motion along the axis normal to an aqueous liquid-organic liquid interface. This is because of an extended interfacial potential with a shallow well (∼1 kBT) that stretches a few μm into the organic phase, in contrast to previous molecular dynamics studies that reported narrow interfaces on the order of ∼1 nm. Spontaneous interfacial emulsification induced by NaOH results in the propagation of water-in-oil nanoemulsions into the organic solvent that creates an equilibrium hybrid-solvent composition that solvates the PTC. A greater mobility and longer residence time of the PTC at the potential well enhance the interfacial phase transfer process and catalytic efficiency.

Dynamical properties of solid and hydrated collagen: Insight from nuclear magnetic resonance relaxometry.

1H spin-lattice Nuclear Magnetic Resonance relaxometry experiments have been performed for collagen and collagen-based artificial tissues in the frequency range of 10 kHz-20MHz. The studies were performed for non-hydrated and hydrated materials. The relaxation data have been interpreted as including relaxation contributions originating from 1H-1H and 1H-14N dipole-dipole interactions, the latter leading to Quadrupole Relaxation Enhancement effects. The 1H-1H relaxation contributions have been decomposed into terms associated with dynamical processes on different time scales. A comparison of the parameters for the non-hydrated and hydrated systems has shown that hydration leads to a decrease in the dipolar relaxation constants without significantly affecting the dynamical processes. In the next step, the relaxation data for the hydrated systems were interpreted in terms of a model assuming two-dimensional translational diffusion of water molecules in the vicinity of the macromolecular surfaces and a sub-diffusive motion leading to a power law of the frequency dependencies of the relaxation rates. It was found that the water diffusion process is slowed down by at least two orders of magnitude compared to bulk water diffusion. The frequency dependencies of the relaxation rates in hydrated tissues and hydrated collagen are characterized by different power laws (ωH-β, where ωH denotes the 1H resonance frequency): the first of about 0.4 and the second close to unity.

Transition from active motion to anomalous diffusion for Bacillus subtilis confined in hydrogel matrices

We investigate the motility of B. subtilis under different degrees of confinement induced by transparent porous hydrogels. The dynamical behavior of the bacteria at short times is linked to characteristic parameters describing the hydrogel porosity. Mean squared displacements (MSDs) reveal that the run-and-tumble dynamics of unconfined B. subtilis progressively turns into sub-diffusive motion with increasing confinement. Correspondingly, the median instantaneous velocity of bacteria decreases and becomes more narrowly distributed, while the reorientation rate increases and reaches a plateau value. Analyzing single-trajectories, we show that the average dynamical behavior is the result of complex displacements, in which active, diffusive and sub-diffusive segments coexist. For small and moderate confinements, the number of active segments reduces, while the diffusive and sub-diffusive segments increase. The alternation of sub-diffusion, diffusion and active motion along the same trajectory can be described as a hopping ad trapping motion, in which hopping events correspond to displacements with an instantaneous velocity exceeding the corresponding mean value along a trajectory. Different from previous observations, escape from local trapping occurs for B. subtilis through active runs but also diffusion. Interestingly, the contribution of diffusion is maximum at intermediate confinements. At sufficiently long times transport coefficients estimated from the experimental MSDs under different degrees of confinement can be reproduced using a recently proposed hopping and trapping model. Finally, we propose a quantitative relationship linking the median velocity of confined and unconfined bacteria through the characteristic confinement length of the hydrogel matrix. Our work provides new insights for the bacterial motility in complex media that mimic natural environments and are relevant to important problems like sterilization, water purification, biofilm formation, membrane permeation and bacteria separation.

Subdiffusive dynamics and hydrodynamic fluctuations: how the latter affect the former.

The characteristics of subdiffusion are influenced by hydrodynamic fluctuations, not the scaling of the mean square displacement (MSD) at the long time limit, but rather the transition of this property from ballistic behaviour at early times to its final scaling at large times. Additionally, since a significant portion of the normalised velocity autocorrelation function (NVAF), a widely used motion descriptor, no longer adequately describes the antipersistent character that is typical of a subdiffusive motion, it is also impacted by these fluctuations. The combined findings can lead to misleading conclusions if hydrodynamic fluctuations are not taken into account. Diffusing and vorticity time scales are crucial for the way the motion turns into the final subdiffusion dynamics, whose exponent is determined by the scaling of the friction.

Subdiffusion in an array of solid obstacles

More than a decade ago, Goychuk reported on a universal behavior of subdiffusive motion (as described by the generalized Langevin equation) in a one-dimensional bounded periodic potential (Goychuk 2009 Phys. Rev. E 80 046125) where the numerical findings show that the long-time behavior of the mean squared displacement is not influenced by the potential, so that the behavior in the potential, under homogenization, is the same as in its absence. This property may break down if the potential is unbounded. In the present work, we report on the results of simulations of subdiffusion in a two-dimensional (2D) periodic array of solid obstacles (i.e. in an unbounded potential) with different packing fractions. It is revealed that the universal subdiffusive behavior at long times is not influenced by the presence of solid scatterers, whose presence influences the behavior at intermediate times only. This result is discussed as having possible relations to the emerging problem of interpretation of results on trajectories of tracers spreading in the brain’s extracellular space.

Random Walks on Comb-like Structures under Stochastic Resetting.

We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker's motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case.

Compartmentalization of androgen receptors at endogenous genes in living cells.

A wide range of nuclear proteins are involved in the spatio-temporal organization of the genome through diverse biological processes such as gene transcription and DNA replication. Upon stimulation by testosterone and translocation to the nucleus, multiple androgen receptors (ARs) accumulate in microscopically discernable foci which are irregularly distributed in the nucleus. Here, we investigated the formation and physical nature of these foci, by combining novel fluorescent labeling techniques to visualize a defined chromatin locus of AR-regulated genes-PTPRN2 or BANP-simultaneously with either AR foci or individual AR molecules. Quantitative colocalization analysis showed evidence of AR foci formation induced by R1881 at both PTPRN2 and BANP loci. Furthermore, single-particle tracking (SPT) revealed three distinct subdiffusive fractional Brownian motion (fBm) states: immobilized ARs were observed near the labeled genes likely as a consequence of DNA-binding, while the intermediate confined state showed a similar spatial behavior but with larger displacements, suggesting compartmentalization by liquid-liquid phase separation (LLPS), while freely mobile ARs were diffusing in the nuclear environment. All together, we show for the first time in living cells the presence of AR-regulated genes in AR foci.

Nonequilibrium diffusion of active particles bound to a semiflexible polymer network: Simulations and fractional Langevin equation.

In a viscoelastic environment, the diffusion of a particle becomes non-Markovian due to the memory effect. An open question concerns quantitatively explaining how self-propulsion particles with directional memory diffuse in such a medium. Based on simulations and analytic theory, we address this issue with active viscoelastic systems where an active particle is connected with multiple semiflexible filaments. Our Langevin dynamics simulations show that the active cross-linker displays superdiffusive and subdiffusive athermal motion with a time-dependent anomalous exponent α. In such viscoelastic feedback, the active particle always exhibits superdiffusion with α = 3/2 at times shorter than the self-propulsion time (τA). At times greater than τA, the subdiffusive motion emerges with α bounded between 1/2 and 3/4. Remarkably, active subdiffusion is reinforced as the active propulsion (Pe) is more vigorous. In the high Pe limit, athermal fluctuation in the stiff filament eventually leads to α = 1/2, which can be misinterpreted with the thermal Rouse motion in a flexible chain. We demonstrate that the motion of active particles cross-linking a network of semiflexible filaments can be governed by a fractional Langevin equation combined with fractional Gaussian noise and an Ornstein-Uhlenbeck noise. We analytically derive the velocity autocorrelation function and mean-squared displacement of the model, explaining their scaling relations as well as the prefactors. We find that there exist the threshold Pe (Pe∗) and crossover times (τ∗ and τ†) above which active viscoelastic dynamics emerge on timescales of τ∗≲ t ≲ τ†. Our study may provide theoretical insight into various nonequilibrium active dynamics in intracellular viscoelastic environments.

Complex Monte Carlo Light-Driven Dynamics of Monomers in Functionalized Bond Fluctuation Model Polymer Chains.

We study Monte Carlo dynamics of the monomers and center of mass of a model polymer chain functionalized with azobenzene molecules in the presence of an inhomogeneous linearly polarized laser light. The simulations use a generalized Bond Fluctuation Model. The mean squared displacements of the monomers and the center of mass are analyzed in a period of Monte Carlo time typical for a build-up of Surface Relief Grating. Approximate scaling laws for mean squared displacements are found and interpreted in terms of sub- and superdiffusive dynamics for the monomers and center of mass. A counterintuitive effect is observed, where the monomers perform subdiffusive motion but the resulting motion of the center of mass is superdiffusive. This result disparages theoretical approaches based on an assumption that the dynamics of single monomers in a chain can be characterized in terms of independent identically distributed random variables.

Multiple Scenarios of Low-Temperature Nucleation in Simple Liquids.

Usually, sufficient supercooling of a liquid is employed to bypass the free energy barrier and speed up crystallization. However, lowering the temperature T induces new issues competing with the crystallization, e.g., slow particle motion, geometric frustration, and the glass formation, which complicates our understanding of crystal growth. Here we systematically study the low-temperature nucleation kinetics discriminated by the maximum nucleation rate temperature T_{d} and the glass transition temperature T_{g}. At T_{d}, the ratio of the precursor and geometrically frustrated particles reaches the maximum. When T_{g}<T<T_{d}, nucleation kinetics is characterized by the subdiffusive slow particle motion, the high degrees of geometric frustration, and the saturation of precursors. In this regime, nucleation can proceed through the diffusionless-like ordering of precursors. Near T_{g}, there is a crossover regime, where geometrically frustrated particles percolate and the glass formation strongly slows down the nucleation. When T<T_{g}, diffusionless nucleation is obstructed due to the weak vibrational motion and the mechanical stability of the glassy state.

Impact of cell-sized confinement on particle transport within a motor-driven cytoskeletal network.

Intracellular transport of small particles and macromolecules is affected by the degree of crowding, confinement, and motor-driven activity within the cell. In vitro studies have examined how transport dynamics depend on each of these factors. But studies have primarily focused on these factors in isolation. Previously, we showed how crowding and confinement synergistically couple to slow the diffusion of DNA molecules. Here, we explore the combined effects of crowding by a composite network of actin and microtubule filaments, activity driven by molecular motors, and confinement by a lipid membrane to cell-sized volumes. We quantify transport of tracer particles within this tunable in vitro environment using differential dynamic microscopy (DDM) and single particle tracking. We simultaneously measure the dynamics and spatial organization of the composite network to link transport dynamics to network properties. To what degree tracers exhibit subdiffusive motion due to caging within the network mesh or superdiffusive motion due to motor activity depends on network composition, motor concentration, and confinement size.

Inferences from FRAP data are model dependent: A subdiffusive analysis

Fluorescence recovery after photobleaching (FRAP) is a widely used biological experiment to study the kinetics of molecules that react and move randomly. Since the development of FRAP in the 1970s, many reaction-diffusion models have been used to interpret FRAP data. However, intracellular molecules are widely observed to move by anomalous subdiffusion instead of normal diffusion. In this article, we extend a popular reaction-diffusion model of FRAP to the case of subdiffusion modeled by a fractional diffusion equation. By analyzing this reaction-subdiffusion model, we show that FRAP data are consistent with both diffusive and subdiffusive motion in many scenarios. We illustrate this general result by fitting our model to FRAP data from glucocorticoid receptors in a cell nucleus. We further show that the assumed model of molecular motion (normal diffusion or subdiffusion) strongly impacts the biological parameter values inferred from a given experimentally observed FRAP curve. We additionally analyze our model in three simplified parameter regimes and discuss parameter identifiability for varying subdiffusion exponents.

Dynamics of axonal β-actin mRNA in live hippocampal neurons.

Localization of mRNA facilitates spatiotemporally controlled protein expression in neurons. In axons, mRNA transport followed by local protein synthesis plays a critical role in axonal growth and guidance. However, it is not yet clearly understood how mRNA is transported to axonal subcellular sites and what regulates axonal mRNA localization. Using a transgenic mouse model in which endogenous β-actin mRNA is fluorescently labeled, we investigated β-actin mRNA movement in axons of hippocampal neurons. We cultured neurons in microfluidic devices to separate axons from dendrites and performed single-particle tracking of axonal β-actin mRNA. Compared with dendritic β-actin mRNA, axonal β-actin mRNA showed less directed motion and exhibited mostly subdiffusive motion, especially near filopodia and boutons in mature dissociated hippocampal neurons. We found that axonal β-actin mRNA was likely to colocalize with actin patches (APs), regions that have a high density of filamentous actin (F-actin) and are known to have a role in branch initiation. Moreover, simultaneous imaging of F-actin and axonal β-actin mRNA in live neurons revealed that moving β-actin mRNA tended to be docked in the APs. Our findings reveal that axonal β-actin mRNA localization is facilitated by actin networks and suggest that localized β-actin mRNA plays a potential role in axon branch formation.

Numerical Study of Drift Influence on Diffusion Transport through the Hybrid Membrane.

Sodium alginate membranes filled with iron oxide nanoparticles consist of a mixture of organic and inorganic phases. This design offers the possibility to combine the polymer’s easy processability and superior separation performance. For a better understanding of the mechanisms of mixture separation, we analyze the diffusion motion of a particle in the hybrid membrane environment. We model structures of two-dimensional heterogenic membranes, which resemble real membrane structures, and then we simulate a random walk on them. We investigate how the additional action of drift changes the motion properties of the diffusing particles through the polymeric membrane filled with inorganic powder. We test the effect of two parameters: the distribution of obstacles (filling) in the membrane and the value of drift on the nature of diffusion. It appears that the synergy between drift, the diffusion, and the membrane structure affect the occurrence of the superdiffusive and subdiffusive character of particle motion as measured by the time-averaged mean square displacement. An important point is the observation that the strong drift supports subdiffusive motion as it increases the chances of particle trapping. Moreover, there exists the optimal value of drift, for which the transport through a membrane speeds up and does not cause trapping.

Gelation Dynamics upon Pressure-Induced Liquid-Liquid Phase Separation in a Water-Lysozyme Solution.

Employing X-ray photon correlation spectroscopy, we measure the kinetics and dynamics of a pressure-induced liquid-liquid phase separation (LLPS) in a water-lysozyme solution. Scattering invariants and kinetic information provide evidence that the system reaches the phase boundary upon pressure-induced LLPS with no sign of arrest. The coarsening slows down with increasing quench depths. The g2 functions display a two-step decay with a gradually increasing nonergodicity parameter typical for gelation. We observe fast superdiffusive (γ ≥ 3/2) and slow subdiffusive (γ < 0.6) motion associated with fast viscoelastic fluctuations of the network and a slow viscous coarsening process, respectively. The dynamics age linearly with time τ ∝ tw, and we observe the onset of viscoelastic relaxation for deeper quenches. Our results suggest that the protein solution gels upon reaching the phase boundary.