Cytoplasmic RNA-Protein Particles Exhibit Non-Gaussian Subdiffusive Behavior

Macromolecular interactions and geometrical confinement determine the 3D diffusion of ribosome-sized particles in live Escherichia coli cells

The crowded bacterial cytoplasm is composed of biomolecules that span several orders of magnitude in size and electrical charge. This complexity has been proposed as the source of the rich spatial organization and apparent anomalous diffusion of intracellular components, although this has not been tested directly. Here, we use biplane microscopy to track the 3D motion of self-assembled bacterial genetically encoded multimeric nanoparticles (bGEMs) with tunable size (20 to 50 nm) and charge (-3,240 to +2,700 e) in live Escherichia coli cells. To probe intermolecular details at spatial and temporal resolutions beyond experimental limits, we also developed a colloidal whole-cell model that explicitly represents the size and charge of cytoplasmic macromolecules and the porous structure of the bacterial nucleoid. Combining these techniques, we show that bGEMs spatially segregate by size, with small 20-nm particles enriched inside the nucleoid, and larger and/or positively charged particles excluded from this region. Localization is driven by entropic and electrostatic forces arising from cytoplasmic polydispersity, nucleoid structure, geometrical confinement, and interactions with other biomolecules including ribosomes and DNA. We observe that at the timescales of traditional single molecule tracking experiments, motion appears subdiffusive for all particle sizes and charges. However, using computer simulations with higher temporal resolution, we find that the apparent anomalous exponents are governed by the region of the cell in which bGEMs are located. Molecular motion does not display anomalous diffusion on short time scales and the apparent subdiffusion arises from geometrical confinement within the nucleoid and by the cell boundary.

Brownian non-Gaussian polymer diffusion in non-static media

In nature, essentially, almost all the particles move irregularly in non-static media. With the advance of observation techniques, various kinds of new dynamical phenomena are detected, e.g., Brownian non-Gaussian diffusion. This paper focuses on the dynamical behavior of the center of mass (CM) of a polymer in non-static media and investigates the effect of polymer size fluctuations on the diffusion behavior. First, we establish a diffusing diffusivity model for polymer size fluctuations, linking the polymer size variation to the birth and death process, and introduce co-moving and physical coordinate systems to characterize the position of the CM for a polymer in non-static media. Next, the important statistical quantities for the CM diffusing diffusivity model in non-static media, such as mean square displacement (MSD) and kurtosis, are obtained by adopting the subordinate process approach, and the long-time asymptotic behavior of the MSD in the media of different types is specifically analyzed. Finally, the bivariate Fokker-Planck equation and the Feynman-Kac equation corresponding to the diffusing diffusivity model are detailedly derived and solved through the deep backward stochastic differential equation (BSDE) method to confirm the correctness of the derived equations.

Generalized Langevin dynamics for single beads in linear elastic networks

We derive generalized Langevin equations (GLEs) for single beads in linear elastic networks. In particular, the derivations of the GLEs are conducted without employing normal modes, resulting in two distinct representations in terms of resistance and mobility kernels. The fluctuation-dissipation relations are also confirmed for both GLEs. Subsequently, we demonstrate that these two representations are interconnected via Laplace transforms. Furthermore, another GLE is derived by utilizing a projection operator method, and it is shown that the equation obtained through the projection scheme is consistent with the GLE with the resistance kernel. As simple examples, the general theory is applied to the Rouse model and the ring polymer, where the GLEs with the resistance and mobility kernels are explicitly derived for arbitrary positions of the tagged bead in these models. Finally, the GLE with the mobility kernel is also derived for the elastic network with hydrodynamic interactions under the pre-averaging approximation.

Distinct Sub- to Superdiffuse Insulin Granule Transport Behaviors in β-Cells Are Strongly Affected by Granule Age

Intracellular transport is a complex process that is difficult to describe by a single general model for motion. Here, we study the transport of insulin containing vesicles, termed granules, in live MIN6 cells. We characterize how the observed heterogeneity is affected by different intracellular factors by constructing a MIN6 cell line by CRISPR-CAS9 that constitutively expresses mCherry fused to insulin and is thus packaged in granules. Confocal microscopy imaging and single particle tracking of the granule transport provide long trajectories of thousands of single granule trajectories for statistical analysis. Mean squared displacement (MSD), angle correlation distribution, and step size distribution analysis allowed identifying five distinct granule transport subpopulations, from nearly immobile and subdiffusive to run-pause and superdiffusive. The subdiffusive subpopulation recapitulates the subordinated random walk we reported earlier (Tabei, 2013; ref 18). We show that the transport characteristics of the five subpopulations have a strong dependence on the age of insulin granules. The five subpopulations also reflect the effect of local microtubule and actin networks on transport in different cellular regions. Our results provide robust metrics to clarify the heterogeneity of granule transport and demonstrate the roles of microtubule versus actin networks with granule age since initial packaging in the Golgi.

Intermittent subdiffusion of short nuclear actin rods due to interactions with chromatin

The interior of cellular nuclei, the nucleoplasm, is a crowded fluid that is pervaded by protein-decorated DNA polymers, the chromatin. Due to the complex architecture of chromatin and a multitude of associated nonequilibrium processes, e.g., DNA repair, the nucleoplasm can be expected to feature nontrivial material properties and hence anomalous transport phenomena. Here, we have used single-particle tracking on nuclear actin rods to probe such transport phenomena. Our analysis reveals that short actin rods in the nucleus show an intermittent, antipersistent subdiffusion with clear signatures of fractional Brownian motion. Moreover, the diffusive motion is heterogeneous with clear signatures of an intermittent switching of trajectories between at least two different mobilities, most likely due to transient associations with chromatin. In line with this interpretation, hyperosmotic stress is seen to stall the motion of nuclear actin rods, whereas hypo-osmotic conditions yield a reptationlike motion. Our data highlights the heterogeneity of transport in the nucleoplasm that needs to be taken into account for an understanding of nucleoplasmic organization and the mechanobiology of nuclei.

Amphibian Segmentation Clock Models Suggest How Large Genome and Cell Sizes Slow Developmental Rate

Evolutionary increases in genome size, cell volume, and nuclear volume have been observed across the tree of life, with positive correlations documented between all three traits. Developmental tempo slows as genomes, nuclei, and cells increase in size, yet the driving mechanisms are poorly understood. To bridge this gap, we use a mathematical model of the somitogenesis clock to link slowed developmental tempo with changes in intra-cellular gene expression kinetics induced by increasing genome size and nuclear volume. We adapt a well-known somitogenesis clock model to two model amphibian species that vary 10-fold in genome size: Xenopus laevis (3.1 Gb) and Ambystoma mexicanum (32 Gb). Based on simulations and backed by analytical derivations, we identify parameter changes originating from increased genome and nuclear size that slow gene expression kinetics. We simulate biological scenarios for which these parameter changes mathematically recapitulate slowed gene expression in A. mexicanum relative to X. laevis, and we consider scenarios for which additional alterations in gene product stability and chromatin packing are necessary. Results suggest that slowed degradation rates as well as changes induced by increasing nuclear volume and intron length, which remain relatively unexplored, are significant drivers of slowed developmental tempo.

Macromolecular interactions and geometrical confinement determine the 3D diffusion of ribosome-sized particles in live Escherichia coli cells

The crowded bacterial cytoplasm is comprised of biomolecules that span several orders of magnitude in size and electrical charge. This complexity has been proposed as the source of the rich spatial organization and apparent anomalous diffusion of intracellular components, although this has not been tested directly. Here, we use biplane microscopy to track the 3D motion of self-assembled bacterial Genetically Encoded Multimeric nanoparticles (bGEMs) with tunable size (20 to 50 nm) and charge (-2160 to +1800 e) in live Escherichia coli cells. To probe intermolecular details at spatial and temporal resolutions beyond experimental limits, we also developed a colloidal whole-cell model that explicitly represents the size and charge of cytoplasmic macromolecules and the porous structure of the bacterial nucleoid. Combining these techniques, we show that bGEMs spatially segregate by size, with small 20-nm particles enriched inside the nucleoid, and larger and/or positively charged particles excluded from this region. Localization is driven by entropic and electrostatic forces arising from cytoplasmic polydispersity, nucleoid structure, geometrical confinement, and interactions with other biomolecules including ribosomes and DNA. We observe that at the timescales of traditional single molecule tracking experiments, motion appears sub-diffusive for all particle sizes and charges. However, using computer simulations with higher temporal resolution, we find that the apparent anomalous exponents are governed by the region of the cell in which bGEMs are located. Molecular motion does not display anomalous diffusion on short time scales and the apparent sub-diffusion arises from geometrical confinement within the nucleoid and by the cell boundary.

Molecular Crowding: The History and Development of a Scientific Paradigm

It is now generally accepted that macromolecules do not act in isolation but "live" in a crowded environment, that is, an environment populated by numerous different molecules. The field of molecular crowding has its origins in the far 80s but became accepted only by the end of the 90s. In the present issue, we discuss various aspects that are influenced by crowding and need to consider its effects. This Review is meant as an introduction to the theme and an analysis of the evolution of the crowding concept through time from colloidal and polymer physics to a more biological perspective. We introduce themes that will be more thoroughly treated in other Reviews of the present issue. In our intentions, each Review may stand by itself, but the complete collection has the aspiration to provide different but complementary perspectives to propose a more holistic view of molecular crowding.

Non-Gaussian Displacements in Active Transport on a Carpet of Motile Cells

We study the dynamics of micron-sized particles on a layer of motile cells. This cell carpet acts as an active bath that propels passive tracer particles via direct mechanical contact. The resulting nonequilibrium transport shows a crossover from superdiffusive to normal-diffusive dynamics. The particle displacement distribution is distinctly non-Gaussian even at macroscopic timescales exceeding the measurement time. We obtain the distribution of diffusion coefficients from the experimental data and introduce a model for the displacement distribution that matches the experimentally observed non-Gaussian statistics. We argue why similar transport properties are expected for many composite active matter systems.

A Novel Fractional Brownian Dynamics Method for Simulating the Dynamics of Confined Bottle-Brush Polymers in Viscoelastic Solution

In crowded fluids, polymer segments can exhibit anomalous subdiffusion due to the viscoelasticity of the surrounding environment. Previous single-particle tracking experiments revealed that such anomalous diffusion in complex fluids (e.g., in bacterial cytoplasm) can be described by fractional Brownian motion (fBm). To investigate how the viscoelastic media affects the diffusive behaviors of polymer segments without resolving single crowders, we developed a novel fractional Brownian dynamics method to simulate the dynamics of polymers under confinement. In this work, instead of using Gaussian random numbers ("white Gaussian noise") to model the Brownian force as in the standard Brownian dynamics simulations, we introduce fractional Gaussian noise (fGn) in our homemade fractional Brownian dynamics simulation code to investigate the anomalous diffusion of polymer segments by using a simple "bottle-brush"-type polymer model. The experimental results of the velocity autocorrelation function and the exponent that characterizes the subdiffusion of the confined polymer segments can be reproduced by this simple polymer model in combination with fractional Gaussian noise (fGn), which mimics the viscoelastic media.

Non-Gaussian anomalous diffusion of optical vortices

Anomalous diffusion of different particlelike entities, the deviation from typical Brownian motion, is ubiquitous in complex physical and biological systems. While optical vortices move randomly in evolving speckle fields, optical vortices have only been observed to exhibit pure Brownian motion in random speckle fields. Here we present direct experimental evidence of the anomalous diffusion of optical vortices in temporally varying speckle patterns from multiple-scattering viscoelastic media. Moreover, we observe two characteristic features, i.e., the self-similarity and the antipersistent correlation of the optical vortex motion, indicating that the mechanism of the observed subdiffusion of optical vortices can only be attributed to fractional Brownian motion (FBM). We further demonstrate that the vortex displacements exhibit a non-Gaussian heavy-tailed distribution. Additionally, we modulate the extent of subdiffusion, such as diffusive scaling exponents, and the non-Gaussianity of optical vortices by altering the viscoelasticity of samples. The discovery of the complex FBM but non-Gaussian subdiffusion of optical vortices may not only offer insight into certain fundamental physics, including the anomalous diffusion of vortices in fluids and the decoupling between Brownianity and Gaussianity, but also suggest a strong potential for utilizing optical vortices as tracers in microrheology instead of the introduced exogenous probe particles in particle tracking microrheology.

Universal scaling of the diffusivity of dendrimers in a semidilute solution of linear polymers

The static and dynamic properties of dendrimers in semidilute solutions of linear chains of comparable size are investigated using Brownian dynamics simulations. The radius of gyration and diffusivity of a wide variety of low generation dendrimers and linear chains in solution follow universal scaling laws independent of their topology. Analysis of the shape functions and internal density of dendrimers shows that they are more spherical than linear chains and have a dense core. At intermediate times, dendrimers become subdiffusive, with an exponent higher than that previously reported for nanoparticles in semidilute polymer solutions. The long-time diffusivity of dendrimers does not follow theoretical predictions for nanoparticles. We propose a new scaling law for the long-time diffusion coefficients of dendrimers which accounts for the fact that, unlike nanoparticles, dendrimers shrink with an increase in background solution concentration. Analysis of the properties of a special case of a higher functionality dendrimer shows a transition from polymer-like to nanoparticle-like behaviour.

Inferring pointwise diffusion properties of single trajectories with deep learning

To characterize the mechanisms governing the diffusion of particles in biological scenarios, it is essential to accurately determine their diffusive properties. To do so, we propose a machine-learning method to characterize diffusion processes with time-dependent properties at the experimental time resolution. Our approach operates at the single-trajectory level predicting the properties of interest, such as the diffusion coefficient or the anomalous diffusion exponent, at every time step of the trajectory. In this way, changes in the diffusive properties occurring along the trajectory emerge naturally in the prediction and thus allow the characterization without any prior knowledge or assumption about the system. We first benchmark the method on synthetic trajectories simulated under several conditions. We show that our approach can successfully characterize both abrupt and continuous changes in the diffusion coefficient or the anomalous diffusion exponent. Finally, we leverage the method to analyze experiments of single-molecule diffusion of two membrane proteins in living cells: the pathogen-recognition receptor DC-SIGN and the integrin α5β1. The analysis allows us to characterize physical parameters and diffusive states with unprecedented accuracy, shedding new light on the underlying mechanisms.

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.

Different glassy characteristics are related to either caging or dynamical heterogeneity

Despite the enormous theoretical and application interests, a fundamental understanding of the glassy dynamics remains elusive. The static properties of glassy and ordinary liquids are similar, but their dynamics are dramatically different. What leads to this difference is the central puzzle of the field. Even the primary defining glassy characteristics, their implications, and if they are related to a single mechanism remain unclear. This lack of clarity is a severe hindrance to theoretical progress. Here, we combine analytical arguments and simulations of various systems in different dimensions and address these questions. Our results suggest that the myriad of glassy features are manifestations of two distinct mechanisms. Particle caging controls the mean, and coexisting slow- and fast-moving regions govern the distribution of particle displacements. All the other glassy characteristics are manifestations of these two mechanisms; thus, the Fickian yet non-Gaussian nature of glassy liquids is not surprising. We discover a crossover, from stretched exponential to a power law, in the behavior of the overlap function. This crossover is prominent in simulation data and forms the basis of our analyses. Our results have crucial implications on how the glassy dynamics data are analyzed, challenge some recent suggestions on the mechanisms governing glassy dynamics, and impose strict constraints that a correct theory of glasses must have.

Heterogeneous anomalous transport in cellular and molecular biology

It is well established that a wide variety of phenomena in cellular and molecular biology involve anomalous transport e.g. the statistics for the motility of cells and molecules are fractional and do not conform to the archetypes of simple diffusion or ballistic transport. Recent research demonstrates that anomalous transport is in many cases heterogeneous in both time and space. Thus single anomalous exponents and single generalised diffusion coefficients are unable to satisfactorily describe many crucial phenomena in cellular and molecular biology. We consider advances in the field ofheterogeneous anomalous transport(HAT) highlighting: experimental techniques (single molecule methods, microscopy, image analysis, fluorescence correlation spectroscopy, inelastic neutron scattering, and nuclear magnetic resonance), theoretical tools for data analysis (robust statistical methods such as first passage probabilities, survival analysis, different varieties of mean square displacements, etc), analytic theory and generative theoretical models based on simulations. Special emphasis is made on high throughput analysis techniques based on machine learning and neural networks. Furthermore, we consider anomalous transport in the context of microrheology and the heterogeneous viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion systems is also considered since it plays an important role in morphogenesis and signalling. In addition, we present specific examples from cellular biology including embryonic cells, leucocytes, cancer cells, bacterial cells, bacterial biofilms, and eukaryotic microorganisms. Case studies from molecular biology include DNA, membranes, endosomal transport, endoplasmic reticula, mucins, globular proteins, and amyloids.

How it feels in a cell

Life emerges from thousands of biochemical processes occurring within a shared intracellular environment. We have gained deep insights from in vitro reconstitution of isolated biochemical reactions. However, the reaction medium in test tubes is typically simple and diluted. The cell interior is far more complex: macromolecules occupy more than a third of the space, and energy-consuming processes agitate the cell interior. Here, we review how this crowded, active environment impacts the motion and assembly of macromolecules, with an emphasis on mesoscale particles (10-1000 nm diameter). We describe methods to probe and analyze the biophysical properties of cells and highlight how changes in these properties can impact physiology and signaling, and potentially contribute to aging, and diseases, including cancer and neurodegeneration.

Simulation Study on the Mechanism of Intermediate Subdiffusion of Diffusive Particles in Crowded Systems

The intermediate subdiffusion of diffusive particles in crowded systems is studied for two model systems: the continuous time random walk (CTRW) model and the obstruction-binding model. For the CTRW model with an arbitrarily given longest waiting time τmax, we find that the diffusive particle exhibits subdiffusion below τmax and recovers normal diffusion above τmax. For the obstruction-binding model with randomly distributed attractive obstacles, the diffusion of the diffusive particle is dependent on the binding energy and the density of obstacles. Interestingly, diffusion curves for different binding strengths can be overlapped by rescaling the simulation time, indicating that the diffusive particle in the obstruction-binding model can change from the intermediate subdiffusion to the normal diffusion at a long-term simulation scale. The results of the two model systems show that the diffusive particles only exhibit intermediate subdiffusion below the longest waiting time. Therefore, long timescale subdiffusion would only be observed in the CTRW model with an infinitely long waiting time and in the obstruction-binding model with an infinitely large binding strength.

The Role of Polymer Chain Stiffness and Guest Nanoparticle Loading in Improving the Glass Transition Temperature of Polymer Nanocomposites

The impact of polymer chain stiffness characterized by the bending modulus (k_θ) on the glass transition temperature (T_g) of pure polymer systems, as well as polymer nanocomposites (PNCs), is investigated using molecular dynamics simulations. At small k_θ values, the pure polymer system and respective PNCs are in an amorphous state, whereas at large k_θ values, both systems are in a semicrystalline state with a glass transition at low temperature. For the pure polymer system, T_g initially increases with k_θ and does not change obviously at large k_θ. However, the T_g of PNCs shows interesting behaviors with the increasing volume fraction of nanoparticles (f_NP) at different k_θ values. T_g tends to increase with f_NP at small k_θ, whereas it becomes suppressed at large k_θ.

Extending the analogy between intracellular motion in mammalian cells and glassy dynamics

How molecules, organelles, and foreign objects move within living cells has been studied in organisms ranging from bacteria to human cells. In mammalian cells, in particular, cellular vesicles move across the cell using motor proteins that carry the vesicle down the cytoskeleton to their destination. We have recently noted several similarities between the motion of such vesicles and that in disordered, "glassy", systems, but the generality of this observation remains unclear. Here we follow the motion of mitochondria, the organelles responsible for cell energy production, in mammalian cells over timescales from 50 ms to 70 s. Qualitative observations show that single mitochondria remain within a spatially limited region for extended periods of time, before moving longer distances relatively quickly. The displacement distribution is roughly Gaussian for shorter distances (≲0.05 μm) but exhibits exponentially decaying tails at longer distances (up to 0.40 μm). This behaviour is well-described by a model developed to describe the motion in glassy systems. These observations are extended to in total 3 different objects (mitochondria, lysosomes and nano-sized beads enclosed in vesicles), 3 different mammalian cell types (HEK 293, HeLa, and HT22), from 2 different organisms (human and mouse). Further evidence that supports glass-like characteristics of the motion is a difference between the time it takes to move a longer distance for the first time and subsequent times, as well as a weak ergodicity breaking of the motion. Overall, we demonstrate the ubiquity of glass-like motion in mammalian cells, providing a different perspective on intracellular motion.

Extracting, quantifying, and comparing dynamical and biomechanical properties of living matter through single particle tracking

A panoply of new tools for tracking single particles and molecules has led to an explosion of experimental data, leading to novel insights into physical properties of living matter governing cellular development and function, health and disease. In this Perspective, we present tools to investigate the dynamics and mechanics of living systems from the molecular to cellular scale via single-particle techniques. In particular, we focus on methods to measure, interpret, and analyse complex data sets that are associated with forces, materials properties, transport, and emergent organisation phenomena within biological and soft-matter systems. Current approaches, challenges, and existing solutions in the associated fields are outlined in order to support the growing community of researchers at the interface of physics and the life sciences. Each section focuses not only on the general physical principles and the potential for understanding living matter, but also on details of practical data extraction and analysis, discussing limitations, interpretation, and comparison across different experimental realisations and theoretical frameworks. Particularly relevant results are introduced as examples. While this Perspective describes living matter from a physical perspective, highlighting experimental and theoretical physics techniques relevant for such systems, it is also meant to serve as a solid starting point for researchers in the life sciences interested in the implementation of biophysical methods.

Chloroplasts in plant cells show active glassy behavior under low-light conditions

Plants have developed intricate mechanisms to adapt to changing light conditions. Besides phototropism and heliotropism (differential growth toward light and diurnal motion with respect to sunlight, respectively), chloroplast motion acts as a fast mechanism to change the intracellular structure of leaf cells. While chloroplasts move toward the sides of the plant cell to avoid strong light, they accumulate and spread out into a layer on the bottom of the cell at low light to increase the light absorption efficiency. Although the motion of chloroplasts has been studied for over a century, the collective organelle motion leading to light-adapting self-organized structures remains elusive. Here, we study the active motion of chloroplasts under dim-light conditions, leading to an accumulation in a densely packed quasi-2D layer. We observe burst-like rearrangements and show that these dynamics resemble systems close to the glass transition by tracking individual chloroplasts. Furthermore, we provide a minimal mathematical model to uncover relevant system parameters controlling the stability of the dense configuration of chloroplasts. Our study suggests that the meta-stable caging close to the glass transition in the chloroplast monolayer serves a physiological relevance: Chloroplasts remain in a spread-out configuration to increase the light uptake but can easily fluidize when the activity is increased to efficiently rearrange the structure toward an avoidance state. Our research opens questions about the role that dynamical phase transitions could play in self-organized intracellular responses of plant cells toward environmental cues.

Theoretical insights into the full description of DNA target search by subdiffusing proteins

DNA binding proteins (DBPs) diffuse in the cytoplasm to recognise and bind with their respective target sites on the DNA to initiate several biologically important processes. The first passage time distributions (FPTDs) of DBPs are useful in quantifying the timescales of the most-probable search paths in addition to the mean value of the distribution which, strikingly, are decades of order apart in time. However, extremely crowded in vivo conditions or the viscoelasticity of the cellular medium among other factors causes biomolecules to exhibit anomalous diffusion which is usually overlooked in most theoretical studies. We have obtained approximate analytical expressions of a general FPTD and the two characteristic timescales that are valid for any single subdiffusing protein searching for its target in vivo. Our results can be applied to single-particle tracking experiments of target search.

Heterogeneous Nanostructures Cause Anomalous Diffusion in Lipid Monolayers

The diffusion and mobility in biomembranes are crucial for various cell functions; however, the mechanisms involved in such processes remain ambiguous due to the complex membrane structures. Herein, we investigate how the heterogeneous nanostructures cause anomalous diffusion in dipalmitoylphosphatidylcholine (DPPC) monolayers. By identifying the existence of condensed nanodomains and clarifying their impact, our findings renew the understanding of the hydrodynamic description and the statistical feature of the diffusion in the monolayers. We find a universal characteristic of the multistage mean square displacement (MSD) with an intermediate crossover, signifying two membrane viscosities at different scales: the short-time scale describes the local fluidity and is independent of the nominal DPPC density, and the long-time scale represents the global continuous phase taking into account nanodomains and increases with DPPC density. The constant short-time viscosity reflects a dynamic equilibrium between the continuous fluid phase and the condensed nanodomains in the molecular scale. Notably, we observe an "anomalous yet Brownian" phenomenon exhibiting an unusual double-peaked displacement probability distribution (DPD), which is attributed to the net dipolar repulsive force from the heterogeneous nanodomains around the microdomains. The findings provide physical insights into the transport of membrane inclusions that underpin various biological functions and drug deliveries.

Heterogeneous run-and-tumble motion accounts for transient non-Gaussian super-diffusion in haematopoietic multi-potent progenitor cells

Multi-potent progenitor (MPP) cells act as a key intermediary step between haematopoietic stem cells and the entirety of the mature blood cell system. Their eventual fate determination is thought to be achieved through migration in and out of spatially distinct niches. Here we first analyze statistically MPP cell trajectory data obtained from a series of long time-course 3D in vivo imaging experiments on irradiated mouse calvaria, and report that MPPs display transient super-diffusion with apparent non-Gaussian displacement distributions. Second, we explain these experimental findings using a run-and-tumble model of cell motion which incorporates the observed dynamical heterogeneity of the MPPs. Third, we use our model to extrapolate the dynamics to time-periods currently inaccessible experimentally, which enables us to quantitatively estimate the time and length scales at which super-diffusion transitions to Fickian diffusion. Our work sheds light on the potential importance of motility in early haematopoietic progenitor function.

Defect-driven anomalous transport in fast-ion conducting solid electrolytes

Solid-state ionic conduction is a key enabler of electrochemical energy storage and conversion. The mechanistic connections between material processing, defect chemistry, transport dynamics and practical performance are of considerable importance but remain incomplete. Here, inspired by studies of fluids and biophysical systems, we re-examine anomalous diffusion in the iconic two-dimensional fast-ion conductors, the β- and β″-aluminas. Using large-scale simulations, we reproduce the frequency dependence of alternating-current ionic conductivity data. We show how the distribution of charge-compensating defects, modulated by processing, drives static and dynamic disorder and leads to persistent subdiffusive ion transport at macroscopic timescales. We deconvolute the effects of repulsions between mobile ions, the attraction between the mobile ions and charge-compensating defects, and geometric crowding on ionic conductivity. Finally, our characterization of memory effects in transport connects atomistic defect chemistry to macroscopic performance with minimal assumptions and enables mechanism-driven 'atoms-to-device' optimization of fast-ion conductors.

Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically.

Molecular and Colloidal Transport in Bacterial Cellulose Hydrogels

Bacterial cellulose biofilms are complex networks of strong interwoven nanofibers that control transport and protect bacterial colonies in the film. The design of diverse applications of these bacterial cellulose films also relies on understanding and controlling transport through the fiber mesh, and transport simulations of the films are most accurate when guided by experimental characterization of the structures and the resultant diffusion inside. Diffusion through such films is a function of their key microstructural length scales, determining how molecules, as well as particles and microorganisms, permeate them. We use microscopy to study the unique bacterial cellulose film via its pore structure and quantify the mobility dynamics of various sizes of tracer particles and macromolecules. Mobility is hindered within the films, as confinement and local movement strongly depend on the void size relative to diffusing tracers. The biofilms have a naturally periodic structure of alternating dense and porous layers of nanofiber mesh, and we tune the magnitude of the spacing via fermentation conditions. Micron-sized particles can diffuse through the porous layers but cannot penetrate the dense layers. Tracer mobility in the porous layers is isotropic, indicating a largely random pore structure there. Molecular diffusion through the whole film is only slightly reduced by the structural tortuosity. Knowledge of transport variations within bacterial cellulose networks can be used to guide the design of symbiotic cultures in these structures and enhance their use in applications like biomedical implants, wound dressings, lab-grown meat, clothing textiles, and sensors.

Motion of an active particle with dynamical disorder

We propose a model for investigating the motion of a single active particle in a heterogeneous environment where the heterogeneity may arise due to crowding, conformational fluctuations and/or slow rearrangement of the surroundings. Describing the active particle in terms of the Ornstein-Uhlenbeck process (OUP) and incorporating heterogeneity in a thermal bath using two separate models, namely "diffusing diffusivity" and "switching diffusion", we explore the essential dynamical properties of the particle for its one-dimensional motion. In addition, we show how the dynamical behavior is controlled by dynamical variables associated with active noise such as strength and persistence time. Our model is relevant in the context of single particle dynamics in a crowded environment, driven by activity.

Diffusion and distal linkages govern interchromosomal dynamics during meiotic prophase

SignificanceEssential for sexual reproduction, meiosis is a specialized cell division required for the production of haploid gametes. Critical to this process are the pairing, recombination, and segregation of homologous chromosomes (homologs). While pairing and recombination are linked, it is not known how many linkages are sufficient to hold homologs in proximity. Here, we reveal that random diffusion and the placement of a small number of linkages are sufficient to establish the apparent "pairing" of homologs. We also show that colocalization between any two loci is more dynamic than anticipated. Our study provides observations of live interchromosomal dynamics during meiosis and illustrates the power of combining single-cell measurements with theoretical polymer modeling.

Morphology and Transport in Eukaryotic Cells

Transport of intracellular components relies on a variety of active and passive mechanisms, ranging from the diffusive spreading of small molecules over short distances to motor-driven motion across long distances. The cell-scale behavior of these mechanisms is fundamentally dependent on the morphology of the underlying cellular structures. Diffusion-limited reaction times can be qualitatively altered by the presence of occluding barriers or by confinement in complex architectures, such as those of reticulated organelles. Motor-driven transport is modulated by the architecture of cytoskeletal filaments that serve as transport highways. In this review, we discuss the impact of geometry on intracellular transport processes that fulfill a broad range of functional objectives, including delivery, distribution, and sorting of cellular components. By unraveling the interplay between morphology and transport efficiency, we aim to elucidate key structure-function relationships that govern the architecture of transport systems at the cellular scale. Expected final online publication date for the Annual Review of Biophysics, Volume 51 is May 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

How does a systematic time-dependence of the diffusion coefficient $D (t)$ affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we examine how the behavior of the...

Dependence of diffusion in Escherichia coli cytoplasm on protein size, environmental conditions, and cell growth

Inside prokaryotic cells, passive translational diffusion typically limits the rates with which cytoplasmic proteins can reach their locations. Diffusion is thus fundamental to most cellular processes, but the understanding of protein mobility in the highly crowded and non-homogeneous environment of a bacterial cell is still limited. Here, we investigated the mobility of a large set of proteins in the cytoplasm of Escherichia coli, by employing fluorescence correlation spectroscopy (FCS) combined with simulations and theoretical modeling. We conclude that cytoplasmic protein mobility could be well described by Brownian diffusion in the confined geometry of the bacterial cell and at the high viscosity imposed by macromolecular crowding. We observed similar size dependence of protein diffusion for the majority of tested proteins, whether native or foreign to E. coli. For the faster-diffusing proteins, this size dependence is well consistent with the Stokes-Einstein relation once taking into account the specific dumbbell shape of protein fusions. Pronounced subdiffusion and hindered mobility are only observed for proteins with extensive interactions within the cytoplasm. Finally, while protein diffusion becomes markedly faster in actively growing cells, at high temperature, or upon treatment with rifampicin, and slower at high osmolarity, all of these perturbations affect proteins of different sizes in the same proportions, which could thus be described as changes of a well-defined cytoplasmic viscosity.

Precise measurements of chromatin diffusion dynamics by modeling using Gaussian processes

The spatiotemporal organization of chromatin influences many nuclear processes: from chromosome segregation to transcriptional regulation. To get a deeper understanding of these processes, it is essential to go beyond static viewpoints of chromosome structures, to accurately characterize chromatin's diffusion properties. We present GP-FBM: a computational framework based on Gaussian processes and fractional Brownian motion to extract diffusion properties from stochastic trajectories of labeled chromatin loci. GP-FBM uses higher-order temporal correlations present in the data, therefore, outperforming existing methods. Furthermore, GP-FBM allows to interpolate incomplete trajectories and account for substrate movement when two or more particles are present. Using our method, we show that average chromatin diffusion properties are surprisingly similar in interphase and mitosis in mouse embryonic stem cells. We observe surprising heterogeneity in local chromatin dynamics, correlating with potential regulatory activity. We also present GP-Tool, a user-friendly graphical interface to facilitate usage of GP-FBM by the research community.

Fingerprints of viscoelastic subdiffusion in random environments: Revisiting some experimental data and their interpretations

Many experimental studies revealed subdiffusion of various nanoparticles in diverse polymer and colloidal solutions, cytosol and plasma membrane of biological cells, which are viscoelastic and, at the same time, highly inhomogeneous randomly fluctuating environments. The observed subdiffusion often combines features of ergodic fractional Brownian motion (reflecting viscoelasticity) and nonergodic jumplike non-Markovian diffusional processes (reflecting disorder). Accordingly, several theories were proposed to explain puzzling experimental findings. Below we show that some of the significant and profound published experimental results are better rationalized within the viscoelastic subdiffusion approach in random environments, which is based on generalized Langevin dynamics in random potentials, than some earlier proposed theories.

Thermodynamic Uncertainty Relation Bounds the Extent of Anomalous Diffusion

In a finite system driven out of equilibrium by a constant external force the thermodynamic uncertainty relation (TUR) bounds the variance of the conjugate current variable by the thermodynamic cost of maintaining the nonequilibrium stationary state. Here we highlight a new facet of the TUR by showing that it also bounds the timescale on which a finite system can exhibit anomalous kinetics. In particular, we demonstrate that the TUR bounds subdiffusion in a single file confined to a ring as well as a dragged Gaussian polymer chain even when detailed balance is satisfied. Conversely, the TUR bounds the onset of superdiffusion in the active comb model. Remarkably, the fluctuations in a comb model evolving from a steady state behave anomalously as soon as detailed balance is broken. Our work establishes a link between stochastic thermodynamics and the field of anomalous dynamics that will fertilize further investigations of thermodynamic consistency of anomalous diffusion models.

Local Analysis of Heterogeneous Intracellular Transport: Slow and Fast Moving Endosomes

Trajectories of endosomes inside living eukaryotic cells are highly heterogeneous in space and time and diffuse anomalously due to a combination of viscoelasticity, caging, aggregation and active transport. Some of the trajectories display switching between persistent and anti-persistent motion, while others jiggle around in one position for the whole measurement time. By splitting the ensemble of endosome trajectories into slow moving subdiffusive and fast moving superdiffusive endosomes, we analyzed them separately. The mean squared displacements and velocity auto-correlation functions confirm the effectiveness of the splitting methods. Applying the local analysis, we show that both ensembles are characterized by a spectrum of local anomalous exponents and local generalized diffusion coefficients. Slow and fast endosomes have exponential distributions of local anomalous exponents and power law distributions of generalized diffusion coefficients. This suggests that heterogeneous fractional Brownian motion is an appropriate model for both fast and slow moving endosomes. This article is part of a Special Issue entitled: "Recent Advances In Single-Particle Tracking: Experiment and Analysis" edited by Janusz Szwabiński and Aleksander Weron.

Single-Particle Tracking Reveals Anti-Persistent Subdiffusion in Cell Extracts

Single-particle tracking (SPT) has become a powerful tool to quantify transport phenomena in complex media with unprecedented detail. Based on the reconstruction of individual trajectories, a wealth of informative measures become available for each particle, allowing for a detailed comparison with theoretical predictions. While SPT has been used frequently to explore diffusive transport in artificial fluids and inside living cells, intermediate systems, i.e., biochemically active cell extracts, have been studied only sparsely. Extracts derived from the eggs of the clawfrog Xenopus laevis, for example, are known for their ability to support and mimic vital processes of cells, emphasizing the need to explore also the transport phenomena of nano-sized particles in such extracts. Here, we have performed extensive SPT on beads with 20 nm radius in native and chemically treated Xenopus extracts. By analyzing a variety of distinct measures, we show that these beads feature an anti-persistent subdiffusion that is consistent with fractional Brownian motion. Chemical treatments did not grossly alter this finding, suggesting that the high degree of macromolecular crowding in Xenopus extracts equips the fluid with a viscoelastic modulus, hence enforcing particles to perform random walks with a significant anti-persistent memory kernel.

Subtle changes in crosslinking drive diverse anomalous transport characteristics in actin-microtubule networks

Anomalous diffusion in crowded and complex environments is widely studied due to its importance in intracellular transport, fluid rheology and materials engineering. Specifically, diffusion through the cytoskeleton, a network comprised of semiflexible actin filaments and rigid microtubules that interact both sterically and via crosslinking, plays a principal role in viral infection, vesicle transport and targeted drug delivery. Here, we elucidate the impact of crosslinking on particle diffusion in composites of actin and microtubules with actin-actin, microtubule-microtubule and actin-microtubule crosslinking. We analyze a suite of transport metrics by coupling single-particle tracking and differential dynamic microscopy. Using these complementary techniques, we find that particles display non-Gaussian and non-ergodic subdiffusion that is markedly enhanced by cytoskeletal crosslinking, which we attribute to suppressed microtubule mobility. However, the extent to which transport deviates from normal Brownian diffusion depends strongly on the crosslinking motif - with actin-microtubule crosslinking inducing the most pronounced anomalous characteristics. Our results reveal that subtle changes to actin-microtubule interactions can have complex impacts on particle diffusion in cytoskeleton composites, and suggest that a combination of reduced filament mobility and more variance in actin mobilities leads to more strongly anomalous particle transport.

Mesoscale Simulation of Bacterial Chromosome and Cytoplasmic Nanoparticles in Confinement

In this study we investigated, using a simple polymer model of bacterial chromosome, the subdiffusive behaviors of both cytoplasmic particles and various loci in different cell wall confinements. Non-Gaussian subdiffusion of cytoplasmic particles as well as loci were obtained in our Langevin dynamic simulations, which agrees with fluorescence microscope observations. The effects of cytoplasmic particle size, locus position, confinement geometry, and density on motions of particles and loci were examined systematically. It is demonstrated that the cytoplasmic subdiffusion can largely be attributed to the mechanical properties of bacterial chromosomes rather than the viscoelasticity of cytoplasm. Due to the randomly positioned bacterial chromosome segments, the surrounding environment for both particle and loci is heterogeneous. Therefore, the exponent characterizing the subdiffusion of cytoplasmic particle/loci as well as Laplace displacement distributions of particle/loci can be reproduced by this simple model. Nevertheless, this bacterial chromosome model cannot explain the different responses of cytoplasmic particles and loci to external compression exerted on the bacterial cell wall, which suggests that the nonequilibrium activity, e.g., metabolic reactions, play an important role in cytoplasmic subdiffusion.

Glass-like characteristics of intracellular motion in human cells

The motion in the cytosol of microorganisms such as bacteria and yeast has been observed to undergo a dramatic slowing down upon cell energy depletion. These observations have been interpreted as the motion being "glassy," but whether this notion is useful also for active, motor-protein-driven transport in eukaryotic cells is less clear. Here, we use fluorescence microscopy of beads in human (HeLa) cells to probe the motion of membrane-surrounded structures that are carried along the cytoskeleton by motor proteins. Evaluating several hallmarks of glassy dynamics, we show that at short length scales, the motion is heterogeneous, is nonergodic, is well described by a model for the displacement distribution in glassy systems, and exhibits a decoupling of the exchange and persistence times. Overall, these results suggest that the short length scale behavior of objects that can be transported actively by motor proteins in human cells shares features with the motion in glassy systems.

Universal Relation between Instantaneous Diffusivity and Radius of Gyration of Proteins in Aqueous Solution

Protein conformational fluctuations are highly complex and exhibit long-term correlations. Here, molecular dynamics simulations of small proteins demonstrate that these conformational fluctuations directly affect the protein's instantaneous diffusivity D_{I}. We find that the radius of gyration R_{g} of the proteins exhibits 1/f fluctuations that are synchronous with the fluctuations of D_{I}. Our analysis demonstrates the validity of the local Stokes-Einstein-type relation D_{I}∝1/(R_{g}+R_{0}), where R_{0}∼0.3  nm is assumed to be a hydration layer around the protein. From the analysis of different protein types with both strong and weak conformational fluctuations, the validity of the Stokes-Einstein-type relation appears to be a general property.

Fluctuating Diffusivity of RNA-Protein Particles: Analogy with Thermodynamics

A formal analogy of fluctuating diffusivity to thermodynamics is discussed for messenger RNA molecules fluorescently fused to a protein in living cells. Regarding the average value of the fluctuating diffusivity of such RNA-protein particles as the analog of the internal energy, the analogs of the quantity of heat and work are identified. The Clausius-like inequality is shown to hold for the entropy associated with diffusivity fluctuations, which plays a role analogous to the thermodynamic entropy, and the analog of the quantity of heat. The change of the statistical fluctuation distribution is also examined from a geometric perspective. The present discussions may contribute to a deeper understanding of the fluctuating diffusivity in view of the laws of thermodynamics.

From diffusion in compartmentalized media to non-Gaussian random walks

In this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the non-Gaussian diffusion that exhibits linear or power-law growth of the mean square displacement joined by the exponential shape of the positional probability density. We explore a microscopic model that gives rise to transient confinement, similar to the one observed for hop-diffusion on top of a cellular membrane. The compartmentalization of the media is achieved by introducing randomly placed, identical barriers. Using this model of a heterogeneous medium we derive a general class of random walks with simple jump rules that are dictated by the geometry of the compartments. Exponential decay of positional probability density is observed and we also quantify the significant decrease of the long time diffusion constant. Our results suggest that the observed exponential decay is a general feature of the transient regime in compartmentalized media.

Superstatistical modelling of protein diffusion dynamics in bacteria

A recent experiment (Sadoon AA, Wang Y. 2018 Phys. Rev. E98, 042411. (doi:10.1103/PhysRevE.98.042411)) has revealed that nucleoid-associated proteins (i.e. DNA-binding proteins) exhibit highly heterogeneous diffusion processes in bacteria where not only the diffusion constant but also the anomalous diffusion exponent fluctuates for the various proteins. The distribution of displacements of such proteins is observed to take a q-Gaussian form, which decays as a power law. Here, a statistical model is developed for the diffusive motion of the proteins within the bacterium, based on a superstatistics with two variables. This model hierarchically takes into account the joint fluctuations of both the anomalous diffusion exponents and the diffusion constants. A fractional Brownian motion is discussed as a possible local model. Good agreement with the experimental data is obtained.

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

We study a two state "jumping diffusivity" model for a Brownian process alternating between two different diffusion constants, D+>D-, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient D-⟶0 is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements P(x,t) for x⟶0. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D+ to a δ-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in P(x,t). We demonstrate how super-statistical framework is a zeroth order short time expansion of P(x,t), in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

Time-dependent classification of protein diffusion types: A statistical detection of mean-squared-displacement exponent transitions

In this paper, we have proposed a statistical procedure for detecting transitions of the mean-square-displacement exponent value within a single trajectory. With this procedure, we have identified three regimes of proteins dynamics on a cell membrane, namely, subdiffusion, free diffusion, and immobility. The fourth considered dynamics type, namely, superdiffusion was not detected. We show that the analyzed protein trajectories are not stationary and not ergodic. Moreover, classification of the dynamics type performed without prior detection of transitions may lead to the overestimation of the proportion of subdiffusive trajectories.

Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement

The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them.