Probing the type of anomalous diffusion with single-particle tracking

Multifractal spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes, characterized by their nonstandard scaling of the mean-squared displacement, pose a unique challenge in classification and characterization. In a previous study [Mangalam et al., Phys. Rev. Res. 5, 023144 (2023)2643-156410.1103/PhysRevResearch.5.023144], we established a comprehensive framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous-time random walk, annealed transient time motion, and Lévy walk. We generate extensive datasets comprising 10^{6} trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory to accomplish this. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. We also explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and p varation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. In summary, our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing the predictive capacity of machine learning to classify anomalous diffusion processes.

Temperature-mediated diffusion of nanoparticles in semidilute polymer solutions

The temperature is often a critical factor affecting the diffusion of nanoparticles in complex physiological media, but its specific effects are still to be fully understood. Here, we constructed a temperature-regulated model of semidilute polymer solution and experimentally investigated the temperature-mediated diffusion of nanoparticles using the particle tracking method. By examining the ensemble-averaged mean square displacements (MSDs), we found that the MSD grows gradually as the temperature increases while the transition time from sublinear to linear stage in MSD decreases. Meanwhile, the temperature-dependent measured diffusivity of the nanoparticles shows an exponential growth. We revealed that these temperature-mediated changes are determined by the composite effect of the macroscale property of polymer solution and the microscale dynamics of polymer chain as well as nanoparticles. Furthermore, the measured non-Gaussian displacement probability distributions were found to exhibit non-Gaussian fat tails, and the tailed distribution is enhanced as the temperature increases. The non-Gaussianity was calculated and found to vary in the same trend with the tailed distribution, suggesting the occurrence of hopping events. This temperature-mediated non-Gaussian feature validates the recent theory of thermally induced activated hopping. Our results highlight the temperature-mediated changes in diffusive transport of nanoparticles in polymer solutions and may provide the possible strategy to improve drug delivery in physiological media.

Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories

Single-particle traces of the diffusive motion of molecules, cells, or animals are by now routinely measured, similar to stochastic records of stock prices or weather data. Deciphering the stochastic mechanism behind the recorded dynamics is vital in understanding the observed systems. Typically, the task is to decipher the exact type of diffusion and/or to determine the system parameters. The tools used in this endeavor are currently being revolutionized by modern machine-learning techniques. In this Perspective we provide an overview of recently introduced methods in machine-learning for diffusive time series, most notably, those successfully competing in the anomalous diffusion challenge. As such methods are often criticized for their lack of interpretability, we focus on means to include uncertainty estimates and feature-based approaches, both improving interpretability and providing concrete insight into the learning process of the machine. We expand the discussion by examining predictions on different out-of-distribution data. We also comment on expected future developments.

Dynamics of a spherical self-propelled tracer in a polymeric medium: interplay of self-propulsion, stickiness, and crowding

We employ computer simulations to study the dynamics of a self-propelled spherical tracer particle in a viscoelastic medium, made of a long polymer chain. Here, the interplay between viscoelasticity, stickiness, and activity (self-propulsion) brings additional complexity to the tracer dynamics. Our simulations show that on increasing the stickiness of the tracer particle to the polymer beads, the dynamics of the tracer particle slows down as it gets stuck to the polymer chain and moves along with it. But with increasing self-propulsion velocity, the dynamics gets enhanced. In the case of increasing stickiness as well as activity, the non-Gaussian parameter (NGP) exhibits non-monotonic behavior, which also shows up in the re-scaled self part of the van-Hove function. Non-Gaussianity results owing to the enhanced binding events and the sticky motion of the tracer along with the chain with increasing stickiness. On the other hand, with increasing activity, initially non-Gaussianity increases as the tracer moves through the heterogeneous polymeric environment but for higher activity, the tracer escapes resulting in a negative NGP. For higher values of stickiness, the trapping time distributions of the passive tracer particle broaden and have long tails. On the other hand, for a given stickiness with increasing self-propulsion force, the trapping time distributions become narrower and have short tails. We believe that our current simulation study will be helpful in elucidating the complex motion of activity-driven probes in viscoelastic media.

Molecular Dynamics Simulations of Rhodamine B Zwitterion Diffusion in Polyelectrolyte Solutions

Polyelectrolytes continue to find wide interest and application in science and engineering, including areas such as water purification, drug delivery, and multilayer thin films. We have been interested in the dynamics of small molecules in a variety of polyelectrolyte (PE) environments; in this paper, we report simulations and analysis of the small dye molecule rhodamine B (RB) in several very simple polyelectrolyte solutions. Translational diffusion of the RB zwitterion has been measured in fully atomistic, 2 μs long molecular dynamics simulations in four different polyelectrolyte solutions. Two solutions contain the common polyanion sodium poly(styrene sulfonate) (PSS), one with a 30-mer chain and the other with 10 trimers. The other two solutions contain the common polycation poly(allyldimethylammonium) chloride (PDDA), one with two 15-mers and the other with 10 trimers. RB diffusion was also simulated in several polymer-free solutions to verify its known experimental value for the translational diffusion coefficient, D_RB, of 4.7 × 10^-6 cm²/s at 300 K. RB diffusion was slowed in all four simulated PE solutions, but to varying degrees. D_RB values of 3.07 × 10^-6 and 3.22 × 10^-6 cm²/s were found in PSS 30-mer and PSS trimer solutions, respectively, whereas PDDA 15-mer and trimer solutions yielded values of 2.19 × 10^-6 and 3.34 × 10^-6 cm²/s. Significant associations between RB and the PEs were analyzed and interpreted via a two-state diffusion model (bound and free diffusion) that describes the data well. Crowder size effects and anomalous diffusion were also analyzed. Finally, RB translation along the polyelectrolytes during association was characterized.

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.

Machine learning classification of trajectories from molecular dynamics simulations of chromosome segregation

In contrast to the well characterized mitotic machinery in eukaryotes it seems as if there is no universal mechanism organizing chromosome segregation in all bacteria. Apparently, some bacteria even use combinations of different segregation mechanisms such as protein machines or rely on physical forces. The identification of the relevant mechanisms is a difficult task. Here, we introduce a new machine learning approach to this problem. It is based on the analysis of trajectories of individual loci in the course of chromosomal segregation obtained by fluorescence microscopy. While machine learning approaches have already been applied successfully to trajectory classification in other areas, so far it has not been possible to use them to discriminate segregation mechanisms in bacteria. A main obstacle for this is the large number of trajectories required to train machine learning algorithms that we overcome here by using trajectories obtained from molecular dynamics simulations. We used these trajectories to train four different machine learning algorithms, two linear models and two tree-based classifiers, to discriminate segregation mechanisms and possible combinations of them. The classification was performed once using the complete trajectories as high-dimensional input vectors as well as on a set of features which were used to transform the trajectories into low-dimensional input vectors for the classifiers. Finally, we tested our classifiers on shorter trajectories with duration times comparable (or even shorter) than typical experimental trajectories and on trajectories measured with varying temporal resolutions. Our results demonstrate that machine learning algorithms are indeed capable of discriminating different segregation mechanisms in bacteria and to even resolve combinations of the mechanisms on rather short time scales.

First-passage statistics of colloids on fractals: Theory and experimental realization

In nature and technology, particle dynamics frequently occur in complex environments, for example in restricted geometries or crowded media. These dynamics have often been modeled invoking a fractal structure of the medium although the fractal structure was only indirectly inferred through the dynamics. Moreover, systematic studies have not yet been performed. Here, colloidal particles moving in a laser speckle pattern are used as a model system. In this case, the experimental observations can be reliably traced to the fractal structure of the underlying medium with an adjustable fractal dimension. First-passage time statistics reveal that the particles explore the speckle in a self-similar, fractal manner at least over four decades in time and on length scales up to 20 times the particle radius. The requirements for fractal diffusion to be applicable are laid out, and methods to extract the fractal dimension are established.

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.

Recursive Deep Prior Video: A super resolution algorithm for time-lapse microscopy of organ-on-chip experiments

Biological experiments based on organ-on-chips (OOCs) exploit light Time-Lapse Microscopy (TLM) for a direct observation of cell movement that is an observable signature of underlying biological processes. A high spatial resolution is essential to capture cell dynamics and interactions from recorded experiments by TLM. Unfortunately, due to physical and cost limitations, acquiring high resolution videos is not always possible. To overcome the problem, we present here a new deep learning-based algorithm that extends the well-known Deep Image Prior (DIP) to TLM Video Super Resolution without requiring any training. The proposed Recursive Deep Prior Video method introduces some novelties. The weights of the DIP network architecture are initialized for each of the frames according to a new recursive updating rule combined with an efficient early stopping criterion. Moreover, the DIP loss function is penalized by two different Total Variation-based terms. The method has been validated on synthetic, i.e., artificially generated, as well as real videos from OOC experiments related to tumor-immune interaction. The achieved results are compared with several state-of-the-art trained deep learning Super Resolution algorithms showing outstanding performances.

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.

Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes

Heterogeneous diffusion processes (HDPs) with space-dependent diffusion coefficients D(x) are found in a number of real-world systems, such as for diffusion of macromolecules or submicron tracers in biological cells. Here, we examine HDPs in quenched-disorder systems with Gaussian colored noise (GCN) characterized by a diffusion coefficient with a power-law dependence on the particle position and with a spatially random scaling exponent. Typically, D(x) is considered to be centerd at the origin and the entire x axis is characterized by a single scaling exponent α. In this work we consider a spatially random scenario: in periodic intervals ("layers") in space D(x) is centerd to the midpoint of each interval. In each interval the scaling exponent α is randomly chosen from a Gaussian distribution. The effects of the variation of the scaling exponents, the periodicity of the domains ("layer thickness") of the diffusion coefficient in this stratified system, and the correlation time of the GCN are analyzed numerically in detail. We discuss the regimes of superdiffusion, subdiffusion, and normal diffusion realisable in this system. We observe and quantify the domains where nonergodic and non-Gaussian behaviors emerge in this system. Our results provide new insights into the understanding of weak ergodicity breaking for HDPs driven by colored noise, with potential applications in quenched layered systems, typical model systems for diffusion in biological cells and tissues, as well as for diffusion in geophysical systems.

Quantifying active diffusion in an agitated fluid

Mixing of reactants in microdroplets predominantly relies on diffusional motion due to small Reynolds numbers and the resulting absence of turbulent flows. Enhancing diffusion in microdroplets by an auxiliary noise source is therefore a topical problem. Here we report on how the diffusional motion of tracer beads is enhanced upon agitating the surrounding aqueous fluid with miniaturized magnetic stir bars that are compatible with microdroplets and microfluidic devices. Using single-particle tracking, we demonstrate via a broad palette of measures that local stirring of the fluid at different frequencies leads to an enhanced but apparently normal and homogenous diffusion process, i.e. diffusional steps follow the anticipated Gaussian distribution and no ballistic motion is observed whereas diffusion coefficients are significantly increased. The signature of stirring is, however, visible in the power-spectral density and in the velocity autocorrelation function of trajectories. Our data therefore demonstrate that diffusive mixing can be locally enhanced with miniaturized stir bars while only moderately affecting the ambient noise properties. The latter may also facilitate the controlled addition of nonequilibrium noise to complex fluids in future applications.

Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x)=D_{0}|x|^{α}. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble- and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y∼|x|^{1/(2/(2-α))}/t^{H} coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P_{HDP-FBM}(y)=e^{-y^{2}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.

Anomalous and heterogeneous DNA transport in biomimetic cytoskeleton networks

The cytoskeleton, a complex network of protein filaments and crosslinking proteins, dictates diverse cellular processes ranging from division to cargo transport. Yet, the role the cytoskeleton plays in the intracellular transport of DNA and other macromolecules remains poorly understood. Here, using single-molecule conformational tracking, we measure the transport and conformational dynamics of linear and relaxed circular (ring) DNA in composite networks of actin and microtubules with variable types of crosslinking. While both linear and ring DNA undergo anomalous, non-Gaussian, and non-ergodic subdiffusion, the detailed dynamics are controlled by both DNA topology (linear vs. ring) and crosslinking motif. Ring DNA swells, exhibiting heterogeneous subdiffusion controlled via threading by cytoskeleton filaments, while linear DNA compacts, exhibiting transport via caging and hopping. Importantly, while the crosslinking motif has little effect on ring DNA, linear DNA in networks with actin-microtubule crosslinking is significantly less ergodic and shows more heterogeneous transport than with actin-actin or microtubule-microtubule crosslinking.

Diffusion of Nanoparticles with Activated Hopping in Crowded Polymer Solutions

A long-distance hop of diffusive nanoparticles (NPs) in crowded environments was commonly considered unlikely, and its characteristics remain unclear. In this work, we experimentally identify the occurrence of the intermittent hops of large NPs in crowded entangled poly(ethylene oxide) (PEO) solutions, which are attributed to thermally induced activated hopping. We show that the diffusion of NPs in crowded solutions is considered as a superposition of the activated hopping and the reptation of the polymer solution. Such activated hopping becomes significant when either the PEO molecular weight is large enough or the NP size is relatively small. We reveal that the time-dependent non-Gaussianity of the NP diffusion is determined by the competition of the short-time relaxation of a polymer entanglement strand, the activated hopping, and the long-time reptation. We propose an exponential scaling law τ_hop/τ_e ∼ exp(d/d_t) to characterize the hopping time scale, suggesting a linear dependence of the activated hopping energy barrier on the dimensionless NP size. The activated hopping motion can only be observed between the onset time scale of the short-time relaxation of local entanglement strands and the termination time scale of the long-time relaxation. Our findings on activated hopping provide new insights into long-distance transportation of NPs in crowded biological environments, which is essential to the delivery and targeting of nanomedicines.

Non-Gaussian subdiffusion of single-molecule tracers in a hydrated polymer network

Single molecule tracking experiments inside a hydrated polymer network have shown that the tracer motion is subdiffusive due to the viscoelastic environment inside the gel-like network. This property can be related to the negative autocorrelation of the instantaneous displacements at short times. Although the displacements of the individual tracers exhibit Gaussian statistics, the displacement distribution of all the trajectories combined from different spatial locations of the polymer network exhibits a non-Gaussian distribution. Here, we analyze many individual tracer trajectories to show that the central portion of the non-Gaussian distribution can be well approximated by an exponential distribution that spreads sublinearly with time. We explain all these features seen in the experiment by a generalized Langevin model for an overdamped particle with algebraically decaying correlations. We show that the degree of non-Gaussianity can change with the extent of heterogeneity, which is controlled in our model by the experimentally observed distributions of the motion parameters.

Filament Rigidity Vies with Mesh Size in Determining Anomalous Diffusion in Cytoskeleton

The diffusion of microscopic particles through the cell, important to processes such as viral infection, gene delivery, and vesicle transport, is largely controlled by the complex cytoskeletal network, comprised of semiflexible actin filaments and rigid microtubules, that pervades the cytoplasm. By varying the relative concentrations of actin and microtubules, the cytoskeleton can display a host of different structural and dynamic properties that, in turn, impact the diffusion of particles through the composite network. Here, we couple single-particle tracking with differential dynamic microscopy to characterize the transport of microsphere tracers diffusing through composite in vitro networks with varying ratios of actin and microtubules. We analyze multiple complementary metrics for anomalous transport to show that particles exhibit anomalous subdiffusion in all networks, which our data suggest arises from caging by networks. Further, subdiffusive characteristics are markedly more pronounced in actin-rich networks, which exhibit similarly more prominent viscoelastic properties compared to microtubule-rich composites. While the smaller mesh size of actin-rich composites compared to microtubule-rich composites plays an important role in these results, the rigidity of the filaments comprising the network also influences the anomalous characteristics that we observe. Our results suggest that as microtubules in our composites are replaced with actin filaments, the decreasing filament rigidity competes with increasing network connectivity to drive anomalous transport.

Classification of diffusion modes in single-particle tracking data: Feature-based versus deep-learning approach

Single-particle trajectories measured in microscopy experiments contain important information about dynamic processes occurring in a range of materials including living cells and tissues. However, extracting that information is not a trivial task due to the stochastic nature of the particles' movement and the sampling noise. In this paper, we adopt a deep-learning method known as a convolutional neural network (CNN) to classify modes of diffusion from given trajectories. We compare this fully automated approach working with raw data to classical machine learning techniques that require data preprocessing and extraction of human-engineered features from the trajectories to feed classifiers like random forest or gradient boosting. All methods are tested using simulated trajectories for which the underlying physical model is known. From the results it follows that CNN is usually slightly better than the feature-based methods, but at the cost of much longer processing times. Moreover, there are still some borderline cases in which the classical methods perform better than CNN.

The influence of spatial and temporal resolutions on the analysis of cell-cell interaction: a systematic study for time-lapse microscopy applications

Cell-cell interactions are an observable manifestation of underlying complex biological processes occurring in response to diversified biochemical stimuli. Recent experiments with microfluidic devices and live cell imaging show that it is possible to characterize cell kinematics via computerized algorithms and unravel the effects of targeted therapies. We study the influence of spatial and temporal resolutions of time-lapse videos on motility and interaction descriptors with computational models that mimic the interaction dynamics among cells. We show that the experimental set-up of time-lapse microscopy has a direct impact on the cell tracking algorithm and on the derived numerical descriptors. We also show that, when comparing kinematic descriptors in two diverse experimental conditions, too low resolutions may alter the descriptors’ discriminative power, and so the statistical significance of the difference between the two compared distributions. The conclusions derived from the computational models were experimentally confirmed by a series of video-microscopy acquisitions of co-cultures of unlabelled human cancer and immune cells embedded in 3D collagen gels within microfluidic devices. We argue that the experimental protocol of acquisition should be adapted to the specific kind of analysis involved and to the chosen descriptors in order to derive reliable conclusions and avoid biasing the interpretation of results.

Anomalous Diffusion in Inverted Variable-Lengthscale Fluorescence Correlation Spectroscopy

Using fluorescence correlation spectroscopy (FCS) to distinguish between different types of diffusion processes is often a perilous undertaking because the analysis of the resulting autocorrelation data is model dependant. Two recently introduced strategies, however, can help move toward a model-independent interpretation of FCS experiments: 1) the obtention of correlation data at different length scales and 2) their inversion to retrieve the mean-squared displacement associated with the process under study. We use computer simulations to examine the signature of several biologically relevant diffusion processes (simple diffusion, continuous-time random walk, caged diffusion, obstructed diffusion, two-state diffusion, and diffusing diffusivity) in variable-length-scale FCS. We show that, when used in concert, length-scale variation and data inversion permit us to identify non-Gaussian processes and, regardless of Gaussianity, to retrieve their mean-squared displacement over several orders of magnitude in time. This makes unbiased discrimination between different classes of diffusion models possible.

Microrheology, advances in methods and insights

Microrheology is an emerging technique that probes mechanical response of soft material at micro-scale. Generally, microrheology technique can be divided into active and passive versions. During last two decades, extensive efforts have been paid to improve both the experiment techniques and data analysis methods, especially about how to link consequential particle positions into trajectories. We review the recent advances in microrheology, including improvements in labeling, imaging, data acquiring, data processing and data interpretation. Some of the recent insights in soft matter and living systems gained by using this technique are given. Before these, we also give a very brief description of the basic principles of both active and passive microrheology techniques, and some details about optical particle tracking and DWS.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Elucidating fluctuating diffusivity in center-of-mass motion of polymer models with time-averaged mean-square-displacement tensor

There have been increasing reports that the diffusion coefficient of macromolecules depends on time and fluctuates randomly. Here a method is developed to elucidate this fluctuating diffusivity from trajectory data. Time-averaged mean-square displacement (MSD), a common tool in single-particle-tracking (SPT) experiments, is generalized to a second-order tensor with which both magnitude and orientation fluctuations of the diffusivity can be clearly detected. This method is used to analyze the center-of-mass motion of four fundamental polymer models: the Rouse model, the Zimm model, a reptation model, and a rigid rodlike polymer. It is found that these models exhibit distinctly different types of magnitude and orientation fluctuations of diffusivity. This is an advantage of the present method over previous ones, such as the ergodicity-breaking parameter and a non-Gaussian parameter, because with either of these parameters it is difficult to distinguish the dynamics of the four polymer models. Also, the present method of a time-averaged MSD tensor could be used to analyze trajectory data obtained in SPT experiments.

Test of the diffusing-diffusivity mechanism using near-wall colloidal dynamics

The mechanism of diffusing diffusivity predicts that, in environments where the diffusivity changes gradually, the displacement distribution becomes non-Gaussian, even though the mean-square displacement grows linearly with time. Here, we report single-particle tracking measurements of the diffusion of colloidal spheres near a planar substrate. Because the local effective diffusivity is known, we have been able to carry out a direct test of this mechanism for diffusion in inhomogeneous media.

Unraveling intermittent features in single-particle trajectories by a local convex hull method

We propose a model-free method to detect change points between distinct phases in a single random trajectory of an intermittent stochastic process. The local convex hull (LCH) is constructed for each trajectory point, while its geometric properties (e.g., the diameter or the volume) are used as discriminators between phases. The efficiency of the LCH method is validated for six models of intermittent motion, including Brownian motion with different diffusivities or drifts, fractional Brownian motion with different Hurst exponents, and surface-mediated diffusion. We discuss potential applications of the method for detection of active and passive phases in the intracellular transport, temporal trapping or binding of diffusing molecules, alternating bulk and surface diffusion, run and tumble (or search) phases in the motion of bacteria and foraging animals, and instantaneous firing rates in neurons.

Resolving mixed mechanisms of protein subdiffusion at the T cell plasma membrane

The plasma membrane is a complex medium where transmembrane proteins diffuse and interact to facilitate cell function. Membrane protein mobility is affected by multiple mechanisms, including crowding, trapping, medium elasticity and structure, thus limiting our ability to distinguish them in intact cells. Here we characterize the mobility and organization of a short transmembrane protein at the plasma membrane of live T cells, using single particle tracking and photoactivated-localization microscopy. Protein mobility is highly heterogeneous, subdiffusive and ergodic-like. Using mobility characteristics, we segment individual trajectories into subpopulations with distinct Gaussian step-size distributions. Particles of low-to-medium mobility consist of clusters, diffusing in a viscoelastic and fractal-like medium and are enriched at the centre of the cell footprint. Particles of high mobility undergo weak confinement and are more evenly distributed. This study presents a methodological approach to resolve simultaneous mixed subdiffusion mechanisms acting on polydispersed samples and complex media such as cell membranes.

Membrane protein diffusion is affected by distinct mechanisms such as molecular crowding and medium elasticity. Here the authors present an analytical approach to analyse single particle trajectories and distinguish mixed subdiffusive processes affecting membrane protein mobility in living cells.

Diffusing diffusivity: Rotational diffusion in two and three dimensions

We consider the problem of calculating the probability distribution function (pdf) of angular displacement for rotational diffusion in a crowded, rearranging medium. We use the diffusing diffusivity model and following our previous work on translational diffusion [R. Jain and K. L. Sebastian, J. Phys. Chem. B 120, 3988 (2016)], we show that the problem can be reduced to that of calculating the survival probability of a particle undergoing Brownian motion, in the presence of a sink. We use the approach to calculate the pdf for the rotational motion in two and three dimensions. We also propose new dimensionless, time dependent parameters, αrot,2D and αrot,3D, which can be used to analyze the experimental/simulation data to find the extent of deviation from the normal behavior, i.e., constant diffusivity, and obtain explicit analytical expressions for them, within our model.

Structure-diffusion relationship of polymer membranes with different texture

Two-dimensional diffusion in heterogenic composite membranes, i.e., materials comprising polymer with dispersed inorganic fillers, composed of ethylcellulose and magnetic powder is studied. In the experimental part, the morphology of membranes is described by the following characteristics: the amount of polymer matrix, the fractal dimension of polymer matrix, the average size of polymer matrix domains, the average number of obstacles in the proximity of each polymer matrix pixel. The simulation work concentrates on the motion of a particle in the membrane environment. The focus is set on the relationship between membranes morphology characterized by polymer matrix density, its fractal dimension, the average size of domains, and the average number of near obstacles and the characteristics of diffusive transport in them. The comparison of diffusion driven by Gaussian random walk and Lévy flights shows that the effective diffusion exponent at long time limits is subdiffusive and it does not depend on the details of the underlying random process causing diffusion. The analysis of the parameters describing the membrane structure shows that the most important factor for the diffusion character is the average size of a domain penetrated by diffusing particles. The presented results may be used in the design and preparation of membrane structures with specific diffusion properties.

Classification and Segmentation of Nanoparticle Diffusion Trajectories in Cellular Micro Environments

Darkfield and confocal laser scanning microscopy both allow for a simultaneous observation of live cells and single nanoparticles. Accordingly, a characterization of nanoparticle uptake and intracellular mobility appears possible within living cells. Single particle tracking allows to measure the size of a diffusing particle close to a cell. However, within the more complex system of a cell’s cytoplasm normal, confined or anomalous diffusion together with directed motion may occur. In this work we present a method to automatically classify and segment single trajectories into their respective motion types. Single trajectories were found to contain more than one motion type. We have trained a random forest with 9 different features. The average error over all motion types for synthetic trajectories was 7.2%. The software was successfully applied to trajectories of positive controls for normal- and constrained diffusion. Trajectories captured by nanoparticle tracking analysis served as positive control for normal diffusion. Nanoparticles inserted into a diblock copolymer membrane was used to generate constrained diffusion. Finally we segmented trajectories of diffusing (nano-)particles in V79 cells captured with both darkfield- and confocal laser scanning microscopy. The software called “TraJClassifier” is freely available as ImageJ/Fiji plugin via https://git.io/v6uz2.

Tracer diffusion in a sea of polymers with binding zones: mobile vs. frozen traps

We use molecular dynamics simulations to investigate the tracer diffusion in a sea of polymers with specific binding zones for the tracer. These binding zones act as traps. Our simulations show that the tracer can undergo normal yet non-Gaussian diffusion under certain circumstances, e.g., when the polymers with traps are frozen in space and the volume fraction and the binding strength of the traps are moderate. In this case, as the tracer moves, it experiences a heterogeneous environment and exhibits confined continuous time random walk (CTRW) like motion resulting in a non-Gaussian behavior. Also the long time dynamics becomes subdiffusive as the number or the binding strength of the traps increases. However, if the polymers are mobile then the tracer dynamics is Gaussian but could be normal or subdiffusive depending on the number and the binding strength of the traps. In addition, with increasing binding strength and number of polymer traps, the probability of the tracer being trapped increases. On the other hand, removing the binding zones does not result in trapping, even at comparatively high crowding. Our simulations also show that the trapping probability increases with the increasing size of the tracer and for a bigger tracer with the frozen polymer background the dynamics is only weakly non-Gaussian but highly subdiffusive. Our observations are in the same spirit as found in many recent experiments on tracer diffusion in polymeric materials and question the validity of using Gaussian theory to describe diffusion in a crowded environment in general.

Controlled Synthesis and Fluorescence Tracking of Highly Uniform Poly(N-isopropylacrylamide) Microgels

Stimuli-sensitive poly(N-isopropylacrylamide) (PNIPAM) microgels have various prospective practical applications and uses in fundamental research. In this work, we use single particle tracking of fluorescently labeled PNIPAM microgels as a showcase for tuning microgel size by a rapid non-stirred precipitation polymerization procedure. This approach is well suited for prototyping new reaction compositions and conditions or for applications that do not require large amounts of product. Microgel synthesis, particle size and structure determination by dynamic and static light scattering are detailed in the protocol. It is shown that the addition of functional comonomers can have a large influence on the particle nucleation and structure. Single particle tracking by wide-field fluorescence microscopy allows for an investigation of the diffusion of labeled tracer microgels in a concentrated matrix of non-labeled microgels, a system not easily investigated by other methods such as dynamic light scattering.

Langevin equation with fluctuating diffusivity: A two-state model

Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though the origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. Namely, the diffusivity follows a dichotomous stochastic process. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a nonequilibrium ensemble, the ensemble-averaged mean-square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time and converges to a Gaussian distribution in a long-time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterizing the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool.

Time- and ensemble-averages in evolving systems: the case of Brownian particles in random potentials

Anomalous diffusion is a ubiquitous phenomenon in complex systems. It is often quantified using time- and ensemble-averages to improve statistics, although time averages represent a non-local measure in time and hence can be difficult to interpret. We present a detailed analysis of the influence of time- and ensemble-averages on dynamical quantities by investigating Brownian particles in a rough potential energy landscape (PEL). Initially, the particle ensemble is randomly distributed, but the occupancy of energy values evolves towards the equilibrium distribution. This relaxation manifests itself in the time evolution of time- and ensemble-averaged dynamical measures. We use Monte Carlo simulations to study particle dynamics in a potential with a Gaussian distribution of energy values, where the long-time limit of the diffusion coefficient is known from theory. In our experiments, individual colloidal particles are exposed to a laser speckle pattern inducing a non-Gaussian roughness and are followed by optical microscopy. The relaxation depends on the kind and degree of roughness of the PEL. It can be followed and quantified by the time- and ensemble-averaged mean squared displacement. Moreover, the heterogeneity of the dynamics is characterized using single-trajectory analysis. The results of this work are relevant for the correct interpretation of single-particle tracking experiments in general.

Ergodicity testing for anomalous diffusion: small sample statistics

The analysis of trajectories recorded in experiments often requires calculating time averages instead of ensemble averages. According to the Boltzmann hypothesis, they are equivalent only under the assumption of ergodicity. In this paper, we implement tools that allow to study ergodic properties. This analysis is conducted in two classes of anomalous diffusion processes: fractional Brownian motion and subordinated Ornstein-Uhlenbeck process. We show that only first of them is ergodic. We demonstrate this by applying rigorous statistical methods: mean square displacement, confidence intervals, and dynamical functional test. Our methodology is universal and can be implemented for analysis of many experimental data not only if a large sample is available but also when there are only few trajectories recorded.

Probing Non-Gaussianity in Confined Diffusion of Nanoparticles

Confined diffusion is ubiquitous in nature. Ever since the "anomalous yet Brownian" motion was observed, the non-Gaussianity in confined diffusion has been unveiled as an important issue. In this Letter, we experimentally investigate the characteristics and source of non-Gaussian behavior in confined diffusion of nanoparticles suspended in polymer solutions. A time-varied and size-dependent non-Gaussianity is reported based on the non-Gaussian parameter and displacement probability distribution, especially when the nanoparticle's size is smaller than the typical polymer mesh size. This non-Gaussianity does not vanish even at the long-time Brownian stage. By inspecting the displacement autocorrelation, we observe that the nanoparticle-structure interaction, indicated by the anticorrelation, is limited in the short-time stage and makes little contribution to the non-Gaussianity in the long-time stage. The main source of the non-Gaussianity can therefore be attributed to hopping diffusion that results in an exponential probability distribution with the large displacements, which may also explain certain processes dominated by rare events in the biological environment.

Non-Brownian diffusion in lipid membranes: Experiments and simulations

The dynamics of constituents and the surface response of cellular membranes-also in connection to the binding of various particles and macromolecules to the membrane-are still a matter of controversy in the membrane biophysics community, particularly with respect to crowded membranes of living biological cells. We here put into perspective recent single particle tracking experiments in the plasma membranes of living cells and supercomputing studies of lipid bilayer model membranes with and without protein crowding. Special emphasis is put on the observation of anomalous, non-Brownian diffusion of both lipid molecules and proteins embedded in the lipid bilayer. While single component, pure lipid bilayers in simulations exhibit only transient anomalous diffusion of lipid molecules on nanosecond time scales, the persistence of anomalous diffusion becomes significantly longer ranged on the addition of disorder-through the addition of cholesterol or proteins-and on passing of the membrane lipids to the gel phase. Concurrently, experiments demonstrate the anomalous diffusion of membrane embedded proteins up to macroscopic time scales in the minute time range. Particular emphasis will be put on the physical character of the anomalous diffusion, in particular, the occurrence of ageing observed in the experiments-the effective diffusivity of the measured particles is a decreasing function of time. Moreover, we present results for the time dependent local scaling exponent of the mean squared displacement of the monitored particles. Recent results finding deviations from the commonly assumed Gaussian diffusion patterns in protein crowded membranes are reported. The properties of the displacement autocorrelation function of the lipid molecules are discussed in the light of their appropriate physical anomalous diffusion models, both for non-crowded and crowded membranes. In the last part of this review we address the upcoming field of membrane distortion by elongated membrane-binding particles. We discuss how membrane compartmentalisation and the particle-membrane binding energy may impact the dynamics and response of lipid membranes. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.

Subdiffusive motion of bacteriophage in mucosal surfaces increases the frequency of bacterial encounters

Bacteriophages (phages) defend mucosal surfaces against bacterial infections. However, their complex interactions with their bacterial hosts and with the mucus-covered epithelium remain mostly unexplored. Our previous work demonstrated that T4 phage with Hoc proteins exposed on their capsid adhered to mucin glycoproteins and protected mucus-producing tissue culture cells in vitro. On this basis, we proposed our bacteriophage adherence to mucus (BAM) model of immunity. Here, to test this model, we developed a microfluidic device (chip) that emulates a mucosal surface experiencing constant fluid flow and mucin secretion dynamics. Using mucus-producing human cells and Escherichia coli in the chip, we observed similar accumulation and persistence of mucus-adherent T4 phage and nonadherent T4∆hoc phage in the mucus. Nevertheless, T4 phage reduced bacterial colonization of the epithelium >4,000-fold compared with T4∆hoc phage. This suggests that phage adherence to mucus increases encounters with bacterial hosts by some other mechanism. Phages are traditionally thought to be completely dependent on normal diffusion, driven by random Brownian motion, for host contact. We demonstrated that T4 phage particles displayed subdiffusive motion in mucus, whereas T4∆hoc particles displayed normal diffusion. Experiments and modeling indicate that subdiffusive motion increases phage-host encounters when bacterial concentration is low. By concentrating phages in an optimal mucus zone, subdiffusion increases their host encounters and antimicrobial action. Our revised BAM model proposes that the fundamental mechanism of mucosal immunity is subdiffusion resulting from adherence to mucus. These findings suggest intriguing possibilities for engineering phages to manipulate and personalize the mucosal microbiome.

Fluctuation analysis of time-averaged mean-square displacement for the Langevin equation with time-dependent and fluctuating diffusivity

The mean-square displacement (MSD) is widely utilized to study the dynamical properties of stochastic processes. The time-averaged MSD (TAMSD) provides some information on the dynamics which cannot be extracted from the ensemble-averaged MSD. In particular, the relative standard deviation (RSD) of the TAMSD can be utilized to study the long-time relaxation behavior. In this work, we consider a class of Langevin equations which are multiplicatively coupled to time-dependent and fluctuating diffusivities. Various interesting dynamics models such as entangled polymers and supercooled liquids can be interpreted as the Langevin equations with time-dependent and fluctuating diffusivities. We derive a general formula for the RSD of the TAMSD for the Langevin equation with the time-dependent and fluctuating diffusivity. We show that the RSD can be expressed in terms of the correlation function of the diffusivity. The RSD exhibits the crossover at the long time region. The crossover time is related to a weighted average relaxation time for the diffusivity. Thus the crossover time gives some information on the relaxation time of fluctuating diffusivity which cannot be extracted from the ensemble-averaged MSD. We discuss the universality and possible applications of the formula via some simple examples.

Chromosomal locus tracking with proper accounting of static and dynamic errors

The mean-squared displacement (MSD) and velocity autocorrelation (VAC) of tracked single particles or molecules are ubiquitous metrics for extracting parameters that describe the object's motion, but they are both corrupted by experimental errors that hinder the quantitative extraction of underlying parameters. For the simple case of pure Brownian motion, the effects of localization error due to photon statistics ("static error") and motion blur due to finite exposure time ("dynamic error") on the MSD and VAC are already routinely treated. However, particles moving through complex environments such as cells, nuclei, or polymers often exhibit anomalous diffusion, for which the effects of these errors are less often sufficiently treated. We present data from tracked chromosomal loci in yeast that demonstrate the necessity of properly accounting for both static and dynamic error in the context of an anomalous diffusion that is consistent with a fractional Brownian motion (FBM). We compare these data to analytical forms of the expected values of the MSD and VAC for a general FBM in the presence of these errors.

Unusual diffusing regimes caused by different adsorbing surfaces

A confined liquid with dispersed neutral particles is theoretically studied when the limiting surfaces present different dynamics for the adsorption-desorption phenomena. The investigation considers different non-singular kernels in the kinetic equations at the walls, where the suitable choice of the kernel can account for the relative importance of physisorption or chemisorption. We find that even a small difference in the adsorption-desorption rate of one surface (relative to the other) can drastically affect the behavior of the whole system. The surface and bulk densities and the dispersion are calculated when several scenarios are considered and anomalous-like behaviors are found. The approach described here is closely related to experimental situations, and can be applied in several contexts such as dielectric relaxation, diffusion-controlled relaxation in liquids, liquid crystals, and amorphous polymers.

The viscoelastic properties of chromatin and the nucleoplasm revealed by scale-dependent protein mobility

The eukaryotic cell nucleus harbours the DNA genome that is organized in a dynamic chromatin network and embedded in a viscous crowded fluid. This environment directly affects enzymatic reactions and target search processes that access the DNA sequence information. However, its physical properties as a reaction medium are poorly understood. Here, we exploit mobility measurements of differently sized inert green fluorescent tracer proteins to characterize the viscoelastic properties of the nuclear interior of a living human cell. We find that it resembles a viscous fluid on small and large scales but appears viscoelastic on intermediate scales that change with protein size. Our results are consistent with simulations of diffusion through polymers and suggest that chromatin forms a random obstacle network rather than a self-similar structure with fixed fractal dimensions. By calculating how long molecules remember their previous position in dependence on their size, we evaluate how the nuclear environment affects search processes of chromatin targets.

Macromolecular crowding impacts on the diffusion and conformation of DNA hairpins

Biochemical reactions in crowded fluids differ significantly from those in dilute solutions. Both, excluded-volume interactions with surrounding macromolecules ("crowders") and an enhanced rebinding of reaction partners due to crowding-induced viscoelasticity and subdiffusion have been hypothesized to shift chemical equilibria towards the associated state. We have explored the impact of both cues in an experimentally tunable system by monitoring the steady-state fraction of open DNA hairpins in crowded fluids with varying viscoelastic characteristics but similar occupied volume fractions. As a result, we observed an increased fraction of closed DNA hairpins in viscoelastic crowded fluids. Our observations compare favorably to a simple statistical model that considers both facets of crowding, while preferential interactions between crowders and DNA hairpins appear to have little influence.

Weak ergodicity breaking and ageing in anomalous diffusion

Modern single particle tracking techniques and many large scale simulations produce time series r(t) of the position of a tracer particle. Standardly these are evaluated in terms of the time averaged mean squared displacement. For ergodic processes such as Brownian motion, one can interpret the results of such an analysis in terms of the known theories for the corresponding ensemble averaged mean squared displacement, if only the measurement time is sufficiently long. In anomalous diffusion processes, that are widely observed over many orders of magnitude, the equivalence between (long) time and ensemble averages may be broken (weak ergodicity breaking). In such cases the time averages may no longer be interpreted in terms of ensemble theories. Here we collect some recent results on weakly non-ergodic systems with respect to the time averaged mean squared displacement and the inherent irreproducibility of individual measurements. We also address the phenomenon of ageing, the dependence of physical observables on the time span between initial preparation of the system and the start of the measurement.

Non-universal tracer diffusion in crowded media of non-inert obstacles

For tracer motion in an array of attractive obstacles we observe transient, non-ergodic anomalous diffusion depending on the obstacle density.

Anomalous diffusion in thermoresponsive polymer-clay composite hydrogels probed by wide-field fluorescence microscopy

Thermoresponsive materials exhibit an enormous potential for tissue engineering, separation systems, and drug delivery. We investigated the diffusion of laponite clay nanoparticles, which serve as physical cross-linkers to achieve improved material properties in poly(N-isopropylacrylamide) (PNIPAM)-clay composite hydrogels close to the gel point. The networks are formed through physical interactions between PNIPAM chains and clay nanoparticles after these two components are mixed. In contrast to previous studies, a covalent labeling strategy was chosen to minimize the amount of free dyes in solution. Single-particle tracking of the labeled clay nanoparticles showed that their diffusion is anomalous at all temperatures used in this study, reflecting the viscoelastic behavior as a cross-linker. Stepwise heating from 24 to 38 °C resulted in a slight increase of the diffusion coefficient and the anomality parameter α up to the volume phase transition temperature of ca. 31 °C, which was followed by a significant drop of both parameters, reflecting strongly hindered motion of the collapsed nanoparticle aggregates.

Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes

We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence D(x) ∼ |x|(α) of the generalized diffusion coefficient encompassing sub- and superdiffusive anomalous diffusion. Based on statistical measures such as the amplitude scatter of the time-averaged mean-squared displacement of individual realizations, the ergodicity breaking and non-Gaussianity parameters, as well as the probability density function P(x,t), we analyze the weakly nonergodic character of the heterogeneous diffusion process and, particularly, the degree of irreproducibility of individual realizations. As we show, the fluctuations between individual realizations increase with growing modulus |α| of the scaling exponent. The fluctuations appear to diverge when the critical value α = 2 is approached, while for even larger α the fluctuations decrease, again. At criticality, the power-law behavior of the mean-squared displacement changes to an exponentially fast growth, and the fluctuations of the time-averaged mean-squared displacement do not converge for increasing number of realizations. From a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power-law waiting time distribution and diverging characteristic waiting time.