Mapping of spatiotemporal heterogeneous particle dynamics in living cells

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.

Universal singularities of anomalous diffusion in the Richardson class

Inhomogeneous environments are rather ubiquitous in nature, often implying anomalies resulting in deviation from Gaussianity of diffusion processes. While sub- and superdiffusion are usually due to contrasting environmental features (hindering or favoring the motion, respectively), they are both observed in systems ranging from the micro- to the cosmological scale. Here we show how a model encompassing sub- and superdiffusion in an inhomogeneous environment exhibits a critical singularity in the normalized generator of the cumulants. The singularity originates directly and exclusively from the asymptotics of the non-Gaussian scaling function of displacement, and the independence from other details confers it a universal character. Our analysis, based on the method first applied by Stella et al. [Phys. Rev. Lett. 130, 207104 (2023)10.1103/PhysRevLett.130.207104], shows that the relation connecting the scaling function asymptotics to the diffusion exponent characteristic of processes in the Richardson class implies a nonstandard extensivity in time of the cumulant generator. Numerical tests fully confirm the results.

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.

Getting around the cell: physical transport in the intracellular world

Eukaryotic cells face the challenging task of transporting a variety of particles through the complex intracellular milieu in order to deliver, distribute, and mix the many components that support cell function. In this review, we explore the biological objectives and physical mechanisms of intracellular transport. Our focus is on cytoplasmic and intra-organelle transport at the whole-cell scale. We outline several key biological functions that depend on physically transporting components across the cell, including the delivery of secreted proteins, support of cell growth and repair, propagation of intracellular signals, establishment of organelle contacts, and spatial organization of metabolic gradients. We then review the three primary physical modes of transport in eukaryotic cells: diffusive motion, motor-driven transport, and advection by cytoplasmic flow. For each mechanism, we identify the main factors that determine speed and directionality. We also highlight the efficiency of each transport mode in fulfilling various key objectives of transport, such as particle mixing, directed delivery, and rapid target search. Taken together, the interplay of diffusion, molecular motors, and flows supports the intracellular transport needs that underlie a broad variety of biological phenomena.

Spherical probes for simultaneous measurement of rotational and translational diffusion in 3 dimensions

Real time visualization and tracking of colloidal particles with 3D resolution is essential for probing the local structure and dynamics in complex fluids. Although tracking translational motion of spherical particles is well-known, accessing rotational dynamics of such particles remains a great challenge. Here, we report a novel approach of using fluorescently labeled raspberry-like colloids with an optical anisotropy to concurrently track translational and rotational dynamics in 3 dimensions. The raspberry-like particles are coated by a silica layer of adjustable thickness, which allows tuning the surface roughness. The synthesis and applicability of the proposed method is demonstrated by two types of probes: rough and smoothened. The accuracies of measuring Mean Squared (Angular) Displacements are also demonstrated by using these 2 probes dispersed in 2 different solvents. The presented 3D trackable colloids offer a high potential for wide range of applications and studies, such as probing the dynamics of crystallization, phase transitions, biological interactions and the effect of surface roughness on diffusion.

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?

Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement

The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.

Viscoelasticity Measurements Reveal Rheological Differences Between Stem-like and Non-stem-like Breast Cancer Cells

Defining the characteristics of cancer stem cells (CSC) has become an important subject in cancer research during the past decade. Although molecular surface expression levels have been used for CSC recognition, the clinical and prognostic impacts of these markers have remained a controversial issue. The finding that cancerous cells are considerably more deformable than normal ones provides the motivation for the hypothesis that the mechanical properties can be used as biomarkers to distinguish between stem-like and non-stem-like cancer cells. In this study, using micropipette aspiration (MA) and intracellular particle tracking (IPT) microrheology, measurements of the whole-cell and local viscoelasticity were made on four breast cancer cell lines with different CSC phenotypes based on their surface markers. Stem-like Hs578T and MDA-MB-231 cell lines were found to be the most deformable, while the non-stem-like MDA-MB-468 line was the least deformable. The non-stem-like BT-20 cell line showed an intermediate deformability. The enhanced deformability for stem-like cells was consistent with the observed lower and more dispersed F-actin content for the stem-like cells. Therefore, the cytoskeleton-related differences in the rheological properties of cancer cells can be a potential biomarker for CSC and eventually lead to novel cancer diagnostic and therapeutic methods.

Cytoplasmic RNA-Protein Particles Exhibit Non-Gaussian Subdiffusive Behavior

The cellular cytoplasm is a complex, heterogeneous environment (both spatially and temporally) that exhibits viscoelastic behavior. To further develop our quantitative insight into cellular transport, we analyze data sets of mRNA molecules fluorescently labeled with MS2-GFP tracked in real time in live Escherichia coli and Saccharomyces cerevisiae cells. As shown previously, these RNA-protein particles exhibit subdiffusive behavior that is viscoelastic in its origin. Examining the ensemble of particle displacements reveals a Laplace distribution at all observed timescales rather than the Gaussian distribution predicted by the central limit theorem. This ensemble non-Gaussian behavior is caused by a combination of an exponential distribution in the time-averaged diffusivities and non-Gaussian behavior of individual trajectories. We show that the non-Gaussian behavior is a consequence of significant heterogeneity between trajectories and dynamic heterogeneity along single trajectories. Informed by theory and simulation, our work provides an in-depth analysis of the complex diffusive behavior of RNA-protein particles in live cells.

Systems-level approach to uncovering diffusive states and their transitions from single-particle trajectories

The stochastic motions of a diffusing particle contain information concerning the particle's interactions with binding partners and with its local environment. However, an accurate determination of the underlying diffusive properties, beyond normal diffusion, has remained challenging when analyzing particle trajectories on an individual basis. Here, we introduce the maximum-likelihood estimator (MLE) for confined diffusion and fractional Brownian motion. We demonstrate that this MLE yields improved estimation over traditional mean-square displacement analyses. We also introduce a model selection scheme (that we call mleBIC) that classifies individual trajectories to a given diffusion mode. We demonstrate the statistical limitations of classification via mleBIC using simulated data. To overcome these limitations, we introduce a version of perturbation expectation-maximization (pEMv2), which simultaneously analyzes a collection of particle trajectories to uncover the system of interactions that give rise to unique normal and/or non-normal diffusive states within the population. We test and evaluate the performance of pEMv2 on various sets of simulated particle trajectories, which transition among several modes of normal and non-normal diffusion, highlighting the key considerations for employing this analysis methodology.

Diffusing colloidal probes of cell surfaces

Measurements and analyses are reported to quantify dynamic and equilibrium interactions between colloidal particles and live cell surfaces using dark field video microscopy. Two-dimensional trajectories of micron-sized polyethylene glycol (PEG)-coated silica colloids relative to adherent epithelial breast cancer cell perimeters are determined allowing measurement of position dependent diffusivities and interaction potentials. PEG was chosen as the material system of interest to assess non-specific interactions with cell surfaces and establishes a basis for investigation of specific interactions in future studies. Analysis of measured potential energies on cell surfaces reveals the spatial dependence in cell topography. With the measured cell topography and models for particle-cell surface hydrodynamic interactions, excellent agreement is obtained between theoretical and measured colloidal transport on cell surfaces. Quantitative analyses of association lifetimes showed that PEG coatings act to stabilize colloids above the cell surface through net repulsive, steric interactions. Our results demonstrate a self-consistent analysis of diffusing colloidal probe interactions due to conservative and non-conservative forces to characterize biophysical cell surface properties.

Revealing nonergodic dynamics in living cells from a single particle trajectory

We propose the improved ergodicity and mixing estimators to identify nonergodic dynamics from a single particle trajectory. The estimators are based on the time-averaged characteristic function of the increments and can thus capture additional information on the process as compared to the conventional time-averaged mean-square displacement. The estimators are first investigated and validated for several models of anomalous diffusion, such as ergodic fractional Brownian motion and diffusion on percolating clusters, and nonergodic continuous-time random walks and scaled Brownian motion. The estimators are then applied to two sets of earlier published trajectories of mRNA molecules inside live Escherichia coli cells and of Kv2.1 potassium channels in the plasma membrane. These statistical tests did not reveal nonergodic features in the former set, while some trajectories of the latter set could be classified as nonergodic. Time averages along such trajectories are thus not representative and may be strongly misleading. Since the estimators do not rely on ensemble averages, the nonergodic features can be revealed separately for each trajectory, providing a more flexible and reliable analysis of single-particle tracking experiments in microbiology.

Minimizing biases associated with tracking analysis of submicron particles in heterogeneous biological fluids

Tracking the dynamic motion of individual nanoparticles or viruses offers quantitative insights into their real-time behavior and fate in different biological environments. Indeed, particle tracking is a powerful tool that has facilitated the development of drug carriers with enhanced penetration of mucus, brain tissues and other extracellular matrices. Nevertheless, heterogeneity is a hallmark of nanoparticle diffusion in such complex environments: identical particles can exhibit strongly hindered or unobstructed diffusion within microns of each other. The common practice in 2D particle tracking, namely analyzing all trackable particle traces with equal weighting, naturally biases towards rapidly diffusing sub-populations at shorter time scales. This in turn results in misrepresentation of particle behavior and a systematic underestimate of the time necessary for a population of nanoparticles to diffuse specific distances. We show here via both computational simulation and experimental data that this bias can be rigorously corrected by weighing the contribution by each particle trace on a 'frame-by-frame' basis. We believe this methodology presents an important step towards objective and accurate assessment of the heterogeneous transport behavior of submicron drug carriers and pathogens in biological environments.

Micro-heterogeneity metrics for diffusion in soft matter

Passive particle tracking of diffusive paths in soft matter, coupled with analysis of the path data, is firmly established as a fundamental methodology for characterization of both diffusive transport properties (the focus here) and linear viscoelasticity. For either focus, particle time series are typically analyzed by ensemble averaging over paths, a perfectly natural protocol for homogeneous materials or for applications where mean properties are sufficient. Many biological materials, however, are heterogeneous over length scales above the probe diameter, and the implications of heterogeneity for biologically relevant transport properties (e.g. diffusive passage times through a complex fluid layer) motivate this paper. Our goals are three-fold: first, to detect heterogeneity as reflected by the ensemble path data; second, to further decompose the ensemble of particle paths into statistically distinct clusters; and third, to fit the path data in each cluster to a model for the underlying stochastic process. After reviewing current best practices for detection and assessment of heterogeneity in diffusive processes, we introduce our strategy toward the first two goals with methods from the statistics and machine learning literature that have not found application thus far to passive particle tracking data. We apply an analysis based solely on the path data that detects heterogeneity and yields a decomposition of particle paths into statistically distinct clusters. After these two goals are achieved, one can then pursue model-fitting. We illustrate these heterogeneity metrics on diverse datasets: for numerically generated and experimental particle paths, with tunable and unknown heterogeneity, on numerical models for simple diffusion and anomalous sub-diffusion, and experimentally on sucrose, hyaluronic acid, agarose, and human lung culture mucus solutions.

Investigating longitudinal changes in the mechanical properties of MCF-7 cells exposed to paclitaxol using particle tracking microrheology

Evidence suggests that compression and shear wave elastography are sensitive to the mechanical property changes occuring in dying cells following chemotherapy, and can hence be used to monitor cancer treatment response. A qualitative and quantitative understanding of the mechanical changes at the cellular level would allow to better infer how these changes affect macroscopic tissue mechanical properties and therefore allow the optimization of elastographic techniques (such as shear wave elastography) for the monitoring of cancer therapy. We used intracellular particle tracking microrheology (PTM) to investigate the mechanical property changes of cells exposed to paclitaxol, a mitotic inhibitor used in cancer chemotherapy. The average elastic and viscous moduli of the cytoplasm of treated MCF-7 breast cancer cells were calculated for frequency ranges between 0.2 and 100 rad s(-1) (corresponding to 0.03 and 15.92 Hz, respectively). A significant increase in the complex shear modulus of the cell cytoplasm was detected at 12 h post treatment. At 24 h after drug exposure, the elastic and viscous moduli increased by a total of 191.3 Pa (>8000×) and 9 Pa (∼9×), respectively for low frequency shear modulus measurements (at 1 rad s(-1)). At higher frequencies (10 rad s(-1)), the elastic and viscous moduli increased by 188.5 Pa (∼60×) and 1.7 Pa (∼1.1×), respectively. Our work demonstrates that PTM can be used to measure changes in the mechanical properties of treated cells and that cell elasticity significantly increases by 24 h after chemotherapy exposure.

Random walk in chemical space of Cantor dust as a paradigm of superdiffusion

We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.

"Targeting" nanoparticles: the constraints of physical laws and physical barriers

In comparison to the complexities of the body, its organs, its normal and aberrant cells, many nanoparticles will appear to be relatively simple objects. This view is deceptive because the physicochemical properties of nanosystems, although quite well understood on the basis of material science, surface science and colloid theory, are far from simple in practice. While their properties are largely controllable in vitro, often purportedly "designed", their administration by any route changing environments conspires to produce additional layers of complexity. Some of the key physical laws and physicochemical parameters governing the fate of nanoparticles on their journey from point of intravenous administration to desired destinations such as tumors are discussed. Much of the science relevant to nanocarrier based targeting has been elaborated in studying purely physical phenomena, but there can be found therein many analogies with biological systems. These include factors that impede quantitative targeting: diffusion in complex media, aggregation and flocculation, hindered behavior of particles in confined spaces, jamming and dispersion in flow. All of these have the ability to influence fate and destination. Most of the critical processes are particle size dependent but not always linearly so. Virtually all processes in vivo involve an element of probability. Particle size and properties can be controlled to a large extent, but stochastic processes cannot by definition. Progress has been made, but the quantitative delivery of a nanocarrier to defined sites in tumors is neither inevitable nor yet predictable.

Magnetic nanomanipulations inside living cells compared with passive tracking of nanoprobes to get consensus for intracellular mechanics

During the last decade, the development of nanomaterials to penetrate inside living cells has been the focus of a large number of studies, with applications for the biomedical field. However, the further dynamics of these nanomaterials inside the cells is dictated by the intracellular environment and in particular its mechanical properties. The mechanical characteristics of the cell interior can be probed with either active or passive microrheological approaches. However, active intracellular microrheology is still in its infancy, owing to the difficulty of inserting probes that can be manipulated by external forces. Here we review recent active microrheology studies using magnetic nanoprobes inserted into endosomes or phagosomes as useful approaches for measuring frequency-dependent viscoelasticity, for mapping the viscoelastic landscape, as well as for identifying the contribution of individual cytoskeleton components and the influence of cell motility. The results of such direct measurements challenge the validity of more typical passive approaches in which the spontaneous displacement of embedded nanoprobes is measured. Here we discuss that one must distinguish probes suitable for use in conditions of thermal equilibrium, whose movements reflect the mechanical environment from probes that interact actively with the cytoplasm and cytoskeleton, in a state of nonequilibrium for which fluctuation-dissipation theorem no longer holds. However, when data on these probes' viscoelastic microenvironment is available, such passive probe movements can yield useful information on the forces responsible for intracellular activity.

Colloidal dynamics near a particle-covered surface

How the diffusive dynamics of colloidal spheres changes in the vicinity of a particle-coated surface is of importance for industrial challenges such as fouling and sedimentation as well as for fundamental studies into confinement effects. We addressed this question by studying colloidal dynamics in a partially coated surface layer, using video microscopy. Particle mean squared displacement (MSD) functions were measured as a function of a (local) effective volume fraction (EVF), which was varied by making use of gravity settling. Comparison of MSDs at the bare and coated surfaces for EVF of 0.2-0.4 revealed that at the latter surface the motion amplitudes are strongly reduced, accompanied by a sharp transition from diffusive to nearly caged motion. This clearly indicates that the surface-attached particles cannot be taken into account via volume fraction and that their immobility has a distinct effect. For EVF > 0.45, the caging becomes dominated by the suspended particles, making the dynamics at the bare and coated surfaces similar.

Probability distribution of the time-averaged mean-square displacement of a Gaussian process

We study the probability distribution of the time-averaged mean-square displacement of a discrete Gaussian process. An empirical approximation for the probability density is suggested and numerically validated for fractional Brownian motion. The optimality of quadratic forms for inferring dynamical and microrheological quantities from individual random trajectories is discussed, with emphasis on a reliable interpretation of single-particle tracking experiments.

Nonlocal fluctuation correlations in active gels

Many active materials and biological systems are driven far from equilibrium by embedded agents that spontaneously generate forces and distort the surrounding material. Probing and characterizing these athermal fluctuations are essential to understand the properties and behaviors of such systems. Here we present a mathematical procedure to estimate the local action of force-generating agents from the observed fluctuating displacement fields. The active agents are modeled as oriented force dipoles or isotropic compression foci, and the matrix on which they act is assumed to be either a compressible elastic continuum or a coupled network-solvent system. Correlations at a single point and between points separated by an arbitrary distance are obtained, giving a total of three independent fluctuation modes that can be tested with microrheology experiments. Since oriented dipoles and isotropic compression foci give different contributions to these fluctuation modes, ratiometric analysis allows us characterize the force generators. We also predict and experimentally find a high-frequency ballistic regime, arising from individual force-generating events in the form of the slow buildup of stress followed by rapid but finite decay. Finally, we provide a quantitative statistical model to estimate the mean filament tension from these athermal fluctuations, which leads to stiffening of active networks.

Influence of confinement by smooth and rough walls on particle dynamics in dense hard-sphere suspensions

We used video microscopy and particle tracking to study the dynamics of confined hard-sphere suspensions. Our fluids consisted of 1.1-microm-diameter silica spheres suspended at volume fractions of 0.33-0.42 in water-dimethyl sulfoxide. Suspensions were confined in a quasiparallel geometry between two glass surfaces: a millimeter-sized rough sphere and a smooth flat wall. First, as the separation distance (H) is decreased from 18 to 1 particle diameter, a transition takes place from a subdiffusive behavior (as in bulk) at large H, to completely caged particle dynamics at small H. These changes are accompanied by a strong decrease in the amplitude of the mean-square displacement (MSD) in the horizontal plane parallel to the confining surfaces. In contrast, the global volume fraction essentially remains constant when H is decreased. Second, measuring the MSD as a function of distance from the confining walls, we found that the MSD is not spatially uniform but smaller close to the walls. This effect is the strongest near the smooth wall where layering takes place. Although confinement also induces local variations in volume fraction, the spatial variations in MSD can be attributed only partially to this effect. The changes in MSD are predominantly a direct effect of the confining surfaces. Hence, both the wall roughness and the separation distance (H) influence the dynamics in confined geometries.

Intracellular particle tracking as a tool for tumor cell characterization

We studied the dynamics of two types of intracellular probe particles, ballistically injected latex spheres and endogenous granules, in tumor cell lines of different metastatic potential: breast tumor cells (MCF-7 malignant, MCF-10A benign) and pancreas adenocarcinoma (PaTu8988T malignant, PaTu8988S benign). For both tissue types and for both probes, the mean squared displacement (MSD) function measured in the malignant cells was substantially larger than in the benign cells. Only a few cells were needed to characterize the tissue as malignant or benign based on their MSD, since variations in MSD within the same cell line were relatively small. These findings suggest that intracellular particle tracking (IPT) can serve as a simple and reliable method for characterization of cell states obtained from a small amount of cell sample. Mechanical analysis of the same cell lines with atomic force microscopy (AFM) in force-distance mode revealed that AFM could distinguish between the benign and malignant breast cancer cells but not the pancreatic tumor cell lines. This underlines the potential value of IPT as a complementary nanomechanical tool for studying cell-state-dependent mechanical properties.