Bayesian deep learning for error estimation in the analysis of anomalous diffusion

Robust Quantification of Live-Cell Single-Molecule Tracking Data for Fluorophores with Different Photophysical Properties

High-speed single-molecule tracking in live cells is becoming an increasingly popular method for quantifying the spatiotemporal behavior of proteins in vivo. The method provides a wealth of quantitative information, but users need to be aware of biases that can skew estimates of molecular mobilities. The range of suitable fluorophores for live-cell single-molecule imaging has grown substantially over the past few years, but it remains unclear to what extent differences in photophysical properties introduce biases. Here, we tested two fluorophores with entirely different photophysical properties, one that photoswitches frequently between bright and dark states (TMR) and one that shows exceptional photostability without photoswitching (JFX650). We used a fusion of the Escherichia coli DNA repair enzyme MutS to the HaloTag and optimized sample preparation and imaging conditions for both types of fluorophore. We then assessed the reliability of two common data analysis algorithms, mean-square displacement (MSD) analysis and Hidden Markov Modeling (HMM), to estimate the diffusion coefficients and fractions of MutS molecules in different states of motion. We introduce a simple approach that removes discrepancies in the data analyses and show that both algorithms yield consistent results, regardless of the fluorophore used. Nevertheless, each dye has its own strengths and weaknesses, with TMR being more suitable for sampling the diffusive behavior of many molecules, while JFX650 enables prolonged observation of only a few molecules per cell. These characterizations and recommendations should help to standardize measurements for increased reproducibility and comparability across studies.

Aging and confinement in subordinated fractional Brownian motion

We study the effects of aging properties of subordinated fractional Brownian motion (FBM) with drift and in harmonic confinement, when the measurement of the stochastic process starts a time t_{a}>0 after its original initiation at t=0. Specifically, we consider the aged versions of the ensemble mean-squared displacement (MSD) and the time-averaged MSD (TAMSD), along with the aging factor. Our results are favorably compared with simulations results. The aging subordinated FBM exhibits a disparity between MSD and TAMSD and is thus weakly nonergodic, while strong aging is shown to effect a convergence of the MSD and TAMSD. The information on the aging factor with respect to the lag time exhibits an identical form to the aging behavior of subdiffusive continuous-time random walks (CTRW). The statistical properties of the MSD and TAMSD for the confined subordinated FBM are also derived. At long times, the MSD in the harmonic potential has a stationary value, that depends on the Hurst index of the parental (nonequilibrium) FBM. The TAMSD of confined subordinated FBM does not relax to a stationary value but increases sublinearly with lag time, analogously to confined CTRW. Specifically, short aging times t_{a} in confined subordinated FBM do not affect the aged MSD, while for long aging times the aged MSD has a power-law increase and is identical to the aged TAMSD.

Diffusion in Porous Rock Is Anomalous

Molecular diffusion of chemical species in subsurface environments─rock formations, soil sediments, marine, river, and lake sediments─plays a critical role in a variety of dynamic processes, many of which affect water chemistry. We investigate and demonstrate the occurrence of anomalous (non-Fickian) diffusion behavior, distinct from classically assumed Fickian diffusion. We measured molecular diffusion through a series of five chalk and dolomite rock samples over a period of about two months. We demonstrate that in all cases, diffusion behavior is significantly different than Fickian. We then analyze the results using a continuous time random walk framework that can describe anomalous diffusion in heterogeneous porous materials such as rock. This methodology shows extreme long-time tailing of tracer advance as compared to conventional Fickian diffusion processes. The finding that distinct anomalous diffusion occurs ubiquitously implies that diffusion-driven processes in subsurface zones should be analyzed using tools that account for non-Fickian diffusion.

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.

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

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

Inferring pointwise diffusion properties of single trajectories with deep learning

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

Compartmentalization of androgen receptors at endogenous genes in living cells

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

Heterogeneous anomalous transport in cellular and molecular biology

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

Effects of crowding on the diffusivity of membrane adhered particles

The lateral diffusion of cell membrane inclusions, such as integral membrane proteins and bound receptors, drives critical biological processes, including the formation of complexes, cell-cell signaling, and membrane trafficking. These diffusive processes are complicated by how concentrated, or "crowded", the inclusions are, which can occupy between 30-50% of the area fraction of the membrane. In this work, we elucidate the effects of increasing concentration of model membrane inclusions in a free-standing artificial cell membrane on inclusion diffusivity and the apparent viscosity of the membrane. By multiple particle tracking of fluorescent microparticles covalently tethered to the bilayer, we show the transition from expected Brownian dynamics, which accurately measure the membrane viscosity, to subdiffusive behavior with decreased diffusion coefficient as the particle area fraction increases from 1% to around 30%, approaching physiological levels of crowding. At high crowding, the onset of non-Gaussian behavior is observed. Using hydrodynamic models relating the 2D diffusion coefficient to the viscosity of a membrane, we determine the apparent viscosity of the bilayer from the particle diffusivity and show an increase in the apparent membrane viscosity with increasing particle area fraction. However, the scaling of this increase is in contrast with the behavior of monolayer inclusion diffusion and bulk suspension rheology. These results demonstrate that physiological levels of model membrane crowding nontrivially alter the dynamics and apparent viscosity of the system, which has implications for understanding membrane protein interactions and particle-membrane transport processes.

Parameter estimation of the fractional Ornstein-Uhlenbeck process based on quadratic variation

Modern experiments routinely produce extensive data of the diffusive dynamics of tracer particles in a large range of systems. Often, the measured diffusion turns out to deviate from the laws of Brownian motion, i.e., it is anomalous. Considerable effort has been put in conceiving methods to extract the exact parameters underlying the diffusive dynamics. Mostly, this has been done for unconfined motion of the tracer particle. Here, we consider the case when the particle is confined by an external harmonic potential, e.g., in an optical trap. The anomalous particle dynamics is described by the fractional Ornstein-Uhlenbeck process, for which we establish new estimators for the parameters. Specifically, by calculating the empirical quadratic variation of a single trajectory, we are able to recover the subordination process governing the particle motion and use it as a basis for the parameter estimation. The statistical properties of the estimators are evaluated from simulations.

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.

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

Anomalous diffusion, non-Gaussianity, and nonergodicity for subordinated fractional Brownian motion with a drift

The stochastic motion of a particle with long-range correlated increments (the moving phase) which is intermittently interrupted by immobilizations (the trapping phase) in a disordered medium is considered in the presence of an external drift. In particular, we consider trapping events whose times follow a scale-free distribution with diverging mean trapping time. We construct this process in terms of fractional Brownian motion with constant forcing in which the trapping effect is introduced by the subordination technique, connecting "operational time" with observable "real time." We derive the statistical properties of this process such as non-Gaussianity and nonergodicity, for both ensemble and single-trajectory (time) averages. We demonstrate nice agreement with extensive simulations for the probability density function, skewness, kurtosis, as well as ensemble and time-averaged mean-squared displacements. We place a specific emphasis on the comparisons between the cases with and without drift.

Heterogeneous biological membranes regulate protein partitioning via fluctuating diffusivity

Cell membranes phase separate into ordered L o and disordered L d domains depending on their compositions. This membrane compartmentalization is heterogeneous and regulates the localization of specific proteins related to cell signaling and trafficking. However, it is unclear how the heterogeneity of the membranes affects the diffusion and localization of proteins in L o and L d domains. Here, using Langevin dynamics simulations coupled with the phase-field (LDPF) method, we investigate several tens of milliseconds-scale diffusion and localization of proteins in heterogeneous biological membrane models showing phase separation into L o and L d domains. The diffusivity of proteins exhibits temporal fluctuations depending on the field composition. Increases in molecular concentrations and domain preference of the molecule induce subdiffusive behavior due to molecular collisions by crowding and confinement effects, respectively. Moreover, we quantitatively demonstrate that the protein partitioning into the L o domain is determined by the difference in molecular diffusivity between domains, molecular preference of domain, and molecular concentration. These results pave the way for understanding how biological reactions caused by molecular partitioning may be controlled in heterogeneous media. Moreover, the methodology proposed here is applicable not only to biological membrane systems but also to the study of diffusion and localization phenomena of molecules in various heterogeneous systems.

Ensemble heterogeneity mimics ageing for endosomal dynamics within eukaryotic cells

Transport processes of many structures inside living cells display anomalous diffusion, such as endosomes in eukaryotic cells. They are also heterogeneous in space and time. Large ensembles of single particle trajectories allow the heterogeneities to be quantified in detail and provide insights for mathematical modelling. The development of accurate mathematical models for heterogeneous dynamics has the potential to enable the design and optimization of various technological applications, for example, the design of effective drug delivery systems. Central questions in the analysis of anomalous dynamics are ergodicity and statistical ageing which allow for selecting the proper model for the description. It is believed that non-ergodicity and ageing occur concurrently. However, we found that the anomalous dynamics of endosomes is paradoxical since it is ergodic but shows ageing. We show that this behaviour is caused by ensemble heterogeneity that, in addition to space-time heterogeneity within a single trajectory, is an inherent property of endosomal motion. Our work introduces novel approaches for the analysis and modelling of heterogeneous dynamics.

Plasmodium sporozoite search strategy to locate hotspots of blood vessel invasion

Plasmodium sporozoites actively migrate in the dermis and enter blood vessels to infect the liver. Despite their importance for malaria infection, little is known about these cutaneous processes. We combine intravital imaging in a rodent malaria model and statistical methods to unveil the parasite strategy to reach the bloodstream. We determine that sporozoites display a high-motility mode with a superdiffusive Lévy-like pattern known to optimize the location of scarce targets. When encountering blood vessels, sporozoites frequently switch to a subdiffusive low-motility behavior associated with probing for intravasation hotspots, marked by the presence of pericytes. Hence, sporozoites present anomalous diffusive motility, alternating between superdiffusive tissue exploration and subdiffusive local vessel exploitation, thus optimizing the sequential tasks of seeking blood vessels and pericyte-associated sites of privileged intravasation.

Weak ergodicity breaking and anomalous diffusion in collective motion of active particles under spatiotemporal disorder

The effects of spatiotemporal disorder, i.e., both the noise and quenched disorder, on the dynamics of active particles in two dimensions are investigated. We demonstrate that within the tailored parameter regime, nonergodic superdiffusion and nonergodic subdiffusion occur in the system, identified by the observable quantities (the mean squared displacement and ergodicity-breaking parameter) averaged over both the noise and realizations of quenched disorder. Their origins are attributed to the competition effects between the neighbor alignment and spatiotemporal disorder on the collective motion of active particles. These results may be helpful for further understanding the nonequilibrium transport process of active particles, as well as for detection of the transport of self-propelled particles in complex and crowded environments.

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

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