Bayesian analysis of single-particle tracking data using the nested-sampling algorithm: maximum-likelihood model selection applied to stochastic-diffusivity data

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.

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.

Hidden Markov modeling of single-particle diffusion with stochastic tethering

The statistics of the diffusive motion of particles often serve as an experimental proxy for their interaction with the environment. However, inferring the physical properties from the observed trajectories is challenging. Inspired by a recent experiment, here we analyze the problem of particles undergoing two-dimensional Brownian motion with transient tethering to the surface. We model the problem as a hidden Markov model where the physical position is observed and the tethering state is hidden. We develop an alternating maximization algorithm to infer the hidden state of the particle and estimate the physical parameters of the system. The crux of our method is a saddle-point-like approximation, which involves finding the most likely sequence of hidden states and estimating the physical parameters from it. Extensive numerical tests demonstrate that our algorithm reliably finds the model parameters and is insensitive to the initial guess. We discuss the different regimes of physical parameters and the algorithm's performance in these regimes. We also provide a free software implementation of our algorithm.

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.

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.

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.

A machine learning approach to discover migration modes and transition dynamics of heterogeneous dendritic cells

Dendritic cell (DC) migration is crucial for mounting immune responses. Immature DCs (imDCs) reportedly sense infections, while mature DCs (mDCs) move quickly to lymph nodes to deliver antigens to T cells. However, their highly heterogeneous and complex innate motility remains elusive. Here, we used an unsupervised machine learning (ML) approach to analyze long-term, two-dimensional migration trajectories of Granulocyte-macrophage colony-stimulating factor (GMCSF)-derived bone marrow-derived DCs (BMDCs). We discovered three migratory modes independent of the cell state: slow-diffusive (SD), slow-persistent (SP), and fast-persistent (FP). Remarkably, imDCs more frequently changed their modes, predominantly following a unicyclic SD→FP→SP→SD transition, whereas mDCs showed no transition directionality. We report that DC migration exhibits a history-dependent mode transition and maturation-dependent motility changes are emergent properties of the dynamic switching of the three migratory modes. Our ML-based investigation provides new insights into studying complex cellular migratory behavior.

Gramian angular fields for leveraging pretrained computer vision models with anomalous diffusion trajectories

Anomalous diffusion is present at all scales, from atomic to large ones. Some exemplary systems are ultracold atoms, telomeres in the nucleus of cells, moisture transport in cement-based materials, arthropods' free movement, and birds' migration patterns. The characterization of the diffusion gives critical information about the dynamics of these systems and provides an interdisciplinary framework with which to study diffusive transport. Thus, the problem of identifying underlying diffusive regimes and inferring the anomalous diffusion exponent α with high confidence is critical to physics, chemistry, biology, and ecology. Classification and analysis of raw trajectories combining machine learning techniques with statistics extracted from them have widely been studied in the Anomalous Diffusion Challenge [Muñoz-Gil et al., Nat. Commun. 12, 6253 (2021)2041-172310.1038/s41467-021-26320-w]. Here we present a new data-driven method for working with diffusive trajectories. This method utilizes Gramian angular fields (GAF) to encode one-dimensional trajectories as images (Gramian matrices), while preserving their spatiotemporal structure for input to computer-vision models. This allows us to leverage two well-established pretrained computer-vision models, ResNet and MobileNet, to characterize the underlying diffusive regime and infer the anomalous diffusion exponent α. Short raw trajectories of lengths between 10 and 50 are commonly encountered in single-particle tracking experiments and are the most difficult ones to characterize. We show that GAF images can outperform the current state-of-the-art while increasing accessibility to machine learning methods in an applied setting.

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.

Variational inference of fractional Brownian motion with linear computational complexity

We introduce a simulation-based, amortized Bayesian inference scheme to infer the parameters of random walks. Our approach learns the posterior distribution of the walks' parameters with a likelihood-free method. In the first step a graph neural network is trained on simulated data to learn optimized low-dimensional summary statistics of the random walk. In the second step an invertible neural network generates the posterior distribution of the parameters from the learned summary statistics using variational inference. We apply our method to infer the parameters of the fractional Brownian motion model from single trajectories. The computational complexity of the amortized inference procedure scales linearly with trajectory length, and its precision scales similarly to the Cramér-Rao bound over a wide range of lengths. The approach is robust to positional noise, and generalizes to trajectories longer than those seen during training. Finally, we adapt this scheme to show that a finite decorrelation time in the environment can furthermore be inferred from individual trajectories.

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

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

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

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

Objective comparison of methods to decode anomalous diffusion

Deviations from Brownian motion leading to anomalous diffusion are found in transport dynamics from quantum physics to life sciences. The characterization of anomalous diffusion from the measurement of an individual trajectory is a challenging task, which traditionally relies on calculating the trajectory mean squared displacement. However, this approach breaks down for cases of practical interest, e.g., short or noisy trajectories, heterogeneous behaviour, or non-ergodic processes. Recently, several new approaches have been proposed, mostly building on the ongoing machine-learning revolution. To perform an objective comparison of methods, we gathered the community and organized an open competition, the Anomalous Diffusion challenge (AnDi). Participating teams applied their algorithms to a commonly-defined dataset including diverse conditions. Although no single method performed best across all scenarios, machine-learning-based approaches achieved superior performance for all tasks. The discussion of the challenge results provides practical advice for users and a benchmark for developers.

Precise measurements of chromatin diffusion dynamics by modeling using Gaussian processes

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

Inertia triggers nonergodicity of fractional Brownian motion

How related are the ergodic properties of the over- and underdamped Langevin equations driven by fractional Gaussian noise? We here find that for massive particles performing fractional Brownian motion (FBM) inertial effects not only destroy the stylized fact of the equivalence of the ensemble-averaged mean-squared displacement (MSD) to the time-averaged MSD (TAMSD) of overdamped or massless FBM, but also dramatically alter the values of the ergodicity-breaking parameter (EB). Our theoretical results for the behavior of EB for underdamped or massive FBM for varying particle mass m, Hurst exponent H, and trace length T are in excellent agreement with the findings of stochastic computer simulations. The current results can be of interest for the experimental community employing various single-particle-tracking techniques and aiming at assessing the degree of nonergodicity for the recorded time series (studying, e.g., the behavior of EB versus lag time). To infer FBM as a realizable model of anomalous diffusion for a set single-particle-tracking data when massive particles are being tracked, the EBs from the data should be compared to EBs of massive (rather than massless) FBM.

NOBIAS: Analyzing Anomalous Diffusion in Single-Molecule Tracks With Nonparametric Bayesian Inference

Single particle tracking (SPT) enables the investigation of biomolecular dynamics at a high temporal and spatial resolution in living cells, and the analysis of these SPT datasets can reveal biochemical interactions and mechanisms. Still, how to make the best use of these tracking data for a broad set of experimental conditions remains an analysis challenge in the field. Here, we develop a new SPT analysis framework: NOBIAS (NOnparametric Bayesian Inference for Anomalous Diffusion in Single-Molecule Tracking), which applies nonparametric Bayesian statistics and deep learning approaches to thoroughly analyze SPT datasets. In particular, NOBIAS handles complicated live-cell SPT data for which: the number of diffusive states is unknown, mixtures of different diffusive populations may exist within single trajectories, symmetry cannot be assumed between the x and y directions, and anomalous diffusion is possible. NOBIAS provides the number of diffusive states without manual supervision, it quantifies the dynamics and relative populations of each diffusive state, it provides the transition probabilities between states, and it assesses the anomalous diffusion behavior for each state. We validate the performance of NOBIAS with simulated datasets and apply it to the diffusion of single outer-membrane proteins in Bacteroides thetaiotaomicron. Furthermore, we compare NOBIAS with other SPT analysis methods and find that, in addition to these advantages, NOBIAS is robust and has high computational efficiency and is particularly advantageous due to its ability to treat experimental trajectories with asymmetry and anomalous diffusion.

Curve Registration of Functional Data for Approximate Bayesian Computation

Approximate Bayesian computation is a likelihood-free inference method which relies on comparing model realisations to observed data with informative distance measures. We obtain functional data that are not only subject to noise along their y axis but also to a random warping along their x axis, which we refer to as the time axis. Conventional distances on functions, such as the L2 distance, are not informative under these conditions. The Fisher–Rao metric, previously generalised from the space of probability distributions to the space of functions, is an ideal objective function for aligning one function to another by warping the time axis. We assess the usefulness of alignment with the Fisher–Rao metric for approximate Bayesian computation with four examples: two simulation examples, an example about passenger flow at an international airport, and an example of hydrological flow modelling. We find that the Fisher–Rao metric works well as the objective function to minimise for alignment; however, once the functions are aligned, it is not necessarily the most informative distance for inference. This means that likelihood-free inference may require two distances: one for alignment and one for parameter inference.

TRAIT2D: a Software for Quantitative Analysis of Single Particle Diffusion Data

Single particle tracking (SPT) is one of the most widely used tools in optical microscopy to evaluate particle mobility in a variety of situations, including cellular and model membrane dynamics. Recent technological developments, such as Interferometric Scattering microscopy, have allowed recording of long, uninterrupted single particle trajectories at kilohertz framerates. The resulting data, where particles are continuously detected and do not displace much between observations, thereby do not require complex linking algorithms. Moreover, while these measurements offer more details into the short-term diffusion behaviour of the tracked particles, they are also subject to the influence of localisation uncertainties, which are often underestimated by conventional analysis pipelines. we thus developed a Python library, under the name of TRAIT2D (Tracking Analysis Toolbox - 2D version), in order to track particle diffusion at high sampling rates, and analyse the resulting trajectories with an innovative approach. The data analysis pipeline introduced is more localisation-uncertainty aware, and also selects the most appropriate diffusion model for the data provided on a statistical basis. A trajectory simulation platform also allows the user to handily generate trajectories and even synthetic time-lapses to test alternative tracking algorithms and data analysis approaches. A high degree of customisation for the analysis pipeline, for example with the introduction of different diffusion modes, is possible from the source code. Finally, the presence of graphical user interfaces lowers the access barrier for users with little to no programming experience.

TRAIT2D: a Software for Quantitative Analysis of Single Particle Diffusion Data

Single particle tracking (SPT) is one of the most widely used tools in optical microscopy to evaluate particle mobility in a variety of situations, including cellular and model membrane dynamics. Recent technological developments, such as Interferometric Scattering microscopy, have allowed recording of long, uninterrupted single particle trajectories at kilohertz framerates. The resulting data, where particles are continuously detected and do not displace much between observations, thereby do not require complex linking algorithms. Moreover, while these measurements offer more details into the short-term diffusion behaviour of the tracked particles, they are also subject to the influence of localisation uncertainties, which are often underestimated by conventional analysis pipelines. we thus developed a Python library, under the name of TRAIT2D (Tracking Analysis Toolbox - 2D version), in order to track particle diffusion at high sampling rates, and analyse the resulting trajectories with an innovative approach. The data analysis pipeline introduced is more localisation-uncertainty aware, and also selects the most appropriate diffusion model for the data provided on a statistical basis. A trajectory simulation platform also allows the user to handily generate trajectories and even synthetic time-lapses to test alternative tracking algorithms and data analysis approaches. A high degree of customisation for the analysis pipeline, for example with the introduction of different diffusion modes, is possible from the source code. Finally, the presence of graphical user interfaces lowers the access barrier for users with little to no programming experience.

Single-particle diffusional fingerprinting: A machine-learning framework for quantitative analysis of heterogeneous diffusion

Single-particle tracking (SPT) analysis of individual biomolecules is an indispensable tool for extracting quantitative information from dynamic biological processes, but often requires some a priori knowledge of the system. Here we present “single-particle diffusional fingerprinting,” a more general approach for extraction of diffusional patterns in SPT independently of the biological system. This method extracts a set of descriptive features for each SPT trajectory, which are ranked upon classification to yield mechanistic insights for the species under comparison. We demonstrate its capacity to yield a dictionary of diffusional traits across multiple systems (e.g., lipases hydrolyzing fat, transcription factors diffusing in cells, and nanoparticles in mucus), supporting its use on multiple biological phenomena (e.g., drug delivery, receptor dynamics, and virology).

Single-particle tracking (SPT) is a key tool for quantitative analysis of dynamic biological processes and has provided unprecedented insights into a wide range of systems such as receptor localization, enzyme propulsion, bacteria motility, and drug nanocarrier delivery. The inherently complex diffusion in such biological systems can vary drastically both in time and across systems, consequently imposing considerable analytical challenges, and currently requires an a priori knowledge of the system. Here we introduce a method for SPT data analysis, processing, and classification, which we term “diffusional fingerprinting.” This method allows for dissecting the features that underlie diffusional behavior and establishing molecular identity, regardless of the underlying diffusion type. The method operates by isolating 17 descriptive features for each observed motion trajectory and generating a diffusional map of all features for each type of particle. Precise classification of the diffusing particle identity is then obtained by training a simple logistic regression model. A linear discriminant analysis generates a feature ranking that outputs the main differences among diffusional features, providing key mechanistic insights. Fingerprinting operates by both training on and predicting experimental data, without the need for pretraining on simulated data. We found this approach to work across a wide range of simulated and experimentally diverse systems, such as tracked lipases on fat substrates, transcription factors diffusing in cells, and nanoparticles diffusing in mucus. This flexibility ultimately supports diffusional fingerprinting’s utility as a universal paradigm for SPT diffusional analysis and prediction.

mRBioM: An Algorithm for the Identification of Potential mRNA Biomarkers From Complete Transcriptomic Profiles of Gastric Adenocarcinoma

Purpose

In this work, an algorithm named mRBioM was developed for the identification of potential mRNA biomarkers (PmBs) from complete transcriptomic RNA profiles of gastric adenocarcinoma (GA).

Methods

mRBioM initially extracts differentially expressed (DE) RNAs (mRNAs, miRNAs, and lncRNAs). Next, mRBioM calculates the total information amount of each DE mRNA based on the coexpression network, including three types of RNAs and the protein-protein interaction network encoded by DE mRNAs. Finally, PmBs were identified according to the variation trend of total information amount of all DE mRNAs. Four PmB-based classifiers without learning and with learning were designed to discriminate the sample types to confirm the reliability of PmBs identified by mRBioM. PmB-based survival analysis was performed. Finally, three other cancer datasets were used to confirm the generalization ability of mRBioM.

Results

mRBioM identified 55 PmBs (41 upregulated and 14 downregulated) related to GA. The list included thirteen PmBs that have been verified as biomarkers or potential therapeutic targets of gastric cancer, and some PmBs were newly identified. Most PmBs were primarily enriched in the pathways closely related to the occurrence and development of gastric cancer. Cancer-related factors without learning achieved sensitivity, specificity, and accuracy of 0.90, 1, and 0.90, respectively, in the classification of the GA and control samples. Average accuracy, sensitivity, and specificity of the three classifiers with machine learning ranged within 0.94–0.98, 0.94–0.97, and 0.97–1, respectively. The prognostic risk score model constructed by 4 PmBs was able to correctly and significantly (^∗∗∗p < 0.001) classify 269 GA patients into the high-risk (n = 134) and low-risk (n = 135) groups. GA equivalent classification performance was achieved using the complete transcriptomic RNA profiles of colon adenocarcinoma, lung adenocarcinoma, and hepatocellular carcinoma using PmBs identified by mRBioM.

Conclusions

GA-related PmBs have high specificity and sensitivity and strong prognostic risk prediction. MRBioM has also good generalization. These PmBs may have good application prospects for early diagnosis of GA and may help to elucidate the mechanism governing the occurrence and development of GA. Additionally, mRBioM is expected to be applied for the identification of other cancer-related biomarkers.

Classification of particle trajectories in living cells: Machine learning versus statistical testing hypothesis for fractional anomalous diffusion

Single-particle tracking (SPT) has become a popular tool to study the intracellular transport of molecules in living cells. Inferring the character of their dynamics is important, because it determines the organization and functions of the cells. For this reason, one of the first steps in the analysis of SPT data is the identification of the diffusion type of the observed particles. The most popular method to identify the class of a trajectory is based on the mean-square displacement (MSD). However, due to its known limitations, several other approaches have been already proposed. With the recent advances in algorithms and the developments of modern hardware, the classification attempts rooted in machine learning (ML) are of particular interest. In this work, we adopt two ML ensemble algorithms, i.e., random forest and gradient boosting, to the problem of trajectory classification. We present a new set of features used to transform the raw trajectories data into input vectors required by the classifiers. The resulting models are then applied to real data for G protein-coupled receptors and G proteins. The classification results are compared to recent statistical methods going beyond MSD.

Detection of Anomalous Diffusion with Deep Residual Networks

Identification of the diffusion type of molecules in living cells is crucial to deduct their driving forces and hence to get insight into the characteristics of the cells. In this paper, deep residual networks have been used to classify the trajectories of molecules. We started from the well known ResNet architecture, developed for image classification, and carried out a series of numerical experiments to adapt it to detection of diffusion modes. We managed to find a model that has a better accuracy than the initial network, but contains only a small fraction of its parameters. The reduced size significantly shortened the training time of the model. Moreover, the resulting network has less tendency to overfitting and generalizes better to unseen data.

Measurements and characterization of the dynamics of tracer particles in an actin network

The underlying physics governing the diffusion of a tracer particle in a viscoelastic material is a topic of some dispute. The long-term memory in the mechanical response of such materials should induce diffusive motion with a memory kernel, such as fractional Brownian motion (fBM). This is the reason that microrheology is able to provide the shear modulus of polymer networks. Surprisingly, the diffusion of a tracer particle in a network of a purified protein, actin, was found to conform to the continuous time random walk type (CTRW). We set out to resolve this discrepancy by studying the tracer particle diffusion using two different tracer particle sizes, in actin networks of different mesh sizes. We find that the ratio of tracer particle size to the characteristic length scale of a bio-polymer network plays a crucial role in determining the type of diffusion it performs. We find that the diffusion of the tracer particles has features of fBm when the particle is large compared to the mesh size, of normal diffusion when the particle is much smaller than the mesh size, and of the CTRW in between these two limits. Based on our findings, we propose and verify numerically a new model for the motion of the tracer in all regimes. Our model suggests that diffusion in actin networks consists of fBm of the tracer particle coupled with caging events with power-law distributed escape times.

Single-trajectory characterization of active swimmers in a flow

We develop a maximum likelihood method to infer relevant physical properties of elongated active particles. Using individual trajectories of advected swimmers as input, we are able to accurately determine their rotational diffusion coefficients and an effective measure of their aspect ratio, also providing reliable estimators for the uncertainties of such quantities. We validate our theoretical construction using numerically generated active trajectories upon no flow, simple shear, and Poiseuille flow, with excellent results. Being designed to rely on single-particle data, our method eases applications in experimental conditions where swimmers exhibit a strong morphological diversity. We briefly discuss some of such ongoing experimental applications, specifically, in the characterization of swimming E. coli in a flow.

Feature Selection for Colon Cancer Detection Using K-Means Clustering and Modified Harmony Search Algorithm

This paper proposes a feature selection method that is effective in distinguishing colorectal cancer patients from normal individuals using K-means clustering and the modified harmony search algorithm. As the genetic cause of colorectal cancer originates from mutations in genes, it is important to classify the presence or absence of colorectal cancer through gene information. The proposed methodology consists of four steps. First, the original data are Z-normalized by data preprocessing. Candidate genes are then selected using the Fisher score. Next, one representative gene is selected from each cluster after candidate genes are clustered using K-means clustering. Finally, feature selection is carried out using the modified harmony search algorithm. The gene combination created by feature selection is then applied to the classification model and verified using 5-fold cross-validation. The proposed model obtained a classification accuracy of up to 94.36%. Furthermore, on comparing the proposed method with other methods, we prove that the proposed method performs well in classifying colorectal cancer. Moreover, we believe that the proposed model can be applied not only to colorectal cancer but also to other gene-related diseases.

Bayesian Inference: The Comprehensive Approach to Analyzing Single-Molecule Experiments

Biophysics experiments performed at single-molecule resolution provide exceptional insight into the structural details and dynamic behavior of biological systems. However, extracting this information from the corresponding experimental data unequivocally requires applying a biophysical model. In this review, we discuss how to use probability theory to apply these models to single-molecule data. Many current single-molecule data analysis methods apply parts of probability theory, sometimes unknowingly, and thus miss out on the full set of benefits provided by this self-consistent framework. The full application of probability theory involves a process called Bayesian inference that fully accounts for the uncertainties inherent to single-molecule experiments. Additionally, using Bayesian inference provides a scientifically rigorous method of incorporating information from multiple experiments into a single analysis and finding the best biophysical model for an experiment without the risk of overfitting the data. These benefits make the Bayesian approach ideal for analyzing any type of single-molecule experiment.

Inverse design of nonequilibrium steady states: A large-deviation approach

The design of small scale nonequilibrium steady states (NESS) is a challenging, open ended question. While similar equilibrium problems are tractable using standard thermodynamics, a generalized description for nonequilibrium systems is lacking, making the design problem particularly difficult. Here we show we can exploit the large-deviation behavior of a Brownian particle and design a variety of geometrically complex steady-state density distributions and flux field flows. We achieve this design target from direct knowledge of the joint large-deviation functional for the empirical density and flow, and a "relaxation" algorithm on the desired target states via adjustable force field parameters. We validate the method by replicating analytical results, and demonstrate its capacity to yield complex prescribed targets, such as rose-curve or polygonal shapes on the plane. We consider this dynamical fluctuation approach a first step towards the design of more complex NESS where general frameworks are otherwise still lacking.

Impact of Feature Choice on Machine Learning Classification of Fractional Anomalous Diffusion

The growing interest in machine learning methods has raised the need for a careful study of their application to the experimental single-particle tracking data. In this paper, we present the differences in the classification of the fractional anomalous diffusion trajectories that arise from the selection of the features used in random forest and gradient boosting algorithms. Comparing two recently used sets of human-engineered attributes with a new one, which was tailor-made for the problem, we show the importance of a thoughtful choice of the features and parameters. We also analyse the influence of alterations of synthetic training data set on the classification results. The trained classifiers are tested on real trajectories of G proteins and their receptors on a plasma membrane.

Gaining insight into cellular cardiac physiology using single particle tracking

Single particle tracking (SPT) is a robust technique to monitor single-molecule behaviors in living cells directly. By this approach, we can uncover the potential biological significance of particle dynamics by statistically characterizing individual molecular behaviors. SPT provides valuable information at the single-molecule level, that could be obscured by simple averaging that is inherent to conventional ensemble measurements. Here, we give a brief introduction to SPT including the commonly used optical implementations, fluorescence labeling strategies, and data analysis methods. We then focus on how SPT has been harnessed to decipher myocardial function. In this context, SPT has provided novel insight into the lateral diffusion of signal receptors and ion channels, the dynamic organization of cardiac nanodomains, subunit composition and stoichiometry of cardiac ion channels, myosin movement along actin filaments, the kinetic features of transcription factors involved in cardiac remodeling, and the intercellular communication by nanotubes. Finally, we speculate on the prospects and challenges of applying SPT to future questions regarding cellular cardiac physiology using SPT.

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.

Bayesian analysis of data from segmented super-resolution images for quantifying protein clustering

Super-resolution imaging techniques have largely improved our capabilities to visualize nanometric structures in biological systems. Their application further permits the quantitation relevant parameters to determine the molecular organization and stoichiometry in cells. However, the inherently stochastic nature of fluorescence emission and labeling strategies imposes the use of dedicated methods to accurately estimate these parameters. Here, we describe a Bayesian approach to precisely quantitate the relative abundance of molecular aggregates of different stoichiometry from segmented images. The distribution of proxies for the number of molecules in a cluster, such as the number of localizations or the fluorescence intensity, is fitted via a nested sampling algorithm to compare mixture models of increasing complexity and thus determine the optimum number of mixture components and their weights. We test the performance of the algorithm on in silico data as a function of the number of data points, threshold, and distribution shape. We compare these results to those obtained with other statistical methods, showing the improved performance of our approach. Our method provides a robust tool for model selection in fitting data extracted from fluorescence imaging, thus improving the precision of parameter determination. Importantly, the largest benefit of this method occurs for small-statistics or incomplete datasets, enabling an accurate analysis at the single image level. We further present the results of its application to experimental data obtained from the super-resolution imaging of dynein in HeLa cells, confirming the presence of a mixed population of cytoplasmic single motors and higher-order structures.

The diffusion of doxorubicin drug molecules in silica nanoslits is non-Gaussian, intermittent and anticorrelated

The motion of the confined doxorubicin drug molecule exhibits an interesting combination of anomalous diffusion features.

Complex Diffusion in Bacteria

Diffusion within bacteria is often thought of as a "simple" random process by which molecules collide and interact with each other. New research however shows that this is far from the truth. Here we shed light on the complexity and importance of diffusion in bacteria, illustrating the similarities and differences of diffusive behaviors of molecules within different compartments of bacterial cells. We first describe common methodologies used to probe diffusion and the associated models and analyses. We then discuss distinct diffusive behaviors of molecules within different bacterial cellular compartments, highlighting the influence of metabolism, size, crowding, charge, binding, and more. We also explicitly discuss where further research and a united understanding of what dictates diffusive behaviors across the different compartments of the cell are required, pointing out new research avenues to pursue.

Measurement of anomalous diffusion using recurrent neural networks

Anomalous diffusion occurs in many physical and biological phenomena, when the growth of the mean squared displacement (MSD) with time has an exponent different from one. We show that recurrent neural networks (RNNs) can efficiently characterize anomalous diffusion by determining the exponent from a single short trajectory, outperforming the standard estimation based on the MSD when the available data points are limited, as is often the case in experiments. Furthermore, the RNNs can handle more complex tasks where there are no standard approaches, such as determining the anomalous diffusion exponent from a trajectory sampled at irregular times, and estimating the switching time and anomalous diffusion exponents of an intermittent system that switches between different kinds of anomalous diffusion. We validate our method on experimental data obtained from subdiffusive colloids trapped in speckle light fields and superdiffusive microswimmers.

Single-Particle Diffusion Characterization by Deep Learning

Diffusion plays a crucial role in many biological processes including signaling, cellular organization, transport mechanisms, and more. Direct observation of molecular movement by single-particle-tracking experiments has contributed to a growing body of evidence that many cellular systems do not exhibit classical Brownian motion but rather anomalous diffusion. Despite this evidence, characterization of the physical process underlying anomalous diffusion remains a challenging problem for several reasons. First, different physical processes can exist simultaneously in a system. Second, commonly used tools for distinguishing between these processes are based on asymptotic behavior, which is experimentally inaccessible in most cases. Finally, an accurate analysis of the diffusion model requires the calculation of many observables because different transport modes can result in the same diffusion power-law α, which is typically obtained from the mean-square displacements (MSDs). The outstanding challenge in the field is to develop a method to extract an accurate assessment of the diffusion process using many short trajectories with a simple scheme that is applicable at the nonexpert level. Here, we use deep learning to infer the underlying process resulting in anomalous diffusion. We implement a neural network to classify single-particle trajectories by diffusion type: Brownian motion, fractional Brownian motion and continuous time random walk. Further, we demonstrate the applicability of our network architecture for estimating the Hurst exponent for fractional Brownian motion and the diffusion coefficient for Brownian motion on both simulated and experimental data. These networks achieve greater accuracy than time-averaged MSD analysis on simulated trajectories while only requiring as few as 25 steps. When tested on experimental data, both net and ensemble MSD analysis converge to similar values; however, the net needs only half the number of trajectories required for ensemble MSD to achieve the same confidence interval. Finally, we extract diffusion parameters from multiple extremely short trajectories (10 steps) using our approach.

A Jump-Distance-Based Parameter Inference Scheme for Particulate Trajectories

This study presents an improved quantitative tool for the analysis of particulate trajectories. Particulate trajectory data appears in several different biological contexts, from the trajectory of chemotaxing bacteria to the nuclear mobility inferred from the trajectory of MS2 spots. Presently, the majority of analyses performed on particulate trajectory data have been limited to mean-squared displacement (MSD) analysis. Although simple, MSD analysis has several pitfalls, including difficulty in selecting between competing methods of motion, handling systems with multiple distinct subpopulations, and parameter extraction from limited time-series data. Here, we provide an alternative to MSD analysis using the jump distance distribution (JDD), which addresses the aforementioned issues. In particular, the method outperforms MSD analysis in the data-poor limit, thereby giving access to a larger range of temporal dynamics. In this work, we construct and validate a derivation of the JDD for different transportation modes and dimensions and implement a parameter estimation and model selection scheme. This scheme is validated, and direct improvements over MSD analysis are shown. Through an analysis of bacterial chemotaxis data, we highlight the JDD's ability to extract parameters at a variety of timescales, as well as extract underlying biological features of interest.

A method for single molecule tracking using a conventional single-focus confocal setup

One way to achieve spatial resolution using fluorescence imaging-and track single molecules-is to use wide-field illumination and collect measurements over multiple sensors (camera pixels). Here we propose another way that uses confocal measurements and a single sensor. Traditionally, confocal microscopy has been used to achieve high temporal resolution at the expense of spatial resolution. This is because it utilizes very few, and commonly just one, sensors to collect data. Yet confocal data encode spatial information. Here we show that non-uniformities in the shape of the confocal excitation volume can be exploited to achieve spatial resolution. To achieve this, we formulate a specialized hidden Markov model and adapt a forward filtering-backward sampling Markov chain Monte Carlo scheme to efficiently handle molecular motion within a symmetric confocal volume characteristically used in fluorescence correlation spectroscopy. Our method can be used for single confocal volume applications or incorporated into larger computational schemes for specialized, multi-confocal volume, optical setups.

Non-Gaussian, non-ergodic, and non-Fickian diffusion of tracers in mucin hydrogels

Native mucus is polymer-based soft-matter material of paramount biological importance. How non-Gaussian and non-ergodic is the diffusive spreading of pathogens in mucus? We study the passive, thermally driven motion of micron-sized tracers in hydrogels of mucins, the main polymeric component of mucus. We report the results of the Bayesian analysis for ranking several diffusion models for a set of tracer trajectories [C. E. Wagner et al., Biomacromolecules, 2017, 18, 3654]. The models with "diffusing diffusivity", fractional and standard Brownian motion are used. The likelihood functions and evidences of each model are computed, ranking the significance of each model for individual traces. We find that viscoelastic anomalous diffusion is often most probable, followed by Brownian motion, while the model with a diffusing diffusion coefficient is only realised rarely. Our analysis also clarifies the distribution of time-averaged displacements, correlations of scaling exponents and diffusion coefficients, and the degree of non-Gaussianity of displacements at varying pH levels. Weak ergodicity breaking is also quantified. We conclude that-consistent with the original study-diffusion of tracers in the mucin gels is most non-Gaussian and non-ergodic at low pH that corresponds to the most heterogeneous networks. Using the Bayesian approach with the nested-sampling algorithm, together with the quantitative analysis of multiple statistical measures, we report new insights into possible physical mechanisms of diffusion in mucin gels.