Fractional kinetics emerging from ergodicity breaking in random media

We present a modelling approach for diffusion in a complex medium characterized by a random length scale. The resulting stochastic process shows subdiffusion with a behavior in qualitative agreement with single particle tracking experiments in living cells, such as ergodicity breaking, p-variation and aging. In particular, this approach recapitulates characteristic features previously described in part by the fractional Brownian motion and in part by the continuous-time random walk. Moreover, for a proper distribution of the length scale, a single parameter controls the ergodic-to-nonergodic transition and, remarkably, also drives the transition of the diffusion equation of the process from non-fractional to fractional, thus demonstrating that fractional kinetics emerges from ergodicity breaking.Comment: Extended version of arXiv:1508.01361: Gianni Pagnini, Daniel Molina-Garc\'ia, Tuan Minh Pham, Carlo Manzo, Paolo Paradisi, Ergodicity breaking in random media and the foundation of fractional kinetic

FLUORESCENCE-BASED INVESTIGATION OF THE JANOSSY EFFECT ANOMALOUS WAVELENGTH DEPENDENCE

In order to explain the wavelength dependence of the"Janossy effect"observed in certain dye-doped liquid crystals, we investigated the fluorescence emitted by 1,8-diamino 4,5-dihydroxy 3,6-diisopentyl anthraquinone dye dissolved in different hosts. We measured the lifetimeτe, rotational timeτr, initial anisotropy r0and fluorescence quantum yieldΦof photo-excited dye molecules versus excitation-light wavelengthλ. The only significantλ-dependence observed was that of the quantum yieldΦ, which showed a marked increase withλ.Φ(λ)appears to be well correlated with the Janossy effect. These results rule out several proposed explanations for the Janossy effect wavelength dependence and suggest a simple alternative explanation

Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity

Molecular transport in living systems regulates numerous processes underlying biological function. Although many cellular components exhibit anomalous diffusion, only recently has the subdiffusive motion been associated with nonergodic behavior. These findings have stimulated new questions for their implications in statistical mechanics and cell biology. Is nonergodicity a common strategy shared by living systems? Which physical mechanisms generate it? What are its implications for biological function? Here, we use single particle tracking to demonstrate that the motion of DC-SIGN, a receptor with unique pathogen recognition capabilities, reveals nonergodic subdiffusion on living cell membranes. In contrast to previous studies, this behavior is incompatible with transient immobilization and therefore it can not be interpreted according to continuous time random walk theory. We show that the receptor undergoes changes of diffusivity, consistent with the current view of the cell membrane as a highly dynamic and diverse environment. Simulations based on a model of ordinary random walk in complex media quantitatively reproduce all our observations, pointing toward diffusion heterogeneity as the cause of DC-SIGN behavior. By studying different receptor mutants, we further correlate receptor motion to its molecular structure, thus establishing a strong link between nonergodicity and biological function. These results underscore the role of disorder in cell membranes and its connection with function regulation. Due to its generality, our approach offers a framework to interpret anomalous transport in other complex media where dynamic heterogeneity might play a major role, such as those found, e.g., in soft condensed matter, geology and ecology.Comment: 27 pages, 5 figure

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 enables one to potentially quantitate relevant parameters to determine the molecular organization and stoichiometry in cells. However, the inherently stochastic nature of the fluorescence emission and labeling strategies imposes the use of dedicated methods to accurately measure these parameters. Here, we describe a Bayesian approach to precisely quantitate the relative abundance of molecular oligomers 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 determine the optimal number of mixture components and their weights. We test the performance of the algorithm on {\it 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 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 cytoplasmatic single motors and higher-order structures.Comment: 17 pages, 6 figure

Learning minimal representations of stochastic processes with variational autoencoders

Stochastic processes have found numerous applications in science, as they are broadly used to model a variety of natural phenomena. Due to their intrinsic randomness and uncertainty, they are however difficult to characterize. Here, we introduce an unsupervised machine learning approach to determine the minimal set of parameters required to effectively describe the dynamics of a stochastic process. Our method builds upon an extended $\beta$-variational autoencoder architecture. By means of simulated datasets corresponding to paradigmatic diffusion models, we showcase its effectiveness in extracting the minimal relevant parameters that accurately describe these dynamics. Furthermore, the method enables the generation of new trajectories that faithfully replicate the expected stochastic behavior. Overall, our approach enables for the autonomous discovery of unknown parameters describing stochastic processes, hence enhancing our comprehension of complex phenomena across various fields.Comment: 9 pages, 5 figures, 1 table. Code available at https://github.com/GabrielFernandezFernandez/SPIVA

Quantifying Protein Copy Number in Super-Resolution Using an Imaging Invariant Calibration

The use of super-resolution microscopy in recent years has revealed that proteins often form small assemblies inside cells and are organized in nanoclusters. However, determining the copy number of proteins within these nanoclusters constitutes a major challenge because of unknown labeling stoichiometries and complex fluorophore photophysics. We previously developed a DNA-origami-based calibration approach to extract protein copy number from super-resolution images. However, the applicability of this approach is limited by the fact that the calibration is dependent on the specific labeling and imaging conditions used in each experiment. Hence, the calibration must be repeated for each experimental condition, which is a formidable task. Here, using cells stably expressing dynein intermediate chain fused to green fluorescent protein (HeLa IC74 cells) as a reference sample, we demonstrate that the DNA-origami-based calibration data we previously generated can be extended to super-resolution images taken under different experimental conditions, enabling the quantification of any green-fluorescent-protein-fused protein of interest. To do so, we first quantified the copy number of dynein motors within nanoclusters in the cytosol and along the microtubules. Interestingly, this quantification showed that dynein motors form assemblies consisting of more than one motor, especially along microtubules. This quantification enabled us to use the HeLa IC74 cells as a reference sample to calibrate and quantify protein copy number independently of labeling and imaging conditions, dramatically improving the versatility and applicability of our approach

Inferring pointwise diffusion properties of single trajectories with deep learning

In order 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 $\alpha5\beta1$. The analysis allows us to characterize physical parameters and diffusive states with unprecedented accuracy, shedding new light on the underlying mechanisms.Comment: 17 pages, 9 figures, 1 table. Code is found in https://github.com/BorjaRequena/ste

High-density single-molecule maps reveal transient membrane receptor interactions within a dynamically varying environment

Over recent years, super-resolution and single-molecule imaging methods have delivered unprecedented details on the nanoscale organization and dynamics of individual molecules in different contexts. Yet, visualizing single-molecule processes in living cells with the required spatial and temporal resolution remains highly challenging. Here, we report on an analytical approach that extracts such information from live-cell single-molecule imaging at high-labeling densities using standard fluorescence probes. Our high-density-mapping (HiDenMap) methodology provides single-molecule nanometric localization accuracy together with millisecond temporal resolution over extended observation times, delivering multi-scale spatiotemporal data that report on the interaction of individual molecules with their dynamic environment. We validated HiDenMaps by simulations of Brownian trajectories in the presence of patterns that restrict free diffusion with different probabilities. We further generated and analyzed HiDenMaps from single-molecule images of transmembrane proteins having different interaction strengths to cortical actin, including the transmembrane receptor CD44. HiDenMaps uncovered a highly heterogenous and multi-scale spatiotemporal organization for all the proteins that interact with the actin cytoskeleton. Notably, CD44 alternated between periods of random diffusion and transient trapping, likely resulting from actin-dependent CD44 nanoclustering. Whereas receptor trapping was dynamic and lasted for hundreds of milliseconds, actin remodeling occurred at the timescale of tens of seconds, coordinating the assembly and disassembly of CD44 nanoclusters rich regions. Together, our data demonstrate the power of HiDenMaps to explore how individual molecules interact with and are organized by their environment in a dynamic fashion.Comment: 33 pages, 5 figure