Time-dependent classification of protein diffusion types: A statistical detection of mean-squared-displacement exponent transitions

Diffusion State Transitions in Single-Particle Trajectories of MET Receptor Tyrosine Kinase Measured in Live Cells

Single-particle tracking enables the analysis of the dynamics of biomolecules in living cells with nanometer spatial and millisecond temporal resolution. This technique reports on the mobility of membrane proteins and is sensitive to the molecular state of a biomolecule and to interactions with other biomolecules. Trajectories describe the mobility of single particles over time and provide information such as the diffusion coefficient and diffusion state. Changes in particle dynamics within single trajectories lead to segmentation, which allows to extract information on transitions of functional states of a biomolecule. Here, mean-squared displacement analysis is developed to classify trajectory segments into immobile, confined diffusing, and freely diffusing states, and to extract the occurrence of transitions between these modes. We applied this analysis to single-particle tracking data of the membrane receptor MET in live cells and analyzed state transitions in single trajectories of the un-activated receptor and the receptor bound to the ligand internalin B. We found that internalin B-bound MET shows an enhancement of transitions from freely and confined diffusing states into the immobile state as compared to un-activated MET. Confined diffusion acts as an intermediate state between immobile and free, as this state is most likely to change the diffusion state in the following segment. This analysis can be readily applied to single-particle tracking data of other membrane receptors and intracellular proteins under various conditions and contribute to the understanding of molecular states and signaling pathways.

Fundamentals of the logarithmic measure for revealing multimodal diffusion

We develop a theoretical foundation for a time-series analysis method suitable for revealing the spectrum of diffusion coefficients in mixed Brownian systems, for which no prior knowledge of particle distinction is required. This method is directly relevant for particle tracking in biological systems, in which diffusion processes are often nonuniform. We transform Brownian data onto the logarithmic domain, in which the coefficients for individual modes of diffusion appear as distinct spectral peaks in the probability density. We refer to the method as the logarithmic measure of diffusion, or simply as the logarithmic measure. We provide a general protocol for deriving analytical expressions for the probability densities on the logarithmic domain. The protocol is applicable for any number of spatial dimensions with any number of diffusive states. The analytical form can be fitted to data to reveal multiple diffusive modes. We validate the theoretical distributions and benchmark the accuracy and sensitivity of the method by extracting multimodal diffusion coefficients from two-dimensional Brownian simulations of polydisperse filament bundles. Bundling the filaments allows us to control the system nonuniformity and hence quantify the sensitivity of the method. By exploiting the anisotropy of the simulated filaments, we generalize the logarithmic measure to rotational diffusion. By fitting the analytical forms to simulation data, we confirm the method's theoretical foundation. An error analysis in the single-mode regime shows that the proposed method is comparable in accuracy to the standard mean-squared displacement approach for evaluating diffusion coefficients. For the case of multimodal diffusion, we compare the logarithmic measure against other, more sophisticated methods, showing that both model selectivity and extraction accuracy are comparable for small data sets. Therefore, we suggest that the logarithmic measure, as a method for multimodal diffusion coefficient extraction, is ideally suited for small data sets, a condition often confronted in the experimental context. Finally, we critically discuss the proposed benefits of the method and its information content.

An Estimation Algorithm for General Linear Single Particle Tracking Models with Time-Varying Parameters

Single Particle Tracking (SPT) is a powerful class of methods for studying the dynamics of biomolecules inside living cells. The techniques reveal the trajectories of individual particles, with a resolution well below the diffraction limit of light, and from them the parameters defining the motion model, such as diffusion coefficients and confinement lengths. Most existing algorithms assume these parameters are constant throughout an experiment. However, it has been demonstrated that they often vary with time as the tracked particles move through different regions in the cell or as conditions inside the cell change in response to stimuli. In this work, we propose an estimation algorithm to determine time-varying parameters of systems that discretely switch between different linear models of motion with Gaussian noise statistics, covering dynamics such as diffusion, directed motion, and Ornstein-Uhlenbeck dynamics. Our algorithm consists of three stages. In the first stage, we use a sliding window approach, combined with Expectation Maximization (EM) to determine maximum likelihood estimates of the parameters as a function of time. These results are only used to roughly estimate the number of model switches that occur in the data to guide the selection of algorithm parameters in the second stage. In the second stage, we use Change Detection (CD) techniques to identify where the models switch, taking advantage of the off-line nature of the analysis of SPT data to create non-causal algorithms with better precision than a purely causal approach. Finally, we apply EM to each set of data between the change points to determine final parameter estimates. We demonstrate our approach using experimental data generated in the lab under controlled conditions.

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.

Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement

The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them.