A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories

Currently used techniques for the analysis of single-molecule trajectories only exploit a small part of the available information stored in the data. Here, we apply a Bayesian inference scheme to trajectories of confined receptors that are targeted by pore-forming toxins to extract the two-dimensional confining potential that restricts the motion of the receptor. The receptor motion is modeled by the overdamped Langevin equation of motion. The method uses most of the information stored in the trajectory and converges quickly onto inferred values, while providing the uncertainty on the determined values. The inference is performed on the polynomial development of the potential and on the diffusivities that have been discretized on a mesh. Numerical simulations are used to test the scheme and quantify the convergence toward the input values for forces, potential, and diffusivity. Furthermore, we show that the technique outperforms the classical mean-square-displacement technique when forces act on confined molecules because the typical mean-square-displacement analysis does not account for them. We also show that the inferred potential better represents input potentials than the potential extracted from the position distribution based on Boltzmann statistics that assumes statistical equilibrium.     

Quantifying Biomolecule Diffusivity Using an Optimal Bayesian Method

We propose a Bayesian method to extract the diffusivity of biomolecules evolving freely or inside membrane microdomains. This approach assumes a model of motion for the particle considered, namely free Brownian motion or confined diffusion. In each framework, a systematic Bayesian scheme is provided for estimating the diffusivity. We show that this method reaches the best performances theoretically achievable. Its efficiency overcomes that of widely used methods based on the analysis of the mean-square displacement. The approach presented here also gives direct access to the uncertainty on the estimation of the diffusivity and predicts the number of steps of the trajectory necessary to achieve any desired precision. Its robustness with respect to noise on the position of the biomolecule is also investigated.     

Nonparametric mixed effects models for unequally sampled noisy curves

We propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients. The method is applicable when the individual curves are sampled at variable and irregularly spaced points. This produces a low-rank, low-frequency approximation to the covariance structure, which can be estimated naturally by the EM algorithm. Smooth curves for individual trajectories are constructed as best linear unbiased predictor (BLUP) estimates, combining data from that individual and the entire collection. This framework leads naturally to methods for examining the effects of covariates on the shapes of the curves. We use model selection techniques--Akaike information criterion (AIC), Bayesian information criterion (BIC), and cross-validation--to select the number of breakpoints for the spline approximation. We believe that the methodology we propose provides a simple, flexible, and computationally efficient means of functional data analysis.     

The Effect of Branch Length Variation on the Selection of Models of Molecular Evolution

Models of sequence evolution play an important role in molecular evolutionary studies. The use of inappropriate models of evolution may bias the results of the analysis and lead to erroneous conclusions. Several procedures for selecting the best-fit model of evolution for the data at hand have been proposed, like the likelihood ratio test (LRT) and the Akaike (AIC) and Bayesian (BIC) information criteria. The relative performance of these model-selecting algorithms has not yet been studied under a range of different model trees. In this study, the influence of branch length variation upon model selection is characterized. This is done by simulating sequence alignments under a known model of nucleotide substitution, and recording how often this true model is recovered by different model-fitting strategies. Results of this study agree with previous simulations and suggest that model selection is reasonably accurate. However, different model selection methods showed distinct levels of accuracy. Some LRT approaches showed better performance than the AIC or BIC information criteria. Within the LRTs, model selection is affected by the complexity of the initial model selected for the comparisons, and only slightly by the order in which different parameters are added to the model. A specific hierarchy of LRTs, which starts from a simple model of evolution, performed overall better than other possible LRT hierarchies, or than the AIC or BIC. Received: 2 October 2000 / Accepted: 4 January 2001     

A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data

In the analysis of data generated by change-point processes, one critical challenge is to determine the number of change-points. The classic Bayes information criterion (BIC) statistic does not work well here because of irregularities in the likelihood function. By asymptotic approximation of the Bayes factor, we derive a modified BIC for the model of Brownian motion with changing drift. The modified BIC is similar to the classic BIC in the sense that the first term consists of the log likelihood, but it differs in the terms that penalize for model dimension. As an example of application, this new statistic is used to analyze array-based comparative genomic hybridization (array-CGH) data. Array-CGH measures the number of chromosome copies at each genome location of a cell sample, and is useful for finding the regions of genome deletion and amplification in tumor cells. The modified BIC performs well compared to existing methods in accurately choosing the number of regions of changed copy number. Unlike existing methods, it does not rely on tuning parameters or intensive computing. Thus it is impartial and easier to understand and to use.     

Detection of temporary lateral confinement of membrane proteins using single-particle tracking analysis.

Techniques such as single-particle tracking allow the characterization of the movements of single or very few molecules. Features of the molecular trajectories, such as confined diffusion or directed transport, can reveal interesting biological interactions, but they can also arise from simple Brownian motion. Careful analysis of the data, therefore, is necessary to identify interesting effects from pure random movements. A method was developed to detect temporary confinement in the trajectories of membrane proteins that cannot be accounted for by Brownian motion. This analysis was applied to trajectories of two lipid-linked members of the immunoglobulin superfamily, Thy-1 and a neural cell adhesion molecule (NCAM 125), and the results were compared with those for simulated random walks. Approximately 28% of the trajectories for both proteins exhibited periods of transient confinement, which were < 0.07% likely to arise from random movements. In contrast to these results, only 1.5% of the simulated trajectories showed confined periods. Transient confinement for both proteins lasted on average 8 s in regions that were approximately 280 nm in diameter.     

Classification of Dynamical Diffusion States in Single Molecule Tracking Microscopy

Single molecule tracking of membrane proteins by fluorescence microscopy is a promising method to investigate dynamic processes in live cells. Translating the trajectories of proteins to biological implications, such as protein interactions, requires the classification of protein motion within the trajectories. Spatial information of protein motion may reveal where the protein interacts with cellular structures, because binding of proteins to such structures often alters their diffusion speed. For dynamic diffusion systems, we provide an analytical framework to determine in which diffusion state a molecule is residing during the course of its trajectory. We compare different methods for the quantification of motion to utilize this framework for the classification of two diffusion states (two populations with different diffusion speed). We found that a gyration quantification method and a Bayesian statistics-based method are the most accurate in diffusion-state classification for realistic experimentally obtained datasets, of which the gyration method is much less computationally demanding. After classification of the diffusion, the lifetime of the states can be determined, and images of the diffusion states can be reconstructed at high resolution. Simulations validate these applications. We apply the classification and its applications to experimental data to demonstrate the potential of this approach to obtain further insights into the dynamics of cell membrane proteins.     

Application of Brownian motion theory to the analysis of membrane channel ionic trajectories calculated by molecular dynamics.

This paper shows how Brownian motion theory can be used to analyze features of individual ion trajectories in channels as calculated by molecular dynamics, and that its use permits more precise determinations of diffusion coefficients than would otherwise be possible. We also show how a consideration of trajectories of single particles can distinguish between effects due to the magnitude of the diffusion coefficient and effects due to barriers and wells in the potential profile, effects which can not be distinguished by consideration of average fluxes.     

Classification of Dynamical Diffusion States in Single Molecule Tracking Microscopy

Single molecule tracking of membrane proteins by fluorescence microscopy is a promising method to investigate dynamic processes in live cells. Translating the trajectories of proteins to biological implications, such as protein interactions, requires the classification of protein motion within the trajectories. Spatial information of protein motion may reveal where the protein interacts with cellular structures, because binding of proteins to such structures often alters their diffusion speed. For dynamic diffusion systems, we provide an analytical framework to determine in which diffusion state a molecule is residing during the course of its trajectory. We compare different methods for the quantification of motion to utilize this framework for the classification of two diffusion states (two populations with different diffusion speed). We found that a gyration quantification method and a Bayesian statistics-based method are the most accurate in diffusion-state classification for realistic experimentally obtained datasets, of which the gyration method is much less computationally demanding. After classification of the diffusion, the lifetime of the states can be determined, and images of the diffusion states can be reconstructed at high resolution. Simulations validate these applications. We apply the classification and its applications to experimental data to demonstrate the potential of this approach to obtain further insights into the dynamics of cell membrane proteins.     

Resolving Cytosolic Diffusive States in Bacteria by Single-Molecule Tracking

The trajectory of a single protein in the cytosol of a living cell contains information about its molecular interactions in its native environment. However, it has remained challenging to accurately resolve and characterize the diffusive states that can manifest in the cytosol using analytical approaches based on simplifying assumptions. Here, we show that multiple intracellular diffusive states can be successfully resolved if sufficient single-molecule trajectory information is available to generate well-sampled distributions of experimental measurements and if experimental biases are taken into account during data analysis. To address the inherent experimental biases in camera-based and MINFLUX-based single-molecule tracking, we use an empirical data analysis framework based on Monte Carlo simulations of confined Brownian motion. This framework is general and adaptable to arbitrary cell geometries and data acquisition parameters employed in two-dimensional or three-dimensional single-molecule tracking. We show that, in addition to determining the diffusion coefficients and populations of prevalent diffusive states, the timescales of diffusive state switching can be determined by stepwise increasing the time window of averaging over subsequent single-molecule displacements. Time-averaged diffusion analysis of single-molecule tracking data may thus provide quantitative insights into binding and unbinding reactions among rapidly diffusing molecules that are integral for cellular functions.     

Translational diffusion of individual class II MHC membrane proteins in cells

Single-molecule epifluorescence microscopy was used to observe the translational motion of GPI-linked and native I-E(k) class II MHC membrane proteins in the plasma membrane of CHO cells. The purpose of the study was to look for deviations from Brownian diffusion that might arise from barriers to this motion. Detergent extraction had suggested that these proteins may be confined to lipid microdomains in the plasma membrane. The individual I-E(k) proteins were visualized with a Cy5-labeled peptide that binds to a specific extracytoplasmic site common to both proteins. Single-molecule trajectories were used to compute a radial distribution of displacements, yielding average diffusion coefficients equal to 0.22 (GPI-linked I-E(k)) and 0.18 microm(2)/s (native I-E(k)). The relative diffusion of pairs of proteins was also studied for intermolecular separations in the range 0.3-1.0 microm, to distinguish between free diffusion of a protein molecule and diffusion of proteins restricted to a rapidly diffusing small domain. Both analyses show that motion is predominantly Brownian. This study finds no strong evidence for significant confinement of either GPI-linked or native I-E(k) in the plasma membrane of CHO cells.     

Accounting for uncertainty in the tree topology has little effect on the decision-theoretic approach to model selection in phylogeny estimation

Currently available methods for model selection used in phylogenetic analysis are based on an initial fixed-tree topology. Once a model is picked based on this topology, a rigorous search of the tree space is run under that model to find the maximum-likelihood estimate of the tree (topology and branch lengths) and the maximum-likelihood estimates of the model parameters. In this paper, we propose two extensions to the decision-theoretic (DT) approach that relax the fixed-topology restriction. We also relax the fixed-topology restriction for the Bayesian information criterion (BIC) and the Akaike information criterion (AIC) methods. We compare the performance of the different methods (the relaxed, restricted, and the likelihood-ratio test [LRT]) using simulated data. This comparison is done by evaluating the relative complexity of the models resulting from each method and by comparing the performance of the chosen models in estimating the true tree. We also compare the methods relative to one another by measuring the closeness of the estimated trees corresponding to the different chosen models under these methods. We show that varying the topology does not have a major impact on model choice. We also show that the outcome of the two proposed extensions is identical and is comparable to that of the BIC, Extended-BIC, and DT. Hence, using the simpler methods in choosing a model for analyzing the data is more computationally feasible, with results comparable to the more computationally intensive methods. Another outcome of this study is that earlier conclusions about the DT approach are reinforced. That is, LRT, Extended-AIC, and AIC result in more complicated models that do not contribute to the performance of the phylogenetic inference, yet cause a significant increase in the time required for data analysis.     

Spatial Structure and Diffusive Dynamics from Single-Particle Trajectories Using Spline Analysis

Single-particle tracking of biomolecular probes has provided a wealth of information about intracellular trafficking and the dynamics of proteins and lipids in the cell membrane. Conventional mean-square displacement (MSD) analysis of single-particle trajectories often assumes that probes are moving in a uniform environment. However, the observed two-dimensional motion of probe particles is influenced by the local three-dimensional geometry of the cell membrane and intracellular structures, which are rarely flat at the submicron scale. This complex geometry can lead to spatially confined trajectories that are difficult to analyze and interpret using conventional two-dimensional MSD analysis. Here we present two methods to analyze spatially confined trajectories: spline-curve dynamics analysis, which extends conventional MSD analysis to measure diffusive motion in confined trajectories; and spline-curve spatial analysis, which measures spatial structures smaller than the limits of optical resolution. We show, using simulated random walks and experimental trajectories of quantum dot probes, that differences in measured two-dimensional diffusion coefficients do not always reflect differences in underlying diffusive dynamics, but can instead be due to differences in confinement geometries of cellular structures.     

Direct observation of mobility state transitions in RNA trajectories by sensitive single molecule feedback tracking

Observation and tracking of fluorescently labeled molecules and particles in living cells reveals detailed information about intracellular processes on the molecular level. Whereas light microscopic particle observation is usually limited to two-dimensional projections of short trajectory segments, we report here image-based real-time three-dimensional single particle tracking in an active feedback loop with single molecule sensitivity. We tracked particles carrying only 1–3 fluorophores deep inside living tissue with high spatio-temporal resolution. Using this approach, we succeeded to acquire trajectories containing several hundred localizations. We present statistical methods to find significant deviations from random Brownian motion in such trajectories. The analysis allowed us to directly observe transitions in the mobility of ribosomal (r)RNA and Balbiani ring (BR) messenger (m)RNA particles in living Chironomus tentans salivary gland cell nuclei. We found that BR mRNA particles displayed phases of reduced mobility, while rRNA particles showed distinct binding events in and near nucleoli.     

Spatial information analysis of chemotactic trajectories

During bacterial chemotaxis, a cell acquires information about its environment by sampling changes in the local concentration of a chemoattractant, and then uses that information to bias its motion relative to the source of the chemoattractant. The trajectory of a chemotaxing bacteria is thus a spatial manifestation of the information gathered by the cell. Here we show that a recently developed approach for computing spatial information using Fourier coefficient probabilities, the k-space information (kSI), can be used to quantify the information in such trajectories. The kSI is shown to capture expected responses to gradients of a chemoattractant. We then extend the k-space approach by developing an experimental probability distribution (EPD) that is computed from chemotactic trajectories collected under a reference condition. The EPD accounts for connectivity and other constraints that the nature of the trajectories imposes on the k-space computation. The EPD is used to compute the spatial information from any trajectory of interest, relative to the reference condition. The EPD-based spatial information also captures the expected responses to gradients of a chemoattractant, although the results differ in significant ways from the original kSI computation. In addition, the entropy calculated from the EPD provides a useful measure of trajectory space. The methods developed are highly general, and can be applied to a wide range of other trajectory types as well as non-trajectory data.     

Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles

We track single toxin receptors on the apical cell membrane of MDCK cells with Eu-doped oxide nanoparticles coupled to two toxins of the pore-forming toxin family: α-toxin of Clostridium septicum and ε-toxin of Clostridium perfringens. These nonblinking and photostable labels do not perturb the motion of the toxin receptors and yield long uninterrupted trajectories with mean localization precision of 30 nm for acquisition times of 51.3 ms. We were thus able to study the toxin-cell interaction at the single-molecule level. Toxins bind to receptors that are confined within zones of mean area 0.40 ± 0.05 μm(2). Assuming that the receptors move according to the Langevin equation of motion and using Bayesian inference, we determined mean diffusion coefficients of 0.16 ± 0.01 μm(2)/s for both toxin receptors. Moreover, application of this approach revealed a force field within the domain generated by a springlike confining potential. Both toxin receptors were found to experience forces characterized by a mean spring constant of 0.30 ± 0.03 pN/μm at 37°C. Furthermore, both toxin receptors showed similar distributions of diffusion coefficient, domain area, and spring constant. Control experiments before and after incubation with cholesterol oxidase and sphingomyelinase show that these two enzymes disrupt the confinement domains and lead to quasi-free motion of the toxin receptors. Our control data showing cholesterol and sphingomyelin dependence as well as independence of actin depolymerization and microtubule disruption lead us to attribute the confinement of both receptors to lipid rafts. These toxins require oligomerization to develop their toxic activity. The confined nature of the toxin receptors leads to a local enhancement of the toxin monomer concentration and may thus explain the virulence of this toxin family.     

Penalized likelihood and multiple testing

The classical multiple testing model remains an important practical area of statistics with new approaches still being developed. In this paper we develop a new multiple testing procedure inspired by a method sometimes used in a problem with a different focus. Namely, the inference after model selection problem. We note that solutions to that problem are often accomplished by making use of a penalized likelihood function. A classic example is the Bayesian information criterion (BIC) method. In this paper we construct a generalized BIC method and evaluate its properties as a multiple testing procedure. The procedure is applicable to a wide variety of statistical models including regression, contrasts, treatment versus control, change point, and others. Numerical work indicates that, in particular, for sparse models the new generalized BIC would be preferred over existing multiple testing procedures.     

A Primer on the Bayesian Approach to High-Density Single-Molecule Trajectories Analysis

Tracking single molecules in living cells provides invaluable information on their environment and on the interactions that underlie their motion. New experimental techniques now permit the recording of large amounts of individual trajectories, enabling the implementation of advanced statistical tools for data analysis. In this primer, we present a Bayesian approach toward treating these data, and we discuss how it can be fruitfully employed to infer physical and biochemical parameters from single-molecule trajectories.     

Step Detection in Single-Molecule Real Time Trajectories Embedded in Correlated Noise

Single-molecule real time trajectories are embedded in high noise. To extract kinetic or dynamic information of the molecules from these trajectories often requires idealization of the data in steps and dwells. One major premise behind the existing single-molecule data analysis algorithms is the Gaussian ‘white’ noise, which displays no correlation in time and whose amplitude is independent on data sampling frequency. This so-called ‘white’ noise is widely assumed but its validity has not been critically evaluated. We show that correlated noise exists in single-molecule real time trajectories collected from optical tweezers. The assumption of white noise during analysis of these data can lead to serious over- or underestimation of the number of steps depending on the algorithms employed. We present a statistical method that quantitatively evaluates the structure of the underlying noise, takes the noise structure into account, and identifies steps and dwells in a single-molecule trajectory. Unlike existing data analysis algorithms, this method uses Generalized Least Squares (GLS) to detect steps and dwells. Under the GLS framework, the optimal number of steps is chosen using model selection criteria such as Bayesian Information Criterion (BIC). Comparison with existing step detection algorithms showed that this GLS method can detect step locations with highest accuracy in the presence of correlated noise. Because this method is automated, and directly works with high bandwidth data without pre-filtering or assumption of Gaussian noise, it may be broadly useful for analysis of single-molecule real time trajectories.