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.     

Bayesian Uncertainty Quantification for Bond Energies and Mobilities Using Path Integral Analysis

Dynamic single-molecule force spectroscopy is often used to distort bonds. The resulting responses, in the form of rupture forces, work applied, and trajectories of displacements, are used to reconstruct bond potentials. Such approaches often rely on simple parameterizations of one-dimensional bond potentials, assumptions on equilibrium starting states, and/or large amounts of trajectory data. Parametric approaches typically fail at inferring complicated bond potentials with multiple minima, while piecewise estimation may not guarantee smooth results with the appropriate behavior at large distances. Existing techniques, particularly those based on work theorems, also do not address spatial variations in the diffusivity that may arise from spatially inhomogeneous coupling to other degrees of freedom in the macromolecule. To address these challenges, we develop a comprehensive empirical Bayesian approach that incorporates data and regularization terms directly into a path integral. All experimental and statistical parameters in our method are estimated directly from the data. Upon testing our method on simulated data, our regularized approach requires less data and allows simultaneous inference of both complex bond potentials and diffusivity profiles. Crucially, we show that the accuracy of the reconstructed bond potential is sensitive to the spatially varying diffusivity and accurate reconstruction can be expected only when both are simultaneously inferred. Moreover, after providing a means for self-consistently choosing regularization parameters from data, we derive posterior probability distributions, allowing for uncertainty quantification.     

Bayesian Approach to MSD-Based Analysis of Particle Motion in Live Cells

Quantitative tracking of particle motion using live-cell imaging is a powerful approach to understanding the mechanism of transport of biological molecules, organelles, and cells. However, inferring complex stochastic motion models from single-particle trajectories in an objective manner is nontrivial due to noise from sampling limitations and biological heterogeneity. Here, we present a systematic Bayesian approach to multiple-hypothesis testing of a general set of competing motion models based on particle mean-square displacements that automatically classifies particle motion, properly accounting for sampling limitations and correlated noise while appropriately penalizing model complexity according to Occam's Razor to avoid over-fitting. We test the procedure rigorously using simulated trajectories for which the underlying physical process is known, demonstrating that it chooses the simplest physical model that explains the observed data. Further, we show that computed model probabilities provide a reliability test for the downstream biological interpretation of associated parameter values. We subsequently illustrate the broad utility of the approach by applying it to disparate biological systems including experimental particle trajectories from chromosomes, kinetochores, and membrane receptors undergoing a variety of complex motions. This automated and objective Bayesian framework easily scales to large numbers of particle trajectories, making it ideal for classifying the complex motion of large numbers of single molecules and cells from high-throughput screens, as well as single-cell-, tissue-, and organism-level studies.     

Molecular dating for phylogenies containing a mix of populations and species by using Bayesian and RelTime approaches

Simultaneous molecular dating of population and species divergences is essential in many biological investigations, including phylogeography, phylodynamics and species delimitation studies. In these investigations, multiple sequence alignments consist of both intra‐ and interspecies samples (mixed samples). As a result, the phylogenetic trees contain interspecies, interpopulation and within‐population divergences. Bayesian relaxed clock methods are often employed in these analyses, but they assume the same tree prior for both inter‐ and intraspecies branching processes and require specification of a clock model for branch rates (independent vs. autocorrelated rates models). We evaluated the impact of a single tree prior on Bayesian divergence time estimates by analysing computer‐simulated data sets. We also examined the effect of the assumption of independence of evolutionary rate variation among branches when the branch rates are autocorrelated. Bayesian approach with coalescent tree priors generally produced excellent molecular dates and highest posterior densities with high coverage probabilities. We also evaluated the performance of a non‐Bayesian method, RelTime, which does not require the specification of a tree prior or a clock model. RelTime's performance was similar to that of the Bayesian approach, suggesting that it is also suitable to analyse data sets containing both populations and species variation when its computational efficiency is needed.     

A comparison of rarefaction and bayesian methods for predicting the allelic richness of future samples on the basis of currently available samples

Rarefaction methods have been introduced into population genetics (from ecology) for predicting and comparing the allelic richness of future samples (or sometimes populations) on the basis of currently available samples, possibly of different sizes. Here, we focus our attention on one such problem: Predicting which population is most likely to yield the future sample having the highest allelic richness. (This problem can arise when we want to construct a core collection from a larger germplasm collection.) We use extensive simulations to compare the performance of the Monte Carlo rarefaction (repeated random subsampling) method with a simple Bayesian approach we have developed-which is based on the Ewens sampling distribution. We found that neither this Bayesian method nor the (Monte Carlo) rarefaction method performed uniformly better than the other. We also examine briefly some of the other motivations offered for these methods and try to make sense of them from a Bayesian point of view.     

Bayesian statistics for parasitologists

Bayesian statistical methods are increasingly being used in the analysis of parasitological data. Here, the basis of differences between the Bayesian method and the classical or frequentist approach to statistical inference is explained. This is illustrated with practical implications of Bayesian analyses using prevalence estimation of strongyloidiasis and onchocerciasis as two relevant examples. The strongyloidiasis example addresses the problem of parasitological diagnosis in the absence of a gold standard, whereas the onchocerciasis case focuses on the identification of villages warranting priority mass ivermectin treatment. The advantages and challenges faced by users of the Bayesian approach are also discussed and the readers pointed to further directions for a more in-depth exploration of the issues raised. We advocate collaboration between parasitologists and Bayesian statisticians as a fruitful and rewarding venture for advancing applied research in parasite epidemiology and the control of parasitic infections.     

Bayesian Decision Tree for the Classification of the Mode of Motion in Single-Molecule Trajectories

Membrane proteins move in heterogeneous environments with spatially (sometimes temporally) varying friction and with biochemical interactions with various partners. It is important to reliably distinguish different modes of motion to improve our knowledge of the membrane architecture and to understand the nature of interactions between membrane proteins and their environments. Here, we present an analysis technique for single molecule tracking (SMT) trajectories that can determine the preferred model of motion that best matches observed trajectories. The method is based on Bayesian inference to calculate the posteriori probability of an observed trajectory according to a certain model. Information theory criteria, such as the Bayesian information criterion (BIC), the Akaike information criterion (AIC), and modified AIC (AICc), are used to select the preferred model. The considered group of models includes free Brownian motion, and confined motion in 2nd or 4th order potentials. We determine the best information criteria for classifying trajectories. We tested its limits through simulations matching large sets of experimental conditions and we built a decision tree. This decision tree first uses the BIC to distinguish between free Brownian motion and confined motion. In a second step, it classifies the confining potential further using the AIC. We apply the method to experimental Clostridium Perfingens -toxin (CPT) receptor trajectories to show that these receptors are confined by a spring-like potential. An adaptation of this technique was applied on a sliding window in the temporal dimension along the trajectory. We applied this adaptation to experimental CPT trajectories that lose confinement due to disaggregation of confining domains. This new technique adds another dimension to the discussion of SMT data. The mode of motion of a receptor might hold more biologically relevant information than the diffusion coefficient or domain size and may be a better tool to classify and compare different SMT experiments.     

A Bayesian approach for applying Haseman-Elston methods

The main goal of this paper is to couple the Haseman-Elston method with a simple yet effective Bayesian factor-screening approach. This approach selects markers by considering a set of multigenic models that include epistasis effects. The markers are ranked based on their marginal posterior probability. A significant improvement over our previously proposed Bayesian variable selection methodology is a simple Metropolis-Hasting algorithm that requires minimum tuning on the prior settings. The algorithm, however, is also flexible enough for us to easily incorporate our hypotheses and avoid computational pitfalls. We apply our approach to the microsatellite data of Collaborative Studies on Genetics of Alcoholism using the coded values for the ALDX1 variable as our response.     

Centrosome-dependent anisotropic random walk of cytoplasmic vesicles

We approach the problem of an apparently random movement of small cytoplasmic vesicles and its relationship to centrosome functioning. Motion of small vesicles in the cytoplasm of BSC-1 cells was quantified using computer-assisted microscopy. The vesicles move across the cytoplasm frequently changing their directions with negligible net displacement. The autocorrelation function for consecutive velocities of individual vesicles becomes indistinguishable from zero in 10s. Variance in the displacement is proportional to time. The motion of vesicles is anisotropic: It has diffusivity along the radii drawn from the centrosome several times higher than the tangential diffusivity. This anisotropy is abolished by ultraviolet microbeam irradiation of the centrosome when the microtubule array loses radial structure. We conclude that the motion of the vesicles in the cytoplasm can be described as diffusion-like random walk with centrosome-dependent anisotropy. The present analysis quantitatively corroborates the 'trial and error' model of vesicular transport.     

A Bayesian model comparison approach to inferring positive selection

A popular approach to detecting positive selection is to estimate the parameters of a probabilistic model of codon evolution and perform inference based on its maximum likelihood parameter values. This approach has been evaluated intensively in a number of simulation studies and found to be robust when the available data set is large. However, uncertainties in the estimated parameter values can lead to errors in the inference, especially when the data set is small or there is insufficient divergence between the sequences. We introduce a Bayesian model comparison approach to infer whether the sequence as a whole contains sites at which the rate of nonsynonymous substitution is greater than the rate of synonymous substitution. We incorporated this probabilistic model comparison into a Bayesian approach to site-specific inference of positive selection. Using simulated sequences, we compared this approach to the commonly used empirical Bayes approach and investigated the effect of tree length on the performance of both methods. We found that the Bayesian approach outperforms the empirical Bayes method when the amount of sequence divergence is small and is less prone to false-positive inference when the sequences are saturated, while the results are indistinguishable for intermediate levels of sequence divergence.     

Hydrodynamic interactions between two swimming bacteria

This article evaluates the hydrodynamic interactions between two swimming bacteria precisely. We assume that each bacterium is force free and torque free, with a Stokes flow field around it. The geometry of each bacterium is modeled as a spherical or spheroidal body with a single helical flagellum. The movements of two interacting bacteria in an infinite fluid otherwise at rest are computed using a boundary element method, and the trajectories of the two interacting bacteria and the stresslet are investigated. The results show that as the two bacteria approach each other, they change their orientations considerably in the near field. The bacteria always avoided each other; no stable pairwise swimming motion was observed in this study. The effects of the hydrodynamic interactions between two bacteria on the rheology and diffusivity of a semidilute bacterial suspension are discussed.     

Bayesian Wavelet Shrinkage of the Haar-Fisz Transformed Wavelet Periodogram

It is increasingly being realised that many real world time series are not stationary and exhibit evolving second-order autocovariance or spectral structure. This article introduces a Bayesian approach for modelling the evolving wavelet spectrum of a locally stationary wavelet time series. Our new method works by combining the advantages of a Haar-Fisz transformed spectrum with a simple, but powerful, Bayesian wavelet shrinkage method. Our new method produces excellent and stable spectral estimates and this is demonstrated via simulated data and on differenced infant electrocardiogram data. A major additional benefit of the Bayesian paradigm is that we obtain rigorous and useful credible intervals of the evolving spectral structure. We show how the Bayesian credible intervals provide extra insight into the infant electrocardiogram data.     

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.     

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.     

A Bayesian network approach incorporating imputation of missing data enables exploratory analysis of complex causal biological relationships

Bayesian networks can be used to identify possible causal relationships between variables based on their conditional dependencies and independencies, which can be particularly useful in complex biological scenarios with many measured variables. Here we propose two improvements to an existing method for Bayesian network analysis, designed to increase the power to detect potential causal relationships between variables (including potentially a mixture of both discrete and continuous variables). Our first improvement relates to the treatment of missing data. When there is missing data, the standard approach is to remove every individual with any missing data before performing analysis. This can be wasteful and undesirable when there are many individuals with missing data, perhaps with only one or a few variables missing. This motivates the use of imputation. We present a new imputation method that uses a version of nearest neighbour imputation, whereby missing data from one individual is replaced with data from another individual, their nearest neighbour. For each individual with missing data, the subsets of variables to be used to select the nearest neighbour are chosen by sampling without replacement the complete data and estimating a best fit Bayesian network. We show that this approach leads to marked improvements in the recall and precision of directed edges in the final network identified, and we illustrate the approach through application to data from a recent study investigating the causal relationship between methylation and gene expression in early inflammatory arthritis patients. We also describe a second improvement in the form of a pseudo-Bayesian approach for upweighting certain network edges, which can be useful when there is prior evidence concerning their directions.     

Stochastic Analysis of the Motion of DNA Nanomechanical Bipeds

In this paper, we formulate and analyze a Markov process modeling the motion of DNA nanomechanical walking devices. We consider a molecular biped restricted to a well-defined one-dimensional track and study its asymptotic behavior. Our analysis allows for the biped legs to be of different molecular composition, and thus to contribute differently to the dynamics. Our main result is a functional central limit theorem for the biped with an explicit formula for the effective diffusivity coefficient in terms of the parameters of the model. A law of large numbers, a recurrence/transience characterization and large deviations estimates are also obtained. Our approach is applicable to a variety of other biological motors such as myosin and motor proteins on polymer filaments.     

Inference on fractal processes using multiresolution approximation

We consider Bayesian inference via Markov chain Monte Carlofor a variety of fractal Gaussian processes on the real line.These models have unknown parameters in the covariance matrix,requiring inversion of a new covariance matrix at each Markovchain Monte Carlo iteration. The processes have no suitableindependence properties so this becomes computationally prohibitive.We surmount these difficulties by developing a computationalalgorithm for likelihood evaluation based on a ‘multiresolutionapproximation’ to the original process. The method iscomputationally very efficient and widely applicable, makinglikelihood-based inference feasible for large datasets. A simulationstudy indicates that this approach leads to accurate estimatesfor underlying parameters in fractal models, including fractionalBrownian motion and fractional Gaussian noise, and functionalparameters in the recently introduced multifractional Brownianmotion. We apply the method to a variety of real datasets andillustrate its application to prediction and to model selection.