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.     

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.     

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.     

Analysis of Single Locus Trajectories for Extracting In Vivo Chromatin Tethering Interactions

Is it possible to extract tethering forces applied on chromatin from the statistics of a single locus trajectories imaged in vivo? Chromatin fragments interact with many partners such as the nuclear membrane, other chromosomes or nuclear bodies, but the resulting forces cannot be directly measured in vivo. However, they impact chromatin dynamics and should be reflected in particular in the motion of a single locus. We present here a method based on polymer models and statistics of single trajectories to extract the force characteristics and in particular when they are generated by the gradient of a quadratic potential well. Using numerical simulations of a Rouse polymer and live cell imaging of the MAT-locus located on the yeast Saccharomyces cerevisiae chromosome III, we recover the amplitude and the distance between the observed and the interacting monomer. To conclude, the confined trajectories we observed in vivo reflect local interaction on chromatin.     

A Hidden Markov Model for Single Particle Tracks Quantifies Dynamic Interactions between LFA-1 and the Actin Cytoskeleton

The extraction of hidden information from complex trajectories is a continuing problem in single-particle and single-molecule experiments. Particle trajectories are the result of multiple phenomena, and new methods for revealing changes in molecular processes are needed. We have developed a practical technique that is capable of identifying multiple states of diffusion within experimental trajectories. We model single particle tracks for a membrane-associated protein interacting with a homogeneously distributed binding partner and show that, with certain simplifying assumptions, particle trajectories can be regarded as the outcome of a two-state hidden Markov model. Using simulated trajectories, we demonstrate that this model can be used to identify the key biophysical parameters for such a system, namely the diffusion coefficients of the underlying states, and the rates of transition between them. We use a stochastic optimization scheme to compute maximum likelihood estimates of these parameters. We have applied this analysis to single-particle trajectories of the integrin receptor lymphocyte function-associated antigen-1 (LFA-1) on live T cells. Our analysis reveals that the diffusion of LFA-1 is indeed approximately two-state, and is characterized by large changes in cytoskeletal interactions upon cellular activation.     

Extracting the Stepping Dynamics of Molecular Motors in Living Cells from Trajectories of Single Particles

Molecular motors are responsible of transporting a wide variety of cargos in the cytoplasm. Current efforts are oriented to characterize the biophysical properties of motors in cells with the aim of elucidating the mechanisms of these nanomachines in the complex cellular environment. In this study, we present an algorithm designed to extract motor step sizes and dwell times between steps from trajectories of motors or cargoes driven by motors in cells. The algorithm is based on finding patterns in the trajectory compatible with the behavior expected for a motor step, i.e., a region of confined motion followed by a jump in the position to another region of confined motion with similar characteristics to the previous one. We show that this algorithm allows the analysis of 2D trajectories even if they present complex motion patterns such as active transport interspersed with diffusion and does not require the assumption of a given step size or dwell period. The confidence on the step detection can be easily obtained and allows the evaluation of the confidence of the dwell and step size distributions. To illustrate the possible applications of this algorithm, we analyzed trajectories of myosin-V driven organelles in living cells.     

Analysis and Interpretation of Superresolution Single-Particle Trajectories

A large number (tens of thousands) of single molecular trajectories on a cell membrane can now be collected by superresolution methods. The data contains information about the diffusive motion of molecule, proteins, or receptors and here we review methods for its recovery by statistical analysis of the data. The information includes the forces, organization of the membrane, the diffusion tensor, the long-time behavior of the trajectories, and more. To recover the long-time behavior and statistics of long trajectories, a stochastic model of their nonequilibrium motion is required. Modeling and data analysis serve extracting novel biophysical features at an unprecedented spatiotemporal resolution. The review presents data analysis, modeling, and stochastic simulations applied in particular on surface receptors evolving in neuronal cells.     

Dynamic modelling for implementation of a right turn in bipedal walking

This paper deals with the development of a conceptual model for the control of a multilink biped during a turning maneuver. The skeletal model is a seven link biped for which the equations of motion are derived. A set of lower limb muscles are idealized by simple force actuators with no co-contraction of agonist-antagonist muscle pairs. A nonlinear control scheme is proposed to guide the model along the desired trajectory and to control ground reaction forces. The input to the system is a desired set of trajectories as functions of time and the patterns of desired ground reaction forces in a turn. One set of such inputs are inferred from the existing literature. With this input, the nonlinear control strategy allows computation of muscular forces needed for the turning maneuver.     

Mechanisms underlying the generation of averaged modified trajectories

In this work we have studied what mechanisms might possibly underlie arm trajectory modification when reaching toward visual targets. The double-step target displacement paradigm was used with inter-stimulus intervals (ISIs) in the range of 10-300 ms. For short ISIs, a high percentage of the movements were found to be initially directed in between the first and second target locations (averaged trajectories). The initial direction of motion was found to depend on the target configuration, and on : the time difference between the presentation of the second stimulus and movement onset. To account for the kinematic features of the averaged trajectories two modification schemes were compared: the superposition scheme and the abort-replan scheme. According to the superposition scheme, the modified trajectories result from the vectorial addition of two elemental motions: one for moving between the initial hand position and an intermediate location, and a second one for moving between that intermediate location and the final target. According to the abort-replan scheme, the initial plan for moving toward the intermediate location is aborted and smoothly replaced by a new plan for moving from the hand position at the time the trajectory is modified to the final target location. In both tested schemes we hypothesized that due to the quick displacement of the stimulus, the internally specified intermediate goal might be influenced by both stimuli and may be different from the location of the first stimulus. It was found that the statistically most successful model in accounting for the measured data is based on the superposition scheme. It is suggested that superposition of simple independent elemental motions might be a general principle for the generation of modified motions, which allows for efficient, parallel planning. For increasing values of the inferred locations of the intermediate targets were found to gradually shift from the first toward the second target locations along a path that curved toward the initial hand position. These inferred locations show a strong resemblance to the intermediate locations of saccades generated in a similar double-step paradigm. These similarities in the specification of target locations used in the generation of eye and hand movements may serve to simplify visuomotor integration. Received: 22 June 1994 / Accepted in revised form: 15 September 1994     

Predicting human perceptual decisions by decoding neuronal information profiles

Perception relies on the response of populations of neurons in sensory cortex. How the response profile of a neuronal population gives rise to perception and perceptual discrimination has been conceptualized in various ways. Here we suggest that neuronal population responses represent information about our environment explicitly as Fisher information (FI), which is a local measure of the variance estimate of the sensory input. We show how this sensory information can be read out and combined to infer from the available information profile which stimulus value is perceived during a fine discrimination task. In particular, we propose that the perceived stimulus corresponds to the stimulus value that leads to the same information for each of the alternative directions, and compare the model prediction to standard models considered in the literature (population vector, maximum likelihood, maximum-a-posteriori Bayesian inference). The models are applied to human performance in a motion discrimination task that induces perceptual misjudgements of a target direction of motion by task irrelevant motion in the spatial surround of the target stimulus (motion repulsion). By using the neurophysiological insight that surround motion suppresses neuronal responses to the target motion in the center, all models predicted the pattern of perceptual misjudgements. The variation of discrimination thresholds (error on the perceived value) was also explained through the changes of the total FI content with varying surround motion directions. The proposed FI decoding scheme incorporates recent neurophysiological evidence from macaque visual cortex showing that perceptual decisions do not rely on the most active neurons, but rather on the most informative neuronal responses. We statistically compare the prediction capability of the FI decoding approach and the standard decoding models. Notably, all models reproduced the variation of the perceived stimulus values for different surrounds, but with different neuronal tuning characteristics underlying perception. Compared to the FI approach the prediction power of the standard models was based on neurons with far wider tuning width and stronger surround suppression. Our study demonstrates that perceptual misjudgements can be based on neuronal populations encoding explicitly the available sensory information, and provides testable neurophysiological predictions on neuronal tuning characteristics underlying human perceptual decisions.     

A family of evolution-entropy hybrid methods for ranking protein residues by importance

In order to identify the amino acids that determine protein structure and function it is useful to rank them by their relative importance. Previous approaches belong to two groups; those that rely on statistical inference, and those that focus on phylogenetic analysis. Here, we introduce a class of hybrid methods that combine evolutionary and entropic information from multiple sequence alignments. A detailed analysis in insulin receptor kinase domain and tests on proteins that are well-characterized experimentally show the hybrids' greater robustness with respect to the input choice of sequences, as well as improved sensitivity and specificity of prediction. This is a further step toward proteome scale analysis of protein structure and function.     

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.     

Variational Bayes Analysis of a Photon-Based Hidden Markov Model for Single-Molecule FRET Trajectories

Single-molecule fluorescence resonance energy transfer (smFRET) measurement is a powerful technique for investigating dynamics of biomolecules, for which various efforts have been made to overcome significant stochastic noise. Time stamp (TS) measurement has been employed experimentally to enrich information within the signals, while data analyses such as the hidden Markov model (HMM) have been successfully applied to recover the trajectories of molecular state transitions from time-binned photon counting signals or images. In this article, we introduce the HMM for TS-FRET signals, employing the variational Bayes (VB) inference to solve the model, and demonstrate the application of VB-HMM-TS-FRET to simulated TS-FRET data. The same analysis using VB-HMM is conducted for other models and the previously reported change point detection scheme. The performance is compared to other analysis methods or data types and we show that our VB-HMM-TS-FRET analysis can achieve the best performance and results in the highest time resolution. Finally, an smFRET experiment was conducted to observe spontaneous branch migration of Holliday-junction DNA. VB-HMM-TS-FRET was successfully applied to reconstruct the state transition trajectory with the number of states consistent with the nucleotide sequence. The results suggest that a single migration process frequently involves rearrangement of multiple basepairs.     

Effect of a pedicle-screw-based motion preservation system on lumbar spine biomechanics: A probabilistic finite element study with subsequent sensitivity analysis

Pedicle-screw-based motion preservation systems are often used to support a slightly degenerated disc. Such implants are intended to reduce intradiscal pressure and facet joints forces, while having a minimal effect on the motion patterns.In a probabilistic finite element study with subsequent sensitivity analysis, the effects of 10 input parameters, such as elastic modulus and diameter of the elastic rod, distraction of the segment, level of bridged segments, etc. on the output parameters intervertebral rotations, intradiscal pressures, and facet joint forces were determined. A validated finite element model of the lumbar spine was employed. Probabilistic studies were performed for seven loading cases: upright standing, flexion, extension, left and right lateral bending and left and right axial rotation.The simulations show that intervertebral rotation angles, intradiscal pressures and facet joint forces are in most cases reduced by a motion preservation system. The influence on intradiscal pressure is small, except in extension. For many input parameter combinations, the values for intervertebral rotations and facet joint forces are very low, which indicates that the implant is too stiff in these cases. The output parameters are affected most by the following input parameters: loading case, elastic modulus and diameter of the elastic rod, distraction of the segment, and angular rigidity of the connection between screws and rod.The designated functions of a motion preservation system can best be achieved when the longitudinal rod has a low stiffness, and when the connection between rod and pedicle screws is rigid.     

Homologous control of protein signaling networks

In a previous paper we introduced a method called augmented sparse reconstruction (ASR) that identifies links among nodes of ordinary differential equation networks, given a small set of observed trajectories with various initial conditions. The main purpose of that technique was to reconstruct intracellular protein signaling networks.In this paper we show that a recursive augmented sparse reconstruction generates artificial networks that are homologous to a large, reference network, in the sense that kinase inhibition of several reactions in the network alters the trajectories of a sizable number of proteins in comparable ways for reference and reconstructed networks. We show this result using a large in-silico model of the epidermal growth factor receptor (EGF-R) driven signaling cascade to generate the data used in the reconstruction algorithm.The most significant consequence of this observed homology is that a nearly optimal combinatorial dosage of kinase inhibitors can be inferred, for many nodes, from the reconstructed network, a result potentially useful for a variety of applications in personalized medicine.     

What can one learn from two-state single-molecule trajectories?

A time trajectory of an observable that fluctuates between two values (say, on and off), stemming from some unknown multisubstate kinetic scheme, is the output of many single-molecule experiments. Here we show that when all successive waiting times along the trajectory are uncorrelated the on and the off waiting time probability density functions contain all the information. By relating the lack of correlation in the trajectory to the topology of kinetic schemes, we can immediately specify those kinetic schemes that are equally consistent with experiment, and cannot be differentiated by any sophisticated analyses of the trajectory. Correlated trajectories, however, contain additional information about the underlying kinetic scheme, and we consider the strategy that one should use to extract it.     

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.     

Stamping Vital Cells—a Force-Based Ligand Receptor Assay

Gaining information about receptor profiles on cells, and subsequently finding the most efficient ligands for these signaling receptors, remain challenging tasks in stem cell and cancer research as well as drug development. We introduce a live-cell method with great potential in both screening for surface receptors and analysing binding forces of different ligands. The technique is based on the molecular force assay, a parallel-format, high-throughput experiment on a single-molecule level. On human red blood cells, we demonstrate the detection of the interaction of N-acetyl-α-D-galactosaminyl residues with the lectin helix pomatia agglutinine and of the CD47 receptor with its antibody. The measurements are performed under nearly physiological conditions and still provide a highly specific binding signal. Moreover, with a detailed comparative force analysis on two cell types with different morphology, we show that our method even allows the determination of a DNA force equivalent for the interaction of the CD47 receptor and its antibody.     

Hydration effects on the electrostatic potential around tuftsin

The electrostatic potential and component dielectric constants from molecular dynamics (MD) trajectories of tuftsin, a tetrapeptide with the amino acid sequence Thr–Lys–Pro–Arg in water and in saline solution are presented. The results obtained from the analysis of the MD trajectories for the total electrostatic potential at points on a grid using the Ewald technique are compared with the solution to the Poisson–Boltzmann (PB) equation. The latter was solved using several sets of dielectric constant parameters. The effects of structural averaging on the PB results were also considered. Solute conformational mobility in simulations gives rise to an electrostatic potential map around the solute dominated by the solute monopole (or lowest order multipole). The detailed spatial variation of the electrostatic potential on the molecular surface brought about by the compounded effects of the distribution of water and ions close to the peptide, solvent mobility, and solute conformational mobility are not qualitatively reproducible from a reparametrization of the input solute and solvent dielectric constants to the PB equation for a single structure or for structurally averaged PB calculations. Nevertheless, by fitting the PB to the MD electrostatic potential surfaces with the dielectric constants as fitting parameters, we found that the values that give the best fit are the values calculated from the MD trajectories. Implications of using such field calculations on the design of tuftsin peptide analogues are discussed. © 1999 John Wiley & Sons, Inc. Biopoly 50: 133–143, 1999