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

Confinement energy landscape classification reveals membrane receptor nano-organization mechanisms

The cell membrane organization has an essential functional role through the control of membrane receptor confinement in micro- or nanodomains. Several mechanisms have been proposed to account for these properties, although some features have remained controversial, notably the nature, size, and stability of cholesterol- and sphingolipid-rich domains or lipid rafts. Here, we probed the effective energy landscape acting on single-nanoparticle-labeled membrane receptors confined in raft nanodomains- epidermal growth factor receptor (EGFR), Clostridium perfringens ε-toxin receptor (CPεTR), and Clostridium septicum α-toxin receptor (CSαTR)-and compared it with hop-diffusing transferrin receptors. By establishing a new analysis pipeline combining Bayesian inference, decision trees, and clustering approaches, we systematically classified single-protein trajectories according to the type of effective confining energy landscape. This revealed the existence of only two distinct organization modalities: confinement in a quadratic energy landscape for EGFR, CPεTR, and CSαTR (A), and free diffusion in confinement domains resulting from the steric hindrance due to F-actin barriers for transferrin receptor (B). The further characterization of effective confinement energy landscapes by Bayesian inference revealed the role of interactions with the domain environment in cholesterol- and sphingolipid-rich domains with (EGFR) or without (CPεTR and CSαTR) interactions with F-actin to regulate the confinement energy depth. These two distinct mechanisms result in the same organization type (A). We revealed that the apparent domain sizes for these receptor trajectories resulted from Brownian exploration of the energy landscape in a steady-state-like regime at a common effective temperature, independently of the underlying molecular mechanisms. These results highlight that confinement domains may be adequately described as interaction hotspots rather than rafts with abrupt domain boundaries. Altogether, these results support a new model for functional receptor confinement in membrane nanodomains and pave the way to the constitution of an atlas of membrane protein organization.

Estimating Position-Dependent and Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories: Existing Methods and Future Outlook

Confinement can substantially alter the physicochemical properties of materials by breaking translational isotropy and rendering all physical properties position-dependent. Molecular dynamics (MD) simulations have proven instrumental in characterizing such spatial heterogeneities and probing the impact of confinement on materials' properties. For static properties, this is a straightforward task and can be achieved via simple spatial binning. Such an approach, however, cannot be readily applied to transport coefficients due to lack of natural extensions of autocorrelations used for their calculation in the bulk. The prime example of this challenge is diffusivity, which, in the bulk, can be readily estimated from the particles' mobility statistics, which satisfy the Fokker-Planck equation. Under confinement, however, such statistics will follow the Smoluchowski equation, which lacks a closed-form analytical solution. This brief review explores the rich history of estimating profiles of the diffusivity tensor from MD simulations and discusses various approximate methods and algorithms developed for this purpose. Besides discussing heuristic extensions of bulk methods, we overview more rigorous algorithms, including kernel-based methods, Bayesian approaches, and operator discretization techniques. Additionally, we outline methods based on applying biasing potentials or imposing constraints on tracer particles. Finally, we discuss approaches that estimate diffusivity from mean first passage time or committor probability profiles, a conceptual framework originally developed in the context of collective variable spaces describing rare events in computational chemistry and biology. In summary, this paper offers a concise survey of diverse approaches for estimating diffusivity from MD trajectories, highlighting challenges and opportunities in this area.

Learning Continuous 2D Diffusion Maps from Particle Trajectories without Data Binning

Diffusion coefficients often vary across regions, such as cellular membranes, and quantifying their variation can provide valuable insight into local membrane properties such as composition and stiffness. Toward quantifying diffusion coefficient spatial maps and uncertainties from particle tracks, we use a Bayesian method and place Gaussian Process (GP) Priors on the maps. For the sake of computational efficiency, we leverage inducing point methods on GPs arising from the mathematical structure of the data giving rise to non-conjugate likelihood-prior pairs. We analyze both synthetic data, where ground truth is known, as well as data drawn from live-cell single-molecule imaging of membrane proteins. The resulting tool provides an unsupervised method to rigorously map diffusion coefficients continuously across membranes without data binning.

Anisotropic molecular diffusion in confinement I: Transport of small particles in potential and density gradients

HYPOTHESIS

Diffusion in confinement is an important fundamental problem with significant implications for applications of supported liquid phases. However, resolving the spatially dependent diffusion coefficient, parallel and perpendicular to interfaces, has been a standing issue. In the vicinity of interfaces, density fluctuations as a consequence of layering locally impose statistical drift, which impedes the analysis of spatially dependent diffusion coefficients even further. We hypothesise, that we can derive a model to spatially resolve interface-perpendicular diffusion coefficients based on local lifetime statistics with an extension to explicitly account for the effect of local drift using the Smoluchowski equation, that allows us to resolve anisotropic and spatially dependent diffusivity landscapes at interfaces.

METHODS AND SIMULATIONS

An analytic relation between local crossing times in system slices and diffusivity as well as an explicit term for calculating drift-induced systematic errors is presented. The method is validated on Molecular Dynamics simulations of bulk water and applied to simulations of water in slit pores.

FINDINGS

After validation on bulk liquids, we clearly demonstrate the anisotropic nature of diffusion coefficients at interfaces. Significant spatial variations in the diffusivities correlate with interface-induced structuring but cannot be solely attributed to the drift induced by local density fluctuations.

Single-molecule tracking to determine the abundances and stoichiometries of freely-diffusing protein complexes in living cells: Past applications and future prospects

Most biological processes in living cells rely on interactions between proteins. Live-cell compatible approaches that can quantify to what extent a given protein participates in homo- and hetero-oligomeric complexes of different size and subunit composition are therefore critical to advance our understanding of how cellular physiology is governed by these molecular interactions. Biomolecular complex formation changes the diffusion coefficient of constituent proteins, and these changes can be measured using fluorescence microscopy-based approaches, such as single-molecule tracking, fluorescence correlation spectroscopy, and fluorescence recovery after photobleaching. In this review, we focus on the use of single-molecule tracking to identify, resolve, and quantify the presence of freely-diffusing proteins and protein complexes in living cells. We compare and contrast different data analysis methods that are currently employed in the field and discuss experimental designs that can aid the interpretation of the obtained results. Comparisons of diffusion rates for different proteins and protein complexes in intracellular aqueous environments reported in the recent literature reveal a clear and systematic deviation from the Stokes-Einstein diffusion theory. While a complete and quantitative theoretical explanation of why such deviations manifest is missing, the available data suggest the possibility of weighing freely-diffusing proteins and protein complexes in living cells by measuring their diffusion coefficients. Mapping individual diffusive states to protein complexes of defined molecular weight, subunit stoichiometry, and structure promises to provide key new insights into how protein-protein interactions regulate protein conformational, translational, and rotational dynamics, and ultimately protein function.

Quantifying postsynaptic receptor dynamics: insights into synaptic function

The molecular composition of presynaptic and postsynaptic neuronal terminals is dynamic, and yet long-term stabilizations in postsynaptic responses are necessary for synaptic development and long-term plasticity. The need to reconcile these concepts is further complicated by learning- and memory-related plastic changes in the molecular make-up of synapses. Advances in single-particle tracking mean that we can now quantify the number and diffusive properties of specific synaptic molecules, while statistical thermodynamics provides a framework to analyse these molecular fluctuations. In this Review, we discuss the use of these approaches to gain quantitative descriptions of the processes underlying the turnover, long-term stability and plasticity of postsynaptic receptors and show how these can help us to understand the balance between local molecular turnover and synaptic structural identity and integrity.

Global diffusion of hydrogen molecules in the heterogeneous structure of polymer electrolytes for fuel cells: Dynamic Monte Carlo combined with molecular dynamics calculations

Using our recently developed dynamic Monte Carlo (MC) method [Nagai et al., J. Chem Phys. 156, 154506 (2022)], we investigated global diffusion of hydrogen molecules over structural heterogeneities of polymer electrolyte membranes in fuel cells. The three-dimensional position-dependent free energies and the diffusion constants of the hydrogen molecules, required by the present dynamic MC calculations, were taken from our previous study [Nagai et al., J. Chem. Phys. 156, 044507 (2022)] and newly evaluated in this work, respectively. The calculations enabled evaluating the hydrogen dynamics over long-time scales, including global diffusion constants. Based on the calculated global diffusion constants and free energies, the permeability of hydrogen molecules was estimated via the solubility-diffusion model. The estimated values were in good agreement with reported experimental data, thus validating the present methodology. The analysis of the Monte Carlo trajectories indicated that the main permeation paths are located in the polymer and interfacial phases, although the water phase may make a non-negligible contribution to mass transport.

Inferring potential landscapes from noisy trajectories of particles within an optical feedback trap

While particle trajectories encode information on their governing potentials, potentials can be challenging to robustly extract from trajectories. Measurement errors may corrupt a particle’s position, and sparse sampling of the potential limits data in higher energy regions such as barriers. We develop a Bayesian method to infer potentials from trajectories corrupted by Markovian measurement noise without assuming prior functional form on the potentials. As an alternative to Gaussian process priors over potentials, we introduce structured kernel interpolation to the Natural Sciences which allows us to extend our analysis to large datasets. Structured-Kernel-Interpolation Priors for Potential Energy Reconstruction (SKIPPER) is validated on 1D and 2D experimental trajectories for particles in a feedback trap.

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A feedback trap was used to generate noisy Langevin microbead trajectories

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The potential energy surface is recovered using a Bayesian formulation

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The formulation uses a structured-kernel-interpolation Gaussian process (SKI-GP) to tractably approximate Gaussian process regression for larger datasets

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Thanks to our adaptation of SKI-GP, we have broadened the use of Gaussian processes for natural science applications

Dynamic Monte Carlo calculation generating particle trajectories that satisfy the diffusion equation for heterogeneous systems with a position-dependent diffusion coefficient and free energy

A series of new Monte Carlo (MC) transition probabilities was investigated that could produce molecular trajectories statistically satisfying the diffusion equation with a position-dependent diffusion coefficient and potential energy. The MC trajectories were compared with the numerical solution of the diffusion equation by calculating the time evolution of the probability distribution and the mean first passage time, which exhibited excellent agreement. The method is powerful when investigating, for example, the long-distance and long-time global transportation of a molecule in heterogeneous systems by coarse-graining them into one-particle diffusive molecular motion with a position-dependent diffusion coefficient and free energy. The method can also be applied to many-particle dynamics.

Momentum removal to obtain the position-dependent diffusion constant in constrained molecular dynamics simulation

The position-dependent diffusion coefficient along with free energy profile are important parameters needed to study mass transport in heterogeneous systems such as biological and polymer membranes, and molecular dynamics (MD) calculation is a popular tool to obtain them. Among many methodologies, the Marrink-Berendsen (MB) method is often employed to calculate the position-dependent diffusion coefficient, in which the autocorrelation function of the force on a fixed molecule is related to the friction on the molecule. However, the diffusion coefficient is shown to be affected by the period of the removal of the center-of-mass velocity, τ v 0 , which is necessary when performing MD calculations using the Ewald method for Coulombic interaction. We have clarified theoretically in this study how this operation affects the diffusion coefficient calculated by the MB method, and the theoretical predictions are proven by MD calculations. Therefore, we succeeded in providing guidance on how to select an appropriate τ v 0 value in estimating the position-dependent diffusion coefficient by the MB method. This guideline is applicable also to the Woolf-Roux method.

Inferring effective forces for Langevin dynamics using Gaussian processes

Effective forces derived from experimental or in silico molecular dynamics time traces are critical in developing reduced and computationally efficient descriptions of otherwise complex dynamical problems. This helps motivate why it is important to develop methods to efficiently learn effective forces from time series data. A number of methods already exist to do this when data are plentiful but otherwise fail for sparse datasets or datasets where some regions of phase space are undersampled. In addition, any method developed to learn effective forces from time series data should be minimally a priori committal as to the shape of the effective force profile, exploit every data point without reducing data quality through any form of binning or pre-processing, and provide full credible intervals (error bars) about the prediction for the entirety of the effective force curve. Here, we propose a generalization of the Gaussian process, a key tool in Bayesian nonparametric inference and machine learning, which meets all of the above criteria in learning effective forces for the first time.

Errors in Energy Landscapes Measured with Particle Tracking

Tracking Brownian particles is often employed to map the energy landscape they explore. Such measurements have been exploited to study many biological processes and interactions in soft materials. Yet video tracking is irremediably contaminated by localization errors originating from two imaging artifacts: the "static" errors come from signal noise, and the "dynamic" errors arise from the motion blur due to finite frame-acquisition time. We show that these errors result in systematic and nontrivial biases in the measured energy landscapes. We derive a relationship between the true and the measured potential that elucidates, among other aberrations, the presence of false double-well minima in the apparent potentials reported in recent studies. We further assess several canonical trapping and pair-interaction potentials by using our analytically derived results and Brownian dynamics simulations. In particular, we show that the apparent spring stiffness of harmonic potentials (such as optical traps) is increased by dynamic errors but decreased by static errors. Our formula allows for the development of efficient corrections schemes, and we also present in this work a provisional method for reconstructing true potentials from the measured ones.

High-Resolution Nanoparticle Sizing with Maximum A Posteriori Nanoparticle Tracking Analysis

The rapid and efficient characterization of polydisperse nanoparticle dispersions remains a challenge within nanotechnology and biopharmaceuticals. Current methods for particle sizing, such as dynamic light scattering, analytical ultracentrifugation, and field-flow fractionation, can suffer from a combination of statistical biases, difficult sample preparation, insufficient sampling, and ill-posed data analysis. As an alternative, we introduce a Bayesian method that we call maximum a posteriori nanoparticle tracking analysis (MApNTA) for estimating the size distributions of nanoparticle samples from high-throughput single-particle tracking experiments. We derive unbiased statistical models for two observable quantities in a typical nanoparticle trajectory-the mean square displacement and the trajectory length-as a function of the particle size and calculate size distributions using maximum a posteriori (MAP) estimation with cross validation to mildly regularize solutions. We show that this approach infers nanoparticle size distributions with high resolution by performing extensive Brownian dynamics simulations and experiments with mono- and polydisperse solutions of gold nanoparticles as well as single-walled carbon nanotubes. We further demonstrate particular utility for characterizing minority components and impurity populations and highlight this ability with the identification of an impurity in a commercially produced gold nanoparticle sample. Modern algorithms such as MApNTA should find widespread use in the routine characterization of complex nanoparticle dispersions, allowing for significant advances in nanoparticle synthesis, separation, and functionalization.

Communication: Coordinate-dependent diffusivity from single molecule trajectories

Single-molecule observations of biomolecular folding are commonly interpreted using the model of one-dimensional diffusion along a reaction coordinate, with a coordinate-independent diffusion coefficient. Recent analysis, however, suggests that more general models are required to account for single-molecule measurements performed with high temporal resolution. Here, we consider one such generalization: a model where the diffusion coefficient can be an arbitrary function of the reaction coordinate. Assuming Brownian dynamics along this coordinate, we derive an exact expression for the coordinate-dependent diffusivity in terms of the splitting probability within an arbitrarily chosen interval and the mean transition path time between the interval boundaries. This formula can be used to estimate the effective diffusion coefficient along a reaction coordinate directly from single-molecule trajectories.

Niemann-Pick C2 protein regulates sterol transport between plasma membrane and late endosomes in human fibroblasts

Niemann-Pick disease type C2 is a lipid storage disorder in which mutations in the NPC2 protein cause accumulation of lipoprotein-derived cholesterol in late endosomes and lysosomes (LE/LYSs). Whether cholesterol delivered by other means to NPC2 deficient cells also accumulates in LE/LYSs is currently unknown. We show that the close cholesterol analog dehydroergosterol (DHE), when delivered to the plasma membrane (PM) accumulates in LE/LYSs of human fibroblasts lacking functional NPC2. We measured two different time scales of sterol diffusion; while DHE rich LE/LYSs moved by slow anomalous diffusion in disease cells (D ∼ 4.6∙10^-4 μm²/sec; α∼0.76), a small pool of sterol could exchange rapidly with D ∼ 3 μm²/s between LE/LYSs, as shown by fluorescence recovery after photobleaching (FRAP). By quantitative lipid mass spectrometry we found that esterification of ¹³C-labeled cholesterol but not of DHE is reduced 10-fold in disease fibroblasts compared to control cells. Internalized NPC2 rescued the sterol storage phenotype and strongly expanded the dynamic sterol pool seen in FRAP experiments. Together, our study shows that cholesterol esterification and trafficking of sterols between the PM and LE/LYSs depends on a functional NPC2 protein. NPC2 likely acts inside LE/LYSs from where it increases non-vesicular sterol exchange with other organelles.

Bayesian inference with information content model check for Langevin equations

The Bayesian data analysis framework has been proven to be a systematic and effective method of parameter inference and model selection for stochastic processes. In this work, we introduce an information content model check that may serve as a goodness-of-fit, like the χ^{2} procedure, to complement conventional Bayesian analysis. We demonstrate this extended Bayesian framework on a system of Langevin equations, where coordinate-dependent mobilities and measurement noise hinder the normal mean-squared displacement approach.

Single-Molecule Imaging of Nav1.6 on the Surface of Hippocampal Neurons Reveals Somatic Nanoclusters

Voltage-gated sodium (Nav) channels are responsible for the depolarizing phase of the action potential in most nerve cells, and Nav channel localization to the axon initial segment is vital to action potential initiation. Nav channels in the soma play a role in the transfer of axonal output information to the rest of the neuron and in synaptic plasticity, although little is known about Nav channel localization and dynamics within this neuronal compartment. This study uses single-particle tracking and photoactivation localization microscopy to analyze cell-surface Nav1.6 within the soma of cultured hippocampal neurons. Mean-square displacement analysis of individual trajectories indicated that half of the somatic Nav1.6 channels localized to stable nanoclusters ∼230 nm in diameter. Strikingly, these domains were stabilized at specific sites on the cell membrane for >30 min, notably via an ankyrin-independent mechanism, indicating that the means by which Nav1.6 nanoclusters are maintained in the soma is biologically different from axonal localization. Nonclustered Nav1.6 channels showed anomalous diffusion, as determined by mean-square-displacement analysis. High-density single-particle tracking of Nav channels labeled with photoactivatable fluorophores in combination with Bayesian inference analysis was employed to characterize the surface nanoclusters. A subpopulation of mobile Nav1.6 was observed to be transiently trapped in the nanoclusters. Somatic Nav1.6 nanoclusters represent a new, to our knowledge, type of Nav channel localization, and are hypothesized to be sites of localized channel regulation.

Unraveling the Thousand Word Picture: An Introduction to Super-Resolution Data Analysis

Super-resolution microscopy provides direct insight into fundamental biological processes occurring at length scales smaller than light's diffraction limit. The analysis of data at such scales has brought statistical and machine learning methods into the mainstream. Here we provide a survey of data analysis methods starting from an overview of basic statistical techniques underlying the analysis of super-resolution and, more broadly, imaging data. We subsequently break down the analysis of super-resolution data into four problems: the localization problem, the counting problem, the linking problem, and what we've termed the interpretation problem.

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.

Subdiffusion in Membrane Permeation of Small Molecules

Within the solubility-diffusion model of passive membrane permeation of small molecules, translocation of the permeant across the biological membrane is traditionally assumed to obey the Smoluchowski diffusion equation, which is germane for classical diffusion on an inhomogeneous free-energy and diffusivity landscape. This equation, however, cannot accommodate subdiffusive regimes, which have long been recognized in lipid bilayer dynamics, notably in the lateral diffusion of individual lipids. Through extensive biased and unbiased molecular dynamics simulations, we show that one-dimensional translocation of methanol across a pure lipid membrane remains subdiffusive on timescales approaching typical permeation times. Analysis of permeant motion within the lipid bilayer reveals that, in the absence of a net force, the mean squared displacement depends on time as t0.7, in stark contrast with the conventional model, which assumes a strictly linear dependence. We further show that an alternate model using a fractional-derivative generalization of the Smoluchowski equation provides a rigorous framework for describing the motion of the permeant molecule on the pico- to nanosecond timescale. The observed subdiffusive behavior appears to emerge from a crossover between small-scale rattling of the permeant around its present position in the membrane and larger-scale displacements precipitated by the formation of transient voids.

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.

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.

Revealing nonergodic dynamics in living cells from a single particle trajectory

We propose the improved ergodicity and mixing estimators to identify nonergodic dynamics from a single particle trajectory. The estimators are based on the time-averaged characteristic function of the increments and can thus capture additional information on the process as compared to the conventional time-averaged mean-square displacement. The estimators are first investigated and validated for several models of anomalous diffusion, such as ergodic fractional Brownian motion and diffusion on percolating clusters, and nonergodic continuous-time random walks and scaled Brownian motion. The estimators are then applied to two sets of earlier published trajectories of mRNA molecules inside live Escherichia coli cells and of Kv2.1 potassium channels in the plasma membrane. These statistical tests did not reveal nonergodic features in the former set, while some trajectories of the latter set could be classified as nonergodic. Time averages along such trajectories are thus not representative and may be strongly misleading. Since the estimators do not rely on ensemble averages, the nonergodic features can be revealed separately for each trajectory, providing a more flexible and reliable analysis of single-particle tracking experiments in microbiology.

Simulation-Based Approaches for Determining Membrane Permeability of Small Compounds

Predicting the rate of nonfacilitated permeation of solutes across lipid bilayers is important to drug design, toxicology, and signaling. These rates can be estimated using molecular dynamics simulations combined with the inhomogeneous solubility-diffusion model, which requires calculation of the potential of mean force and position-dependent diffusivity of the solute along the transmembrane axis. In this paper, we assess the efficiency and accuracy of several methods for the calculation of the permeability of a model DMPC bilayer to urea, benzoic acid, and codeine. We compare umbrella sampling, replica exchange umbrella sampling, adaptive biasing force, and multiple-walker adaptive biasing force for the calculation of the transmembrane PMF. No definitive advantage for any of these methods in their ability to predict the membrane permeability coefficient Pm was found, provided that a sufficiently long equilibration is performed. For diffusivities, a Bayesian inference method was compared to a generalized Langevin method, both being sensitive to chosen parameters and the slow relaxation of membrane defects. Agreement within 1.5 log units of the computed Pm with experiment is found for all permeants and methods. Remaining discrepancies can likely be attributed to limitations of the force field as well as slowly relaxing collective movements within the lipid environment. Numerical calculations based on model profiles show that Pm can be reliably estimated from only a few data points, leading to recommendations for calculating Pm from simulations.

Calculation of Lipid-Bilayer Permeabilities Using an Average Force

Calculations of lipid bilayer permeabilities from first principles, using molecular simulations, would be valuable to rapidly assess the bioavailability of drug candidates, as well as to decipher, at the atomic level, the mechanisms that underlie the translocation of permeants. The most common theoretical approach, the solubility-diffusion model, requires determination of the free energy and the diffusivity as functions of the position of the permeant. Quantitative predictions of permeability have, however, been stymied by acute difficulties in calculating the diffusivity, inadequate sampling, and, most insidiously, systematic biases due to imperfections in the force field, simulation parameters, and the inherent limitations of the diffusive model. In the present work, we combine importance-sampling simulations employing an adaptive biasing force with a Bayesian-inference algorithm to determine the free energy and diffusivity with noteworthy precision and spatial resolution. In multimicrosecond simulations, we probe the sensitivity of the permeability estimates to different aspects of the methodology, including the truncation of short-range interactions, the thermostat, the force-field parameters of the permeant, the time scale over which the diffusivity is estimated, and the size of the simulated system. The force-field parameters and time scale dependence of the diffusivities impose the greatest uncertainties on the permeability estimates. Our simulations highlight the importance of membrane distortion due to the presence of the permeant, which may be partially suppressed if the bilayer patch is not large enough. We suggest that improvements to force fields and more robust kinetic models may be needed to reduce systematic errors below a factor of two.

Diffusive Models of Membrane Permeation with Explicit Orientational Freedom

Accurate calculation of permeabilities from first-principles has been a long-standing challenge for computer simulations, notably in the context of drug discovery, as a route to predict the propensity of small, organic molecules to spontaneously translocate biological membranes. Of equal importance is the understanding of the permeation process and the pathway followed by the permeant from the aqueous medium to the interior of the lipid bilayer, and back out again. A convenient framework for the computation of permeabilities is provided by the solubility-diffusion model, which requires knowledge of the underlying free-energy and diffusivity landscapes. Here, we develop a formalism that includes an explicit description of the orientational motion of the solute as it diffuses across the membrane. Toward this end, we have generalized a recently proposed method that reconciles thermodynamics and kinetics in importance-sampling simulations by means of a Bayesian-inference scheme to reverse-solve the underlying Smoluchowski equation. Performance of the proposed formalism is examined in the model cases of a water and an ethanol molecule crossing a fully hydrated lipid bilayer. Our analysis reveals a conspicuous dependence of the free-energy and rotational diffusivity on the orientation of ethanol when it lies within the headgroup region of the bilayer. Specifically, orientations for which the hydroxyl group lies among the polar lipid head groups, while the ethyl group recedes toward the hydrophobic interior are associated with free-energy minima ∼2kBT deep, as well as significantly slower orientational kinetics compared to the bulk solution or the core of the bilayer. The conspicuous orientational anisotropy of ethanol at the aqueous interface is suggestive of a complete rotation of the permeant as it crosses the hydrophobic interior of the membrane.

A review of progress in single particle tracking: from methods to biophysical insights

Optical microscopy has for centuries been a key tool to study living cells with minimum invasiveness. The advent of single molecule techniques over the past two decades has revolutionized the field of cell biology by providing a more quantitative picture of the complex and highly dynamic organization of living systems. Amongst these techniques, single particle tracking (SPT) has emerged as a powerful approach to study a variety of dynamic processes in life sciences. SPT provides access to single molecule behavior in the natural context of living cells, thereby allowing a complete statistical characterization of the system under study. In this review we describe the foundations of SPT together with novel optical implementations that nowadays allow the investigation of single molecule dynamic events with increasingly high spatiotemporal resolution using molecular densities closer to physiological expression levels. We outline some of the algorithms for the faithful reconstruction of SPT trajectories as well as data analysis, and highlight biological examples where the technique has provided novel insights into the role of diffusion regulating cellular function. The last part of the review concentrates on different theoretical models that describe anomalous transport behavior and ergodicity breaking observed from SPT studies in living cells.

Dividing organelle tracks into Brownian and motor-driven intervals by variational maximization of the Bayesian evidence

Many organelles and vesicles in live cells move in a start-stop manner when observed for ~10 s by optical microscopy. Changes in velocity and directional persistence of such particles are a potentially rich source of insight into the mechanisms leading to the start and stop states. Unbiased assessment of the most probable number of states, the properties of each state, and the most probable state for the particle at each moment can be accomplished by variational Bayesian methods combined with a hidden Markov model and a Gaussian mixture model. Our track analysis method, "vbTRACK", applied this combination of methods to particle velocity v or changes in the direction of travel evaluated from simulated tracks and from tracks of peroxisomes in live cells. When tested with numerical data, vbTRACK reliably determined the number of states, the mean and variance of the velocity or the direction of travel for each state, and the most probable state during each frame. When applied to the tracks of peroxisomes in live cells, some tracks separated into two states, one with high velocity and directionality, the other approximately Brownian. Other tracks of particles in live cells separated into several diffusive states with distinct diffusion constants.

The Myelin and Lymphocyte Protein MAL Is Required for Binding and Activity of Clostridium perfringens ε-Toxin

Clostridium perfringens ε-toxin (ETX) is a potent pore-forming toxin responsible for a central nervous system (CNS) disease in ruminant animals with characteristics of blood-brain barrier (BBB) dysfunction and white matter injury. ETX has been proposed as a potential causative agent for Multiple Sclerosis (MS), a human disease that begins with BBB breakdown and injury to myelin forming cells of the CNS. The receptor for ETX is unknown. Here we show that both binding of ETX to mammalian cells and cytotoxicity requires the tetraspan proteolipid Myelin and Lymphocyte protein (MAL). While native Chinese Hamster Ovary (CHO) cells are resistant to ETX, exogenous expression of MAL in CHO cells confers both ETX binding and susceptibility to ETX-mediated cell death. Cells expressing rat MAL are ~100 times more sensitive to ETX than cells expressing similar levels of human MAL. Insertion of the FLAG sequence into the second extracellular loop of MAL abolishes ETX binding and cytotoxicity. ETX is known to bind specifically and with high affinity to intestinal epithelium, renal tubules, brain endothelial cells and myelin. We identify specific binding of ETX to these structures and additionally show binding to retinal microvasculature and the squamous epithelial cells of the sclera in wild-type mice. In contrast, there is a complete absence of ETX binding to tissues from MAL knockout (MAL-/-) mice. Furthermore, MAL-/- mice exhibit complete resistance to ETX at doses in excess of 1000 times the symptomatic dose for wild-type mice. We conclude that MAL is required for both ETX binding and cytotoxicity.

Single-virus force spectroscopy unravels molecular details of virus infection

Virus infection is a multistep process that has significant effects on the structure and function of both the virus and the host cell. The first steps of virus replication include cell binding, entry and release of the viral genome. Single-virus force spectroscopy (SVFS) has become a promising tool to understand the molecular details of those steps. SVFS data complemented by biochemical and biophysical, including theoretical modeling approaches provide valuable insights into molecular events that accompany virus infection. Properties of virus-cell interaction as well as structural alterations of the virus essential for infection can be investigated on a quantitative level. Here we review applications of SVFS to virus binding, structure and mechanics. We demonstrate that SVFS offers unexpected new insights not accessible by other methods.

Tracking molecular dynamics without tracking: image correlation of photo-activation microscopy

Measuring protein dynamics in the plasma membrane can provide insights into the mechanisms of receptor signaling and other cellular functions. To quantify protein dynamics on the single molecule level over the entire cell surface, sophisticated approaches such as single particle tracking (SPT), photo-activation localization microscopy (PALM) and fluctuation-based analysis have been developed. However, analyzing molecular dynamics of fluorescent particles with intermittent excitation and low signal-to-noise ratio present at high densities has remained a challenge. We overcame this problem by applying spatio-temporal image correlation spectroscopy (STICS) analysis to photo-activated (PA) microscopy time series. In order to determine under which imaging conditions this approach is valid, we simulated PA images of diffusing particles in a homogeneous environment and varied photo-activation, reversible blinking and irreversible photo-bleaching rates. Further, we simulated data with high particle densities that populated mobile objects (such as adhesions and vesicles) that often interfere with STICS and fluctuation-based analysis. We demonstrated in experimental measurements that the diffusion coefficient of the epidermal growth factor receptor (EGFR) fused to PAGFP in live COS-7 cells can be determined in the plasma membrane and revealed differences in the time-dependent diffusion maps between wild-type and mutant Lck in activated T cells. In summary, we have developed a new analysis approach for live cell photo-activation microscopy data based on image correlation spectroscopy to quantify the spatio-temporal dynamics of single proteins.

Extracting the diffusion tensor from molecular dynamics simulation with Milestoning

We propose an algorithm to extract the diffusion tensor from Molecular Dynamics simulations with Milestoning. A Kramers-Moyal expansion of a discrete master equation, which is the Markovian limit of the Milestoning theory, determines the diffusion tensor. To test the algorithm, we analyze overdamped Langevin trajectories and recover a multidimensional Fokker-Planck equation. The recovery process determines the flux through a mesh and estimates local kinetic parameters. Rate coefficients are converted to the derivatives of the potential of mean force and to coordinate dependent diffusion tensor. We illustrate the computation on simple models and on an atomically detailed system-the diffusion along the backbone torsions of a solvated alanine dipeptide.

Multifunctional rare-Earth vanadate nanoparticles: luminescent labels, oxidant sensors, and MRI contrast agents

Collecting information on multiple pathophysiological parameters is essential for understanding complex pathologies, especially given the large interindividual variability. We report here multifunctional nanoparticles which are luminescent probes, oxidant sensors, and contrast agents in magnetic resonance imaging (MRI). Eu(3+) ions in an yttrium vanadate matrix have been demonstrated to emit strong, nonblinking, and stable luminescence. Time- and space-resolved optical oxidant detection is feasible after reversible photoreduction of Eu(3+) to Eu(2+) and reoxidation by oxidants, such as H2O2, leading to a modulation of the luminescence emission. The incorporation of paramagnetic Gd(3+) confers in addition proton relaxation enhancing properties to the system. We synthesized and characterized nanoparticles of either 5 or 30 nm diameter with compositions of GdVO4 and Gd0.6Eu0.4VO4. These particles retain the luminescence and oxidant detection properties of YVO4:Eu. Moreover, the proton relaxivity of GdVO4 and Gd0.6Eu0.4VO4 nanoparticles of 5 nm diameter is higher than that of the commercial Gd(3+) chelate compound Dotarem at 20 MHz. Nuclear magnetic resonance dispersion spectroscopy showed a relaxivity increase above 10 MHz. Complexometric titration indicated that rare-earth leaching is negligible. The 5 nm nanoparticles injected in mice were observed with MRI to concentrate in the liver and the bladder after 30 min. Thus, these multifunctional rare-earth vanadate nanoparticles pave the way for simultaneous optical and magnetic resonance detection, in particular, for in vivo localization evolution and reactive oxygen species detection in a broad range of physiological and pathophysiological conditions.

Cell-surface translational dynamics of nicotinic acetylcholine receptors

Synapse efficacy heavily relies on the number of neurotransmitter receptors available at a given time. In addition to the equilibrium between the biosynthetic production, exocytic delivery and recycling of receptors on the one hand, and the endocytic internalization on the other, lateral diffusion and clustering of receptors at the cell membrane play key roles in determining the amount of active receptors at the synapse. Mobile receptors traffic between reservoir compartments and the synapse by thermally driven Brownian motion, and become immobilized at the peri-synaptic region or the synapse by: (a) clustering mediated by homotropic inter-molecular receptor–receptor associations; (b) heterotropic associations with non-receptor scaffolding proteins or the subjacent cytoskeletal meshwork, leading to diffusional “trapping,” and (c) protein-lipid interactions, particularly with the neutral lipid cholesterol. This review assesses the contribution of some of these mechanisms to the supramolecular organization and dynamics of the paradigm neurotransmitter receptor of muscle and neuronal cells -the nicotinic acetylcholine receptor (nAChR). Currently available information stemming from various complementary biophysical techniques commonly used to interrogate the dynamics of cell-surface components is critically discussed. The translational mobility of nAChRs at the cell surface differs between muscle and neuronal receptors in terms of diffusion coefficients and residence intervals at the synapse, which cover an ample range of time regimes. A peculiar feature of brain α7 nAChR is its ability to spend much of its time confined peri-synaptically, vicinal to glutamatergic (excitatory) and GABAergic (inhibitory) synapses. An important function of the α7 nAChR may thus be visiting the territories of other neurotransmitter receptors, differentially regulating the dynamic equilibrium between excitation and inhibition, depending on its residence time in each domain.

The adaptive biasing force method: everything you always wanted to know but were afraid to ask

In the host of numerical schemes devised to calculate free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an average force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy estimates. Much of the appeal of the algorithm to the practitioner is in its physically intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth mathematical analysis of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chemistry and biology. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a critical eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calculations of prototypical molecular systems, many methodological aspects are examined, from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a number of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.

Mapping the energy and diffusion landscapes of membrane proteins at the cell surface using high-density single-molecule imaging and Bayesian inference: application to the multiscale dynamics of glycine receptors in the neuronal membrane

Protein mobility is conventionally analyzed in terms of an effective diffusion. Yet, this description often fails to properly distinguish and evaluate the physical parameters (such as the membrane friction) and the biochemical interactions governing the motion. Here, we present a method combining high-density single-molecule imaging and statistical inference to separately map the diffusion and energy landscapes of membrane proteins across the cell surface at ~100 nm resolution (with acquisition of a few minutes). Upon applying these analytical tools to glycine neurotransmitter receptors at inhibitory synapses, we find that gephyrin scaffolds act as shallow energy traps (~3 kBT) for glycine neurotransmitter receptors, with a depth modulated by the biochemical properties of the receptor-gephyrin interaction loop. In turn, the inferred maps can be used to simulate the dynamics of proteins in the membrane, from the level of individual receptors to that of the population, and thereby, to model the stochastic fluctuations of physiological parameters (such as the number of receptors at synapses). Overall, our approach provides a powerful and comprehensive framework with which to analyze biochemical interactions in living cells and to decipher the multiscale dynamics of biomolecules in complex cellular environments.

Receptor displacement in the cell membrane by hydrodynamic force amplification through nanoparticles

We introduce an intrinsically multiplexed and easy to implement method to apply an external force to a biomolecule and thus probe its interaction with a second biomolecule or, more generally, its environment (for example, the cell membrane). We take advantage of the hydrodynamic interaction with a controlled fluid flow within a microfluidic channel to apply a force. By labeling the biomolecule with a nanoparticle that acts as a kite and increases the hydrodynamic interaction with the fluid, the drag induced by convection becomes important. We use this approach to track the motion of single membrane receptors, the Clostridium perfringens ε-toxin (CPεT) receptors that are confined in lipid raft platforms, and probe their interaction with the environment. Under external force, we observe displacements over distances up to 10 times the confining domain diameter due to elastic deformation of a barrier and return to the initial position after the flow is stopped. Receptors can also jump over such barriers. Analysis of the receptor motion characteristics before, during, and after a force is applied via the flow indicates that the receptors are displaced together with their confining raft platform. Experiments before and after incubation with latrunculin B reveal that the barriers are part of the actin cytoskeleton and have an average spring constant of 2.5 ± 0.6 pN/μm before vs. 0.6 ± 0.2 pN/μm after partial actin depolymerization. Our data, in combination with our previous work demonstrating that the ε-toxin receptor confinement is not influenced by the cytoskeleton, imply that it is the raft platform and its constituents rather than the receptor itself that encounters and deforms the barriers formed by the actin cytoskeleton.

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 [Formula: see text]-toxin (CP[Formula: see text]T) 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 CP[Formula: see text]T 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.

Calibrating optical tweezers with Bayesian inference

We present a new method for calibrating an optical-tweezer setup that does not depend on input parameters and is less affected by systematic errors like drift of the setup. It is based on an inference approach that uses Bayesian probability to infer the diffusion coefficient and the potential felt by a bead trapped in an optical or magnetic trap. It exploits a much larger amount of the information stored in the recorded bead trajectory than standard calibration approaches. We demonstrate that this method outperforms the equipartition method and the power-spectrum method in input information required (bead radius and trajectory length) and in output accuracy.

Propranolol restricts the mobility of single EGF-receptors on the cell surface before their internalization

The epidermal growth factor receptor is involved in morphogenesis, proliferation and cell migration. Its up-regulation during tumorigenesis makes this receptor an interesting therapeutic target. In the absence of the ligand, the inhibition of phosphatidic acid phosphohydrolase activity by propranolol treatment leads to internalization of empty/inactive receptors. The molecular events involved in this endocytosis remain unknown. Here, we quantified the effects of propranolol on the mobility of single quantum-dot labelled receptors before the actual internalization took place. The single receptors showed a clear stop-and-go motion; their diffusive tracks were continuously interrupted by sub-second stalling events, presumably caused by transient clustering. In the presence of propranolol we found that: i) the diffusion rate reduced by 22 %, which indicates an increase in drag of the receptor. Atomic force microscopy measurements did not show an increase of the effective membrane tension, such that clustering of the receptor remains the likely mechanism for its reduced mobility. ii) The receptor got frequently stalled for longer periods of multiple seconds, which may signal the first step of the internalization process.

Diagnosing heterogeneous dynamics in single-molecule/particle trajectories with multiscale wavelets

We describe a simple automated method to extract and quantify transient heterogeneous dynamical changes from large data sets generated in single-molecule/particle tracking experiments. Based on wavelet transform, the method transforms raw data to locally match dynamics of interest. This is accomplished using statistically adaptive universal thresholding, whose advantage is to avoid a single arbitrary threshold that might conceal individual variability across populations. How to implement this multiscale method is described, focusing on local confined diffusion separated by transient transport periods or hopping events, with three specific examples: in cell biology, biotechnology, and glassy colloid dynamics. The discussion is generalized within the framework of continuous time random walk. This computationally efficient method can run routinely on hundreds of millions of data points analyzed within an hour on a desktop personal computer.