Single-particle tracking reveals switching of the HIV fusion peptide between two diffusive modes in membranes

Hidden Markov modeling of single-particle diffusion with stochastic tethering

The statistics of the diffusive motion of particles often serve as an experimental proxy for their interaction with the environment. However, inferring the physical properties from the observed trajectories is challenging. Inspired by a recent experiment, here we analyze the problem of particles undergoing two-dimensional Brownian motion with transient tethering to the surface. We model the problem as a hidden Markov model where the physical position is observed and the tethering state is hidden. We develop an alternating maximization algorithm to infer the hidden state of the particle and estimate the physical parameters of the system. The crux of our method is a saddle-point-like approximation, which involves finding the most likely sequence of hidden states and estimating the physical parameters from it. Extensive numerical tests demonstrate that our algorithm reliably finds the model parameters and is insensitive to the initial guess. We discuss the different regimes of physical parameters and the algorithm's performance in these regimes. We also provide a free software implementation of our algorithm.

Receptor tyrosine kinase MET ligand-interaction classified via machine learning from single-particle tracking data

Internalin B–mediated activation of the membrane-bound receptor tyrosine kinase MET is accompanied by a change in receptor mobility. Conversely, it should be possible to infer from receptor mobility whether a cell has been treated with internalin B. Here, we propose a method based on hidden Markov modeling and explainable artificial intelligence that machine-learns the key differences in MET mobility between internalin B–treated and –untreated cells from single-particle tracking data. Our method assigns receptor mobility to three diffusion modes (immobile, slow, and fast). It discriminates between internalin B–treated and –untreated cells with a balanced accuracy of >99% and identifies three parameters that are most affected by internalin B treatment: a decrease in the mobility of slow molecules (1) and a depopulation of the fast mode (2) caused by an increased transition of fast molecules to the slow mode (3). Our approach is based entirely on free software and is readily applicable to the analysis of other membrane receptors.

Membrane anchoring facilitates colocalization of enzymes in plant cytochrome P450 redox systems

Plant metabolism depends on cascade reactions mediated by dynamic enzyme assemblies known as metabolons. In this context, the cytochrome P450 (P450) superfamily catalyze key reactions underpinning the unique diversity of bioactive compounds. In contrast to their soluble bacterial counterparts, eukaryotic P450s are anchored to the endoplasmic reticulum membrane and serve as metabolon nucleation sites. Hence, membrane anchoring appears to play a pivotal role in the evolution of complex biosynthetic pathways. Here, a model membrane assay enabled characterization of membrane anchor dynamics by single molecule microscopy. As a model system, we reconstituted the membrane anchor of cytochrome P450 oxidoreductase (POR), the ubiquitous electron donor to all microsomal P450s. The transmembrane segment in the membrane anchor of POR is relatively conserved, corroborating its functional importance. We observe dynamic colocalization of the POR anchors in our assay suggesting that membrane anchoring might promote intermolecular interactions and in this way impact assembly of metabolic multienzyme complexes.

Maximum likelihood estimates of diffusion coefficients from single-particle tracking experiments

Single-molecule localization microscopy allows practitioners to locate and track labeled molecules in biological systems. When extracting diffusion coefficients from the resulting trajectories, it is common practice to perform a linear fit on mean-squared-displacement curves. However, this strategy is suboptimal and prone to errors. Recently, it was shown that the increments between the observed positions provide a good estimate for the diffusion coefficient, and their statistics are well-suited for likelihood-based analysis methods. Here, we revisit the problem of extracting diffusion coefficients from single-particle tracking experiments subject to static noise and dynamic motion blur using the principle of maximum likelihood. Taking advantage of an efficient real-space formulation, we extend the model to mixtures of subpopulations differing in their diffusion coefficients, which we estimate with the help of the expectation-maximization algorithm. This formulation naturally leads to a probabilistic assignment of trajectories to subpopulations. We employ the theory to analyze experimental tracking data that cannot be explained with a single diffusion coefficient. We test how well a dataset conforms to the assumptions of a diffusion model and determine the optimal number of subpopulations with the help of a quality factor of known analytical distribution. To facilitate use by practitioners, we provide a fast open-source implementation of the theory for the efficient analysis of multiple trajectories in arbitrary dimensions simultaneously.

Fundamentals of the logarithmic measure for revealing multimodal diffusion

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

Single-molecule analysis of dynamics and interactions of the SecYEG translocon

Protein translocation and insertion into the bacterial cytoplasmic membrane are the essential processes mediated by the Sec machinery. The core machinery is composed of the membrane-embedded translocon SecYEG that interacts with the secretion-dedicated ATPase SecA and translating ribosomes. Despite the simplicity and the available structural insights on the system, diverse molecular mechanisms and functional dynamics have been proposed. Here, we employ total internal reflection fluorescence microscopy to study the oligomeric state and diffusion of SecYEG translocons in supported lipid bilayers at the single-molecule level. Silane-based coating ensured the mobility of lipids and reconstituted translocons within the bilayer. Brightness analysis suggested that approx. 70% of the translocons were monomeric. The translocons remained in a monomeric form upon ribosome binding, but partial oligomerization occurred in the presence of nucleotide-free SecA. Individual trajectories of SecYEG in the lipid bilayer revealed dynamic heterogeneity of diffusion, as translocons commonly switched between slow and fast mobility modes with corresponding diffusion coefficients of 0.03 and 0.7 µm² ·s^-1 . Interactions with SecA ATPase had a minor effect on the lateral mobility, while bound ribosome:nascent chain complexes substantially hindered the diffusion of single translocons. Notably, the mobility of the translocon:ribosome complexes was not affected by the solvent viscosity or macromolecular crowding modulated by Ficoll PM 70, so it was largely determined by interactions within the lipid bilayer and at the interface. We suggest that the complex mobility of SecYEG arises from the conformational dynamics of the translocon and protein:lipid interactions.

Analysis of the Diffusivity Change from Single-Molecule Trajectories on Living Cells

With the wide application of live-cell single-molecule imaging and tracking of biomolecules at work, deriving diffusion state changes of individual molecules is of particular interest as these changes reflect molecular oligomerization or interaction with other cellular components and thus correlate with functional changes. We have developed a Rayleigh mixture distribution-based hidden Markov model method to analyze time-lapse diffusivity change of single molecules, especially membrane proteins, with unknown dynamic states in living cells. With this method, the diffusion parameters, including diffusion state number, state transition probability, diffusion coefficient, and state mixture ratio, can be extracted from the single-molecule diffusion trajectories accurately via easy computation. The validity of our method has been demonstrated with not only experiments on synthetic trajectories but also single-molecule fluorescence imaging data of two typical membrane receptors. Our method offers a new analytical tool for the investigation of molecular interaction kinetics at the single-molecule level.

Estimation of diffusive states from single-particle trajectory in heterogeneous medium using machine-learning methods

We propose a novel approach to analyze random walks in heterogeneous medium using a hybrid machine-learning method based on a gamma mixture and a hidden Markov model. A gamma mixture and a hidden Markov model respectively provide the number and the most probable sequence of diffusive states from the time series position data of particles/molecules obtained by single-particle/molecule tracking (SPT/SMT) method. We evaluate the performance of our proposed method for numerically generated trajectories. It is shown that our proposed method can correctly extract the number of diffusive states when each trajectory is long enough to be frame averaged. We also indicate that our method can provide an indicator whether the assumption of a medium consisting of discrete diffusive states is appropriate or not based on the available amount of trajectory data. Then, we demonstrate an application of our method to the analysis of experimentally obtained SPT data.

A Hidden Markov Model for Detecting Confinement in Single-Particle Tracking Trajectories

State-of-the-art single-particle tracking (SPT) techniques can generate long trajectories with high temporal and spatial resolution. This offers the possibility of mechanistically interpreting particle movements and behavior in membranes. To this end, a number of statistical techniques have been developed that partition SPT trajectories into states with distinct diffusion signatures, allowing a statistical analysis of diffusion state dynamics and switching behavior. Here, we develop a confinement model, within a hidden Markov framework, that switches between phases of free diffusion and confinement in a harmonic potential well. By using a Markov chain Monte Carlo algorithm to fit this model, automated partitioning of individual SPT trajectories into these two phases is achieved, which allows us to analyze confinement events. We demonstrate the utility of this algorithm on a previously published interferometric scattering microscopy data set, in which gold-nanoparticle-tagged ganglioside GM1 lipids were tracked in model membranes. We performed a comprehensive analysis of confinement events, demonstrating that there is heterogeneity in the lifetime, shape, and size of events, with confinement size and shape being highly conserved within trajectories. Our observations suggest that heterogeneity in confinement events is caused by both individual nanoparticle characteristics and the binding-site environment. The individual nanoparticle heterogeneity ultimately limits the ability of interferometric scattering microscopy to resolve molecule dynamics to the order of the tag size; homogeneous tags could potentially allow the resolution to be taken below this limit by deconvolution methods. In a wider context, the presented harmonic potential well confinement model has the potential to detect and characterize a wide variety of biological phenomena, such as hop diffusion, receptor clustering, and lipid rafts.

Resolving mixed mechanisms of protein subdiffusion at the T cell plasma membrane

The plasma membrane is a complex medium where transmembrane proteins diffuse and interact to facilitate cell function. Membrane protein mobility is affected by multiple mechanisms, including crowding, trapping, medium elasticity and structure, thus limiting our ability to distinguish them in intact cells. Here we characterize the mobility and organization of a short transmembrane protein at the plasma membrane of live T cells, using single particle tracking and photoactivated-localization microscopy. Protein mobility is highly heterogeneous, subdiffusive and ergodic-like. Using mobility characteristics, we segment individual trajectories into subpopulations with distinct Gaussian step-size distributions. Particles of low-to-medium mobility consist of clusters, diffusing in a viscoelastic and fractal-like medium and are enriched at the centre of the cell footprint. Particles of high mobility undergo weak confinement and are more evenly distributed. This study presents a methodological approach to resolve simultaneous mixed subdiffusion mechanisms acting on polydispersed samples and complex media such as cell membranes.

Membrane protein diffusion is affected by distinct mechanisms such as molecular crowding and medium elasticity. Here the authors present an analytical approach to analyse single particle trajectories and distinguish mixed subdiffusive processes affecting membrane protein mobility in living cells.

Systems-level approach to uncovering diffusive states and their transitions from single-particle trajectories

The stochastic motions of a diffusing particle contain information concerning the particle's interactions with binding partners and with its local environment. However, an accurate determination of the underlying diffusive properties, beyond normal diffusion, has remained challenging when analyzing particle trajectories on an individual basis. Here, we introduce the maximum-likelihood estimator (MLE) for confined diffusion and fractional Brownian motion. We demonstrate that this MLE yields improved estimation over traditional mean-square displacement analyses. We also introduce a model selection scheme (that we call mleBIC) that classifies individual trajectories to a given diffusion mode. We demonstrate the statistical limitations of classification via mleBIC using simulated data. To overcome these limitations, we introduce a version of perturbation expectation-maximization (pEMv2), which simultaneously analyzes a collection of particle trajectories to uncover the system of interactions that give rise to unique normal and/or non-normal diffusive states within the population. We test and evaluate the performance of pEMv2 on various sets of simulated particle trajectories, which transition among several modes of normal and non-normal diffusion, highlighting the key considerations for employing this analysis methodology.

Repetitive formation of optically-observable planar lipid bilayers by rotating chambers on a microaperture

Optical observation of a planar lipid bilayer is an effective method of lipid bilayer characterization. However, previous methods for optically observable lipid bilayer formation are unsuitable for repetitive formation of lipid bilayers. In this paper, we propose a system that facilitates repetitive formation of horizontal lipid bilayers via mechanical rotation of the rotating part. We show that multiple bilayers can be observed within a short period, and that the electrical and optical characteristics of a bilayer can be analyzed simultaneously.

Single-Molecule Tracking and Its Application in Biomolecular Binding Detection

In the past two decades significant advances have been made in single-molecule detection, which enables the direct observation of single biomolecules at work in real time and under physiological conditions. In particular, the development of single-molecule tracking (SMT) microscopy allows us to monitor the motion paths of individual biomolecules in living systems, unveiling the localization dynamics and transport modalities of the biomolecules that support the development of life. Beyond the capabilities of traditional camera-based tracking techniques, state-of-the-art SMT microscopies developed in recent years can record fluorescence lifetime while tracking a single molecule in the 3D space. This multiparameter detection capability can open the door to a wide range of investigations at the cellular or tissue level, including identification of molecular interaction hotspots and characterization of association/dissociation kinetics between molecules. In this review, we discuss various SMT techniques developed to date, with an emphasis on our recent development of the next generation 3D tracking system that not only achieves ultrahigh spatiotemporal resolution but also provides sufficient working depth suitable for live animal imaging. We also discuss the challenges that current SMT techniques are facing and the potential strategies to tackle those challenges.

Analysis of single particle diffusion with transient binding using particle filtering

Diffusion with transient binding occurs in a variety of biophysical processes, including movement of transmembrane proteins, T cell adhesion, and caging in colloidal fluids. We model diffusion with transient binding as a Brownian particle undergoing Markovian switching between free diffusion when unbound and diffusion in a quadratic potential centered around a binding site when bound. Assuming the binding site is the last position of the particle in the unbound state and Gaussian observational error obscures the true position of the particle, we use particle filtering to predict when the particle is bound and to locate the binding sites. Maximum likelihood estimators of diffusion coefficients, state transition probabilities, and the spring constant in the bound state are computed with a stochastic Expectation-Maximization (EM) algorithm.

Extracting Diffusive States of Rho GTPase in Live Cells: Towards In Vivo Biochemistry

Resolving distinct biochemical interaction states when analyzing the trajectories of diffusing proteins in live cells on an individual basis remains challenging because of the limited statistics provided by the relatively short trajectories available experimentally. Here, we introduce a novel, machine-learning based classification methodology, which we call perturbation expectation-maximization (pEM), that simultaneously analyzes a population of protein trajectories to uncover the system of diffusive behaviors which collectively result from distinct biochemical interactions. We validate the performance of pEM in silico and demonstrate that pEM is capable of uncovering the proper number of underlying diffusive states with an accurate characterization of their diffusion properties. We then apply pEM to experimental protein trajectories of Rho GTPases, an integral regulator of cytoskeletal dynamics and cellular homeostasis, in vivo via single particle tracking photo-activated localization microcopy. Remarkably, pEM uncovers 6 distinct diffusive states conserved across various Rho GTPase family members. The variability across family members in the propensities for each diffusive state reveals non-redundant roles in the activation states of RhoA and RhoC. In a resting cell, our results support a model where RhoA is constantly cycling between activation states, with an imbalance of rates favoring an inactive state. RhoC, on the other hand, remains predominantly inactive.

Single particle tracking is a powerful tool that captures the diffusive dynamics of proteins as they undergo various interactions in living cells. Uncovering different biochemical interactions by analyzing the diffusive behaviors of individual protein trajectories, however, is challenging due to the limited statistics provided by short trajectories and experimental noise sources which are intimately coupled into each protein’s localization. Here, we introduce a novel, unsupervised, machine-learning based classification methodology, which we call perturbation expectation-maximization (pEM), that simultaneously analyzes a population of protein trajectories to uncover the system of diffusive behaviors which collectively result from distinct biochemical interactions. We validate the performance of pEM in silico and in vivo on the biological system of Rho GTPase, a signal transduction protein responsible for regulating cytoskeletal dynamics. We envision that the presented methodology will be applicable to a wide range of single protein tracking data where different biochemical interactions result in distinct diffusive behaviors. More generally, this study brings us an important step closer to the possibility of monitoring the endogenous biochemistry of diffusing proteins within live cells with single molecule resolution.

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.

Calculating hydrodynamic interactions for membrane-embedded objects

A recently introduced numerical scheme for calculating self-diffusion coefficients of solid objects embedded in lipid bilayer membranes is extended to enable calculation of hydrodynamic interactions between multiple objects. The method is used to validate recent analytical predictions by Oppenheimer and Diamant [Biophys. J. 96, 3041 2009] related to the coupled diffusion of membrane embedded proteins and is shown to converge to known near-field lubrication results as objects closely approach one another; however, the present methodology also applies outside of the limiting regimes where analytical results are available. Multiple different examples involving pairs of disk-like objects with various constraints imposed on their relative motions demonstrate the importance of hydrodynamic interactions in the dynamics of proteins and lipid domains on membrane surfaces. It is demonstrated that the relative change in self-diffusion of a membrane embedded object upon perturbation by a similar proximal solid object displays a maximum for object sizes comparable to the Saffman-Delbrück length of the membrane.

pH-dependent vesicle fusion induced by the ectodomain of the human immunodeficiency virus membrane fusion protein gp41: Two kinetically distinct processes and fully-membrane-associated gp41 with predominant β sheet fusion peptide conformation

The gp41 protein of the Human Immunodeficiency Virus (HIV) catalyzes fusion between HIV and host cell membranes. The ~180-residue ectodomain of gp41 is outside the virion and is the most important gp41 region for membrane fusion. The ectodomain consists of an apolar fusion peptide (FP) region hypothesized to bind to the host cell membrane followed by N-heptad repeat (NHR), loop, and C-heptad repeat (CHR) regions. The present study focuses on the large gp41 ectodomain constructs "Hairpin" (HP) containing NHR+loop+CHR and "FP-Hairpin" (FP-HP) containing FP+NHR+loop+CHR. Both proteins induce rapid and extensive fusion of anionic vesicles at pH4 where the protein is positively-charged but do not induce fusion at pH7 where the protein is negatively charged. This observation, along with lack of fusion of neutral vesicles at either pH supports the significance of attractive protein/membrane electrostatics in fusion. There are two kinetically distinct fusion processes at pH4: (1) a faster ~100 ms⁻¹ process with rate strongly positively correlated with vesicle charge; and (2) a slower ~5 ms⁻¹ process with extent strongly inversely correlated with this charge. The slower process may be more physiologically relevant because HIV/host cell fusion occurs at physiologic pH with gp41 restricted to the narrow region between the two membranes. Previous solid-state NMR (SSNMR) of membrane-associated FP-HP has supported protein oligomers with FP's in an intermolecular antiparallel sheet. There was an additional population of molecules with α helical FPs and the samples likely contained a mixture of membrane-bound and -unbound proteins. For the present study, samples were prepared with fully membrane-bound FP-HP and subsequent SSNMR showed dominant β FP conformation at both low and neutral pH. SSNMR also showed close contact of the FP with the lipid headgroups at both low and neutral pH whereas the NHR+CHR regions had contact at low pH and were more distant at neutral pH, consistent with the protein/membrane electrostatics.

Dark-field-based observation of single-nanoparticle dynamics on a supported lipid bilayer for in situ analysis of interacting molecules and nanoparticles

Observation of single plasmonic nanoparticles in reconstituted biological systems allows us to obtain snapshots of dynamic processes between molecules and nanoparticles with unprecedented spatiotemporal resolution and single-molecule/single-particle-level data acquisition. This Concept is intended to introduce nanoparticle-tethered supported lipid bilayer platforms that allow for the dynamic confinement of nanoparticles on a two-dimensional fluidic surface. The dark-field-based long-term, stable, real-time observation of freely diffusing plasmonic nanoparticles on a lipid bilayer enables one to extract a broad range of information about interparticle and molecular interactions throughout the entire reaction period. Herein, we highlight important developments in this context to provide ideas on how molecular interactions can be interpreted by monitoring dynamic behaviors and optical signals of laterally mobile nanoparticles.