Single Particle Tracking Experiments Research Articles

Biophysics of high density nanometer regions extracted from super-resolution single particle trajectories: application to voltage-gated calcium channels and phospholipids

The cellular membrane is very heterogenous and enriched with high-density regions forming microdomains, as revealed by single particle tracking experiments. However the organization of these regions remain unexplained. We determine here the biophysical properties of these regions, when described as a basin of attraction. We develop two methods to recover the dynamics and local potential wells (field of force and boundary). The first method is based on the local density of points distribution of trajectories, which differs inside and outside the wells. The second method focuses on recovering the drift field that is convergent inside wells and uses the transient field to determine the boundary. Finally, we apply these two methods to the distribution of trajectories recorded from voltage gated calcium channels and phospholipid anchored GFP in the cell membrane of hippocampal neurons and obtain the size and energy of high-density regions with a nanometer precision.

Lamperti transformation of scaled Brownian motion and related Langevin equations

Recently, scaled Brownian motion has attracted considerable attention in the context of single particle tracking experiments displaying anomalous fractional dynamics. Its probability density function coincides with the one for fractional Brownian motion. On the other hand, scaled Brownian motion displays weak ergodicity breaking. In this paper we show that by applying the so-called Lamperti transformation we are able to transform the scaled Brownian motion into ergodic process. This allows us to estimate the distribution and moments of the studied process having only one, appropriately long trajectory. We apply the same method to the scaled fractional Brownian motion and stable Lévy motion. It appears that the method works also in the case of long-range correlations and heavy-tailed distributions. We confirm our theoretical results using numerical simulations.

Photoactivation of silicon rhodamines via a light-induced protonation

Photoactivatable fluorophores are important for single-particle tracking and super-resolution microscopy. Here we present a photoactivatable fluorophore that forms a bright silicon rhodamine derivative through a light-dependent protonation. In contrast to other photoactivatable fluorophores, no caging groups are required, nor are there any undesired side-products released. Using this photoactivatable fluorophore, we create probes for HaloTag and actin for live-cell single-molecule localization microscopy and single-particle tracking experiments. The unusual mechanism of photoactivation and the fluorophore’s outstanding spectroscopic properties make it a powerful tool for live-cell super-resolution microscopy.

Antibody-Mediated Immobilization of Virions in Mucus

Antibodies have been shown to hinder the movement of herpes simplex virus virions in cervicovaginal mucus, as well as other viruses in other mucus secretions. However, it has not been possible to directly observe the mechanisms underlying this phenomenon, so the nature of virion–antibody–mucin interactions remain poorly understood. In this work, we analyzed thousands of virion traces from single particle tracking experiments to explicate how antibodies must cooperate to immobilize virions for relatively long time periods. First, using a clustering analysis, we observed a clear separation between two classes of virion behavior: freely diffusing and immobilized. While the proportion of freely diffusing virions decreased with antibody concentration, the magnitude of their diffusivity did not, implying an all-or-nothing dichotomy in the pathwise effect of the antibodies. Proceeding under the assumption that all binding events are reversible, we used a novel switch-point detection method to conclude that there are very few, if any, state switches on the experimental timescale of 20 s. To understand this slow state switching, we analyzed a recently proposed continuous-time Markov chain model for binding kinetics and virion movement. Model analysis implied that virion immobilization requires cooperation by multiple antibodies that are simultaneously bound to the virion and mucin matrix and that there is an entanglement phenomenon that accelerates antibody–mucin binding when a virion is immobilized. In addition to developing a widely applicable framework for analyzing multistate particle behavior, this work substantially enhances our mechanistic understanding of how antibodies can reinforce a mucus barrier against passive invasive species.

Rouse model with transient intramolecular contacts on a timescale of seconds recapitulates folding and fluctuation of yeast chromosomes.

DNA folding and dynamics along with major nuclear functions are determined by chromosome structural properties, which remain, thus far, elusive in vivo. Here, we combine polymer modeling and single particle tracking experiments to determine the physico-chemical parameters of chromatin in vitro and in living yeast. We find that the motion of reconstituted chromatin fibers can be recapitulated by the Rouse model using mechanical parameters of nucleosome arrays deduced from structural simulations. Conversely, we report that the Rouse model shows some inconsistencies to analyze the motion and structural properties inferred from yeast chromosomes determined with chromosome conformation capture techniques (specifically, Hi-C). We hence introduce the Rouse model with Transient Internal Contacts (RouseTIC), in which random association and dissociation occurs along the chromosome contour. The parametrization of this model by fitting motion and Hi-C data allows us to measure the kinetic parameters of the contact formation reaction. Chromosome contacts appear to be transient; associated to a lifetime of seconds and characterized by an attractive energy of –0.3 to –0.5 kBT. We suggest attributing this energy to the occurrence of histone tail-DNA contacts and notice that its amplitude sets chromosomes in ‘theta’ conditions, in which they are poised for compartmentalization and phase separation.

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.

The Devil Is in the Details: What Do We Really Track in Single-Particle Tracking Experiments of Diffusion in Biological Membranes?

Single-particle tracking (SPT) is an experimental technique that allows one to follow the dynamics of individual molecules in biological membranes with unprecedented precision. Given the importance of lipid and membrane protein diffusion in the formation of nanoscale functional complexes, it is critical to understand what exactly is measured in SPT experiments. To clarify this issue, we employed nanoscale computer simulations designed to match SPT experiments that exploit streptavidin-functionalized Au nanoparticles (AuNPs). The results show that lipid labeling interferes critically with the diffusion process; thus, the diffusion measured in SPT is a far more complex process than what has been assumed. It turns out that the influence of AuNP-based labels on the dynamics of probe lipids includes not only the AuNP-induced viscous drag that is the more significant the larger the NP but, more importantly, also the effects related to the interactions of the streptavidin linker with membrane lipids. Due to these effects, the probe lipid moves in a concerted manner as a complex with the linker protein and numerous unlabeled lipids, which can slow down the motion of the probe by almost an order of magnitude. Furthermore, our simulations show that nonlinker streptavidin tetramers on the AuNP surface are able to interact with the membrane lipids, which could potentially lead to multivalent labeling of the NPs by the probe lipids. Our results further demonstrate that in the submicrosecond time domain the motion of the probe lipid is uncorrelated with the motion of the AuNP, showing that there is a 1 μs limit for the temporal resolution of the SPT technique. However, this limit for the temporal resolution depends on the nanoparticle size and increases rapidly with growing AuNPs. Overall, the results provide a molecular-scale framework to accurately interpret SPT data and to design protocols that minimize label-induced artifacts.

Evidence for an Interaction between Influenza Hemagglutinin and PIP2

The influenza viral protein hemagglutinin (HA) forms dense nanoscale clusters on the plasma membrane of infected cells. HA clusters mediate fusion, which is crucial to infectivity. Previous studies have shown a connection between influenza infection and phosphoinositide-related cellular processes; however, no interaction between HA and the major plasma membrane phosphoinositide PIP2 (phosphatidylinositol-(4,5)-biphosphate) has yet been shown. Super-resolution imaging in fixed cells reveals that HA and PIP2 clusters are colocalized with similar radial distribution functions. Furthermore, single particle tracking experiments in living cells indicate that PIP2 molecules have reduced mobility and increased confinement in the presence of HA clusters. This evidence suggests an underlying interaction between HA and PIP2 and merits further investigation. Understanding cellular interactions with influenza viral proteins may lead to new anti-viral therapies that are strain independent.

Inhomogeneous membrane receptor diffusion explained by a fractional heteroscedastic time series model.

Single particle tracking experiments have recently uncovered that the motion of cell membrane components can undergo changes of diffusivity as a result of the heterogeneous environment, producing subdiffusion and nonergodic behavior. In this paper, we show that an autoregressive fractionally integrated moving average (ARFIMA) with noise given by generalized autoregressive conditional heteroscedasticity (GARCH) can describe inhomogeneous diffusion in the cell membrane, where the ARFIMA process models anomalous diffusion and the GARCH process explains a fluctuating diffusion parameter.

Fluid heterogeneity detection based on the asymptotic distribution of the time-averaged mean squared displacement in single particle tracking experiments

A tracer particle is called anomalously diffusive if its mean squared displacement grows approximately as as a function of time t for some constant , where the diffusion exponent satisfies . In this article, we use recent results on the asymptotic distribution of the time-averaged mean squared displacement (Didier and Zhang 2017 J. Time Ser. Anal. 38 395–416) to construct statistical tests for detecting physical heterogeneity in viscoelastic fluid samples starting from one or multiple observed anomalously diffusive paths. The methods are asymptotically valid for the range and involve a mathematical characterization of time-averaged mean squared displacement bias and the effect of correlated disturbance errors. The assumptions on particle motion cover a broad family of fractional Gaussian processes, including fractional Brownian motion and many fractional instances of the generalized Langevin equation framework. We apply the proposed methods in experimental data from treated Pseudomonas aeruginosa biofilms generated by the collaboration of the Hill and Schoenfisch Labs at UNC-Chapel Hill.

Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity.

How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and out-of-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.

Power spectral density of a single Brownian trajectory: what one can and cannot learn from it

The power spectral density (PSD) of any time-dependent stochastic process Xt is a meaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of Xt over an infinitely large observation time T, that is, it is defined as an ensemble-averaged property taken in the limit . A legitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation time T. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is a fluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.

Robust model-based analysis of single-particle tracking experiments with Spot-On.

Single-particle tracking (SPT) has become an important method to bridge biochemistry and cell biology since it allows direct observation of protein binding and diffusion dynamics in live cells. However, accurately inferring information from SPT studies is challenging due to biases in both data analysis and experimental design. To address analysis bias, we introduce 'Spot-On', an intuitive web-interface. Spot-On implements a kinetic modeling framework that accounts for known biases, including molecules moving out-of-focus, and robustly infers diffusion constants and subpopulations from pooled single-molecule trajectories. To minimize inherent experimental biases, we implement and validate stroboscopic photo-activation SPT (spaSPT), which minimizes motion-blur bias and tracking errors. We validate Spot-On using experimentally realistic simulations and show that Spot-On outperforms other methods. We then apply Spot-On to spaSPT data from live mammalian cells spanning a wide range of nuclear dynamics and demonstrate that Spot-On consistently and robustly infers subpopulation fractions and diffusion constants.

Mucus-penetrating solid lipid nanoparticles for the treatment of cystic fibrosis: Proof of concept, challenges and pitfalls

Nanocarrier-mediated transmucosal drug delivery based on conventional mucoadhesive, muco-inert or mucus-penetrating nanoparticles (NPs) is a growing field especially in challenging diseases like cystic fibrosis (CF). Efficacy of such systems dictates profound investigation of particle-mucus interaction and factors governing the whole process. Although variable techniques studying particle diffusion in mucus have been introduced, standardized procedures are lacking. The study comprised different methods based on micro- and macro-displacement as well as colloidal stability and turbidimetric experiments. Artificial sputum medium (ASM), CF sputum and mucus-secreting cell line (Calu-3 air interface culture, AIC) were applied. Solid lipid nanoparticles (SLNs) coated with variable hydrophilic sheath (poloxamer, Tween 80 or PVA) represented the nanocarriers under investigation. Both micro-displacement studies based on single particle tracking and macro-displacement experiments based on 3D-time laps confocal imaging revealed faster diffusion of poloxamer- > Tween- > PVA-coated SLNs. Compared to ASM, CF sputum showed not only lower diffusion rates but also remarkable discrepancies in particle-mucus diffusion rate due to sputum heterogenicity. Meanwhile, in case of Calu-3 AIC, thickness of the mucosal layer as well as density of mucus network were key determinants in the diffusion process. The points emphasized in this study highlight the road towards in vivo relevant particle-mucus interaction research.

Dynamics of Nanoparticles in Entangled Polymer Solutions.

The mean square displacement ⟨r2⟩ of nanoparticle probes dispersed in simple isotropic liquids and in polymer solutions is interrogated using fluorescence correlation spectroscopy and single-particle tracking (SPT) experiments. Probe dynamics in different regimes of particle diameter (d), relative to characteristic polymer length scales, including the correlation length (ξ), the entanglement mesh size (a), and the radius of gyration (Rg), are investigated. In simple fluids and for polymer solutions in which d ≫ Rg, long-time particle dynamics obey random-walk statistics ⟨r2⟩:t, with the bulk zero-shear viscosity of the polymer solution determining the frictional resistance to particle motion. In contrast, in polymer solutions with d < Rg, polymer molecules in solution exert noncontinuum resistances to particle motion and nanoparticle probes appear to interact hydrodynamically only with a local fluid medium with effective drag comparable to that of a solution of polymer chain segments with sizes similar to those of the nanoparticle probes. Under these conditions, the nanoparticles exhibit orders of magnitude faster dynamics than those expected from continuum predictions based on the Stokes-Einstein relation. SPT measurements further show that when d > a, nanoparticle dynamics transition from diffusive to subdiffusive on long timescales, reminiscent of particle transport in a field with obstructions. This last finding is in stark contrast to the nanoparticle dynamics observed in entangled polymer melts, where X-ray photon correlation spectroscopy measurements reveal faster but hyperdiffusive dynamics. We analyze these results with the help of the hopping model for particle dynamics in polymers proposed by Cai et al. and, on that basis, discuss the physical origins of the local drag experienced by the nanoparticles in entangled polymer solutions.

Scale-invariant Green-Kubo relation for time-averaged diffusivity.

In recent years it was shown both theoretically and experimentally that in certain systems exhibiting anomalous diffusion the time- and ensemble-averaged mean-squared displacement are remarkably different. The ensemble-averaged diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale-invariant nonstationary velocity correlation function with the transport coefficient. Here we obtain the relation between time-averaged diffusivity, usually recorded in single-particle tracking experiments, and the underlying scale-invariant velocity correlation function. The time-averaged mean-squared displacement is given by 〈δ^{2}[over ¯]〉∼2D_{ν}t^{β}Δ^{ν-β}, where t is the total measurement time and Δ is the lag time. Here ν is the anomalous diffusion exponent obtained from ensemble-averaged measurements 〈x^{2}〉∼t^{ν}, while β≥-1 marks the growth or decline of the kinetic energy 〈v^{2}〉∼t^{β}. Thus, we establish a connection between exponents that can be read off the asymptotic properties of the velocity correlation function and similarly for the transport constant D_{ν}. We demonstrate our results with nonstationary scale-invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. If the averaged kinetic energy is finite, β=0, the time scaling of 〈δ^{2}[over ¯]〉 and 〈x^{2}〉 are identical; however, the time-averaged transport coefficient D_{ν} is not identical to the corresponding ensemble-averaged diffusion constant.

Elucidating fluctuating diffusivity in center-of-mass motion of polymer models with time-averaged mean-square-displacement tensor.

There have been increasing reports that the diffusion coefficient of macromolecules depends on time and fluctuates randomly. Here a method is developed to elucidate this fluctuating diffusivity from trajectory data. Time-averaged mean-square displacement (MSD), a common tool in single-particle-tracking (SPT) experiments, is generalized to a second-order tensor with which both magnitude and orientation fluctuations of the diffusivity can be clearly detected. This method is used to analyze the center-of-mass motion of four fundamental polymer models: the Rouse model, the Zimm model, a reptation model, and a rigid rodlike polymer. It is found that these models exhibit distinctly different types of magnitude and orientation fluctuations of diffusivity. This is an advantage of the present method over previous ones, such as the ergodicity-breaking parameter and a non-Gaussian parameter, because with either of these parameters it is difficult to distinguish the dynamics of the four polymer models. Also, the present method of a time-averaged MSD tensor could be used to analyze trajectory data obtained in SPT experiments.

Ergodic properties of Lévy flights coexisting with subdiffusion and related models

We analyze ergodic properties of two different stationary processes resulting from a combination of Lévy flights and subdiffusion. The first one comes from the Lamperti transformation of subordinated γ-stable process. The second one is a sequence of increments of subordinated Lévy process. We prove that both processes are mixing and ergodic. We also study increments and the Lamperti transformation of time-changed fractional Brownian motion. It appears that these models of anomalous diffusion are also ergodic and mixing. We believe that the derived results will help to verify ergodicity in single particle tracking experiments.

Mean perimeter and mean area of the convex hull over planar random walks

We investigate the geometric properties of the convex hull over n successive positions of a planar random walk, with a symmetric continuous jump distribution. We derive the large n asymptotic behavior of the mean perimeter. In addition, we compute the mean area for the particular case of isotropic Gaussian jumps. While the leading terms of these asymptotics are universal, the subleading (correction) terms depend on the finer details of the jump distribution and describe a ‘finite size effect’ of discrete-time jump processes, allowing one to accurately compute the mean perimeter and the mean area even for small n, as verified by Monte Carlo simulations. This is particularly valuable for applications dealing with discrete-time jumps processes and ranging from the statistical analysis of single-particle tracking experiments in microbiology to home range estimations in ecology.

Probing Membrane Receptor-Ligand Specificity with Surface- and Tip- Enhanced Raman Scattering.

The specific interaction between a ligand and a protein is a key component in minimizing off-target effects in drug discovery. Investigating these interactions with membrane protein receptors can be quite challenging. In this report, we show how spectral variance observed in surface-enhanced Raman scattering (SERS) and tip-enhanced Raman scattering (TERS) can be correlated with ligand specificity in affinity-based assays. Variations in the enhanced Raman spectra of three peptide ligands (i.e., cyclic-RGDFC, cyclic-isoDGRFC, and CisoDGRC), which have different binding affinity to αvβ3 integrin, are reported from isolated proteins and from receptors in intact cancer cell membranes. The SERS signal from the purified proteins provides basis spectra to analyze the signals in cells. Differences in the spectral variance within the SERS and TERS data for each ligand indicate larger variance for nonspecific ligand–receptor interactions. The SERS and TERS results are correlated with single particle tracking experiments of the ligand-functionalized nanoparticles with purified receptors on glass surfaces and living cells. These results demonstrate the ability to elucidate protein–ligand recognition using the observed vibrational spectra and provide perspective on binding specificity for small-molecule ligands in intact cell membranes, demonstrating a new approach for investigating drug specificity.