Detection of non-Brownian diffusion in the cell membrane in single molecule tracking

Fast Single-Particle Tracking of Membrane Proteins Combined with Super-Resolution Imaging of Actin Nanodomains

Membrane protein dynamics regulates cell functions by initiating downstream signaling cascades. The cell membrane is compartmentalized into nanodomains by actin structures, restricting lateral protein diffusion. Single-particle tracking offers high spatiotemporal resolution for studying protein dynamics in living cells. However, directly observing actin filaments that form barriers of nanodomains for fast protein diffusion is challenging due to their size being below the diffraction limit. Single-molecule localization microscopy resolves these structures but requires imaging in fixed cells. We integrated fast single-particle tracking with single-molecule localization microscopy to generate a dataset of membrane protein dynamics and actin filaments within the same cells at the nanoscales. Optimizing the fixation protocol enabled transition from live-cell tracking to fixed-cell super-resolution imaging. Data for the high-affinity IgE receptor, a transmembrane protein, and the GPI-anchored protein, an outer leaflet protein, was collected at 490 Hz. After fixation, actin filaments were imaged using dSTORM. The treatment of actin structures with phalloidin and PMA generated a dataset of distinct actin architectures for studying their potential influence on membrane protein dynamics.

Self-diffusion is temperature independent on active membranes

Molecular transport maintains cellular structures and functions. For example, lipid and protein diffusion sculpts the dynamic shapes and structures on the cell membrane that perform essential cellular functions, such as cell signaling. Temperature variations in thermal equilibrium rapidly change molecular transport properties. The coefficient of lipid self-diffusion increases exponentially with temperature in thermal equilibrium, for example. Hence, maintaining cellular homeostasis through molecular transport is hard in thermal equilibrium in the noisy cellular environment, where temperatures can fluctuate widely due to local heat generation. In this paper, using both molecular and lattice-based modeling of membrane transport, we show that the presence of active transport originating from the cell's cytoskeleton can make the self-diffusion of the molecules on the membrane robust to temperature fluctuations. The resultant temperature-independence of self-diffusion keeps the precision of cellular signaling invariant over a broad range of ambient temperatures, allowing cells to make robust decisions. We have also found that the Kawasaki algorithm, the widely used model of lipid transport on lattices, predicts incorrect temperature dependence of lipid self-diffusion in equilibrium. We propose a new algorithm that correctly captures the equilibrium properties of lipid self-diffusion and reproduces experimental observations.

Trajectory Analysis in Single-Particle Tracking: From Mean Squared Displacement to Machine Learning Approaches

Single-particle tracking is a powerful technique to investigate the motion of molecules or particles. Here, we review the methods for analyzing the reconstructed trajectories, a fundamental step for deciphering the underlying mechanisms driving the motion. First, we review the traditional analysis based on the mean squared displacement (MSD), highlighting the sometimes-neglected factors potentially affecting the accuracy of the results. We then report methods that exploit the distribution of parameters other than displacements, e.g., angles, velocities, and times and probabilities of reaching a target, discussing how they are more sensitive in characterizing heterogeneities and transient behaviors masked in the MSD analysis. Hidden Markov Models are also used for this purpose, and these allow for the identification of different states, their populations and the switching kinetics. Finally, we discuss a rapidly expanding field-trajectory analysis based on machine learning. Various approaches, from random forest to deep learning, are used to classify trajectory motions, which can be identified by motion models or by model-free sets of trajectory features, either previously defined or automatically identified by the algorithms. We also review free software available for some of the analysis methods. We emphasize that approaches based on a combination of the different methods, including classical statistics and machine learning, may be the way to obtain the most informative and accurate results.

A non-toxic equinatoxin-II reveals the dynamics and distribution of sphingomyelin in the cytosolic leaflet of the plasma membrane

Sphingomyelin (SM) is a major sphingolipid in mammalian cells. SM is enriched in the extracellular leaflet of the plasma membrane (PM). Besides this localization, recent electron microscopic and biochemical studies suggest the presence of SM in the cytosolic leaflet of the PM. In the present study, we generated a non-toxic SM-binding variant (NT-EqtII) based on equinatoxin-II (EqtII) from the sea anemone Actinia equina, and examined the dynamics of SM in the cytosolic leaflet of living cell PMs. NT-EqtII with two point mutations (Leu26Ala and Pro81Ala) had essentially the same specificity and affinity to SM as wild-type EqtII. NT-EqtII expressed in the cytosol was recruited to the PM in various cell lines. Super-resolution microscopic observation revealed that NT-EqtII formed tiny domains that were significantly colocalized with cholesterol and N-terminal Lyn. Meanwhile, single molecule observation at high resolutions down to 1 ms revealed that all the examined lipid probes including NT-EqtII underwent apparent fast simple Brownian diffusion, exhibiting that SM and other lipids in the cytosolic leaflet rapidly moved in and out of domains. Thus, the novel SM-binding probe demonstrated the presence of the raft-like domain in the cytosolic leaflet of living cell PMs.

Freestanding bilayer microscope for single-molecule imaging of membrane proteins

Integral membrane proteins (IMPs) constitute a large fraction of organismal proteomes, playing fundamental roles in physiology and disease. Despite their importance, the mechanisms underlying dynamic features of IMPs, such as anomalous diffusion, protein-protein interactions, and protein clustering, remain largely unknown due to the high complexity of cell membrane environments. Available methods for in vitro studies are insufficient to study IMP dynamics systematically. This publication introduces the freestanding bilayer microscope (FBM), which combines the advantages of freestanding bilayers with single-particle tracking. The FBM, based on planar lipid bilayers, enables the study of IMP dynamics with single-molecule resolution and unconstrained diffusion. This paper benchmarks the FBM against total internal reflection fluorescence imaging on supported bilayers and is used here to estimate ion channel open probability and to examine the diffusion behavior of an ion channel in phase-separated bilayers. The FBM emerges as a powerful tool to examine membrane protein/lipid organization and dynamics to understand cell membrane processes.

Nanoparticle Anisotropy Increases Targeting Interactions on Live-Cell Membranes

This paper describes how branch lengths of anisotropic nanoparticles can affect interactions between grafted ligands and cell-membrane receptors. Using live-cell, single-particle tracking, we found that DNA aptamer-gold nanostar nanoconstructs with longer branches showed improved binding efficacy to human epidermal growth factor receptor 2 (HER2) on cancer cell membranes. Inhibiting nanoconstruct-HER2 binding promoted nonspecific interactions, which increased the rotational speed of long-branched nanoconstructs but did not affect that of short-branched constructs. Bivariate analysis of the rotational and translational dynamics showed that longer branch lengths increased the ratio of targeting to nontargeting interactions. We also found that longer branches increased the nanoconstruct-cell interaction times before internalization and decreased intracellular trafficking velocities. Differences in binding efficacy revealed by single-particle dynamics can be attributed to the distinct protein corona distributions on short- and long-branched nanoconstructs, as validated by transmission electron microscopy. Minimal protein adsorption at the high positive curvature tips of long-branched nanoconstructs facilitated binding of DNA aptamer ligands to HER2. Our study reveals the significance of nanoparticle branch length in regulating local chemical environment and interactions with live cells at the single-particle level.

Impact of temporal resolution in single particle tracking analysis

Temporal resolution is a key parameter in the observation of dynamic processes, as in the case of single molecules motions visualized in real time in two-dimensions by wide field (fluorescence) microscopy, but a systematic investigation of its effects in all the single particle tracking analysis steps is still lacking. Here we present tools to quantify its impact on the estimation of diffusivity and of its distribution using one of the most popular tracking software for biological applications on simulated data and movies. We found important shifts and different widths for diffusivity distributions, depending on the interplay of temporal sampling conditions with various parameters, such as simulated diffusivity, density of spots, signal-to-noise ratio, lengths of trajectories, and kind of boundaries in the simulation. We examined conditions starting from the ones of experiments on the fluorescently labelled receptor p75NTR, a relatively fast-diffusing membrane receptor (diffusivity around 0.5-1 µm2/s), visualized by TIRF microscopy on the basal membrane of living cells. From the analysis of the simulations, we identified the best conditions in cases similar to these ones; considering also the experiments, we could confirm a range of values of temporal resolution suitable for obtaining reliable diffusivity results. The procedure we present can be exploited in different single particle/molecule tracking applications to find an optimal temporal resolution.

Multifractal spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes, characterized by their nonstandard scaling of the mean-squared displacement, pose a unique challenge in classification and characterization. In a previous study [Mangalam et al., Phys. Rev. Res. 5, 023144 (2023)2643-156410.1103/PhysRevResearch.5.023144], we established a comprehensive framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous-time random walk, annealed transient time motion, and Lévy walk. We generate extensive datasets comprising 10^{6} trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory to accomplish this. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. We also explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and p varation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. In summary, our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing the predictive capacity of machine learning to classify anomalous diffusion processes.

Coherent light scattering from cellular dynamics in living tissues

This review examines the biological physics of intracellular transport probed by the coherent optics of dynamic light scattering from optically thick living tissues. Cells and their constituents are in constant motion, composed of a broad range of speeds spanning many orders of magnitude that reflect the wide array of functions and mechanisms that maintain cellular health. From the organelle scale of tens of nanometers and upward in size, the motion inside living tissue is actively driven rather than thermal, propelled by the hydrolysis of bioenergetic molecules and the forces of molecular motors. Active transport can mimic the random walks of thermal Brownian motion, but mean-squared displacements are far from thermal equilibrium and can display anomalous diffusion through Lévy or fractional Brownian walks. Despite the average isotropic three-dimensional environment of cells and tissues, active cellular or intracellular transport of single light-scattering objects is often pseudo-one-dimensional, for instance as organelle displacement persists along cytoskeletal tracks or as membranes displace along the normal to cell surfaces, albeit isotropically oriented in three dimensions. Coherent light scattering is a natural tool to characterize such tissue dynamics because persistent directed transport induces Doppler shifts in the scattered light. The many frequency-shifted partial waves from the complex and dynamic media interfere to produce dynamic speckle that reveals tissue-scale processes through speckle contrast imaging and fluctuation spectroscopy. Low-coherence interferometry, dynamic optical coherence tomography, diffusing-wave spectroscopy, diffuse-correlation spectroscopy, differential dynamic microscopy and digital holography offer coherent detection methods that shed light on intracellular processes. In health-care applications, altered states of cellular health and disease display altered cellular motions that imprint on the statistical fluctuations of the scattered light. For instance, the efficacy of medical therapeutics can be monitored by measuring the changes they induce in the Doppler spectra of livingex vivocancer biopsies.

Fractional and scaled Brownian motion on the sphere: The effects of long-time correlations on navigation strategies

We analyze fractional Brownian motion and scaled Brownian motion on the two-dimensional sphere S^{2}. We find that the intrinsic long-time correlations that characterize fractional Brownian motion collude with the specific dynamics (navigation strategies) carried out on the surface giving rise to rich transport properties. We focus our study on two classes of navigation strategies: one induced by a specific set of coordinates chosen for S^{2} (we have chosen the spherical ones in the present analysis), for which we find that contrary to what occurs in the absence of such long-time correlations, nonequilibrium stationary distributions are attained. These results resemble those reported in confined flat spaces in one and two dimensions [Guggenberger et al. New J. Phys. 21, 022002 (2019)1367-263010.1088/1367-2630/ab075f; Vojta et al. Phys. Rev. E 102, 032108 (2020)2470-004510.1103/PhysRevE.102.032108]; however, in the case analyzed here, there are no boundaries that affect the motion on the sphere. In contrast, when the navigation strategy chosen corresponds to a frame of reference moving with the particle (a Frenet-Serret reference system), then the equilibrium distribution on the sphere is recovered in the long-time limit. For both navigation strategies, the relaxation times toward the stationary distribution depend on the particular value of the Hurst parameter. We also show that on S^{2}, scaled Brownian motion, distinguished by a time-dependent diffusion coefficient with a power-scaling, is independent of the navigation strategy finding a good agreement between the analytical calculations obtained from the solution of a time-dependent diffusion equation on S^{2}, and the numerical results obtained from our numerical method to generate ensemble of trajectories.

Implications of different membrane compartmentalization models in particle-based in silico studies

Studying membrane dynamics is important to understand the cellular response to environmental stimuli. A decisive spatial characteristic of the plasma membrane is its compartmental structure created by the actin-based membrane-skeleton (fences) and anchored transmembrane proteins (pickets). Particle-based reaction-diffusion simulation of the membrane offers a suitable temporal and spatial resolution to analyse its spatially heterogeneous and stochastic dynamics. Fences have been modelled via hop probabilities, potentials or explicit picket fences. Our study analyses the different approaches' constraints and their impact on simulation results and performance. Each of the methods comes with its own constraints; the picket fences require small timesteps, potential fences might induce a bias in diffusion in crowded systems, and probabilistic fences, in addition to carefully scaling the probability with the timesteps, induce higher computational costs for each propagation step.

Recently developed glycosphingolipid probes and their dynamic behavior in cell plasma membranes as revealed by single-molecule imaging

Glycosphingolipids, including gangliosides, are representative lipid raft markers that perform a variety of physiological roles in cell membranes. However, studies aimed at revealing their dynamic behavior in living cells are rare, mostly due to a lack of suitable fluorescent probes. Recently, the ganglio-series, lacto-series, and globo-series glycosphingolipid probes, which mimic the behavior of the parental molecules in terms of partitioning to the raft fraction, were developed by conjugating hydrophilic dyes to the terminal glycans of glycosphingolipids using state-of-art entirely chemical-based synthetic techniques. High-speed, single-molecule observation of these fluorescent probes revealed that gangliosides were scarcely trapped in small domains (100 nm in diameter) for more than 5 ms in steady-state cells, suggesting that rafts including gangliosides were always moving and very small. Furthermore, dual-color, single-molecule observations clearly showed that homodimers and clusters of GPI-anchored proteins were stabilized by transiently recruiting sphingolipids, including gangliosides, to form homodimer rafts and the cluster rafts, respectively. In this review, we briefly summarize recent studies, the development of a variety of glycosphingolipid probes as well as the identification of the raft structures including gangliosides in living cells by single-molecule imaging.

Single-Molecule Tracking of Reagent Diffusion during Chemical Reactions

Recent experiments have shown that the diffusion of reagent molecules is inconsistent with what the Stokes-Einstein equation predicts during a chemical reaction. Here, we used single-molecule tracking to observe the diffusion of reactive reagent molecules during click and Diels-Alder (DA) reactions. We found that the diffusion coefficient of the reagents remained unchanged within the experimental uncertainty upon the DA reaction. Yet, diffusion of reagent molecules is faster than predicted during the click reaction when the reagent concentration and catalyst concentration exceed a threshold. A stepwise analysis suggested that the fast diffusion scenario is due to the reaction but not the involvement of the tracer with the reaction itself. The present results provide experimental evidence on the faster-than-expected reagent diffusion during a CuAAC reaction in specific conditions and propose new insights into understanding this unexpected behavior.

Single-Molecule Imaging of Ganglioside Probes in Living Cell Plasma Membranes

Gangliosides play a variety of physiological roles and are one of the most important lipid raft constituents. However, their dynamic behaviors have scarcely been investigated in living cells because of the lack of fluorescent probes that behave like their parental molecules. Recently, fluorescent ganglioside probes that mimic native ganglioside behaviors have been developed. In this chapter, I discuss the recent advances in research related to the lateral localization and dynamic behaviors of gangliosides in the plasma membranes of living cells.

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.

A method for single particle tracking through a multimode fiber

Multimode optical fiber (MMF) endoscopes have recently gained widespread attention as a novel tool for imaging deep within tissue using light microscopy. We here present a method for particle tracking through the MMF, which overcomes the lack of a fast enough wide-field fluorescence imaging modality for this type of endoscope, namely a discrete implementation of orbital particle tracking. We achieve biologically relevant tracking speeds (up to 1.2 μm/s) despite using a slow SLM for the wavefront shaping. We demonstrate a tracking accuracy of λ/50 for a 0.3 NA fiber and show tracking of a pinhole moving to mimic Brownian motion with diffusion rates of up to 0.3 μm²/s.

Brownian motion on an out-of-thermal-equilibrium surface

The motion of colloidal species on an out-of-thermal equilibrium surface is studied experimentally by optical microscopy. Water droplets of size in the micrometer range, spontaneously formed at a spherical-like interface between water and oil, are the colloidal species. The interface appears as a convex meniscus when putting water on oil with an added nonionic surfactant. Since the water density is greater than that of oil, the interface is produced into the oil. The spontaneously formed water droplets move attached to the interface while still growing from submicrometer sizes to a few micrometers. Although the dynamic nature of the process, with both the interface and the particles still changing, produces heterogeneities in the system, anomalous diffusion was not observed. The motion of the droplets has a well-identified Brownian component with a Gaussian distribution of steps due to the thermal agitation of the media surrounding the droplets and a drift component due to the effect of gravity.

Effect of Protein–Protein Interactions on Translational Diffusion of Spheroidal Proteins

One of the commonly accepted approaches to estimate protein–protein interactions (PPI) in aqueous solutions is the analysis of their translational diffusion. The present review article observes a phenomenological approach to analyze PPI effects via concentration dependencies of self- and collective translational diffusion coefficient for several spheroidal proteins derived from the pulsed field gradient NMR (PFG NMR) and dynamic light scattering (DLS), respectively. These proteins are rigid globular α-chymotrypsin (ChTr) and human serum albumin (HSA), and partly disordered α-casein (α-CN) and β-lactoglobulin (β-Lg). The PPI analysis enabled us to reveal the dominance of intermolecular repulsion at low ionic strength of solution (0.003–0.01 M) for all studied proteins. The increase in the ionic strength to 0.1–1.0 M leads to the screening of protein charges, resulting in the decrease of the protein electrostatic potential. The increase of the van der Waals potential for ChTr and α-CN characterizes their propensity towards unstable weak attractive interactions. The decrease of van der Waals interactions for β-Lg is probably associated with the formation of stable oligomers by this protein. The PPI, estimated with the help of interaction potential and idealized spherical molecular geometry, are in good agreement with experimental data.

Modeling Receptor Motility along Advecting Lipid Membranes

This work aims to overview multiphysics mechanobiological computational models for receptor dynamics along advecting cell membranes. Continuum and statistical models of receptor motility are the two main modeling methodologies identified in reviewing the state of the art. Within the former modeling class, a further subdivision based on different biological purposes and processes of proteins’ motion is recognized; cell adhesion, cell contractility, endocytosis, and receptor relocations on advecting membranes are the most relevant biological processes identified in which receptor motility is pivotal. Numerical and/or experimental methods and approaches are highlighted in the exposure of the reviewed works provided by the literature, pertinent to the topic of the present manuscript. With a main focus on the continuum models of receptor motility, we discuss appropriate multiphyisics laws to model the mass flux of receptor proteins in the reproduction of receptor relocation and recruitment along cell membranes to describe receptor–ligand chemical interactions, and the cell’s structural response. The mass flux of receptor modeling is further supported by a discussion on the methodology utilized to evaluate the protein diffusion coefficient developed over the years.

Electrostatically Controlled ex Situ and in Situ Polymerization of Diacetylene-Containing Peptide Amphiphiles in Living Cells

Precise control of diacetylene-containing peptide amphiphile (DPA) based supramolecular architectures is important for their in cellulo polymerization behaviors and biomedical applications. Herein, we reported two DPAs (cationic PA-NH₂ and zwitterionic PA-OH) with a similar molecular structure, which exhibited completely opposite polymerization behaviors in aqueous solution and living cells. Specifically, PA-NH₂ was unpolymerizable in aqueous solution but underwent in cellulo polymerization to respond to the intracellular microenvironment. On the contrary, zwitterionic PA-OH was polymerized in solution, rather than inside living cells. Based on the results of cell viability and total internal reflection fluorescent microscopy measurement, PA-OH exhibited higher affinity with cell membranes and lower cytotoxicity than those of PA-NH₂. Therefore, it is suggested that the in cellulo polymerization of PA-NH₂ should be responsive for greater cytotoxicity, rather than the membrane affinity. This study provides an in-depth understanding of the role of charge properties in the polymerization behavior of DPAs and seeks their potential biomedical applications.

Heterogeneity of cell membrane structure studied by single molecule tracking

Heterogeneity in cell membrane structure, typified by microdomains with different biophysical and biochemical properties, is thought to impact on a variety of cell functions. Integral membrane proteins act as nanometre-sized probes of the lipid environment and their thermally-driven movements can be used to report local variations in membrane properties. In the current study, we have used total internal reflection fluorescence microscopy (TIRFM) combined with super-resolution tracking of multiple individual molecules, in order to create high-resolution maps of local membrane viscosity. We used a quadrat sampling method and show how statistical tests for membrane heterogeneity can be conducted by analysing the paths of many molecules that pass through the same unit area of membrane. We describe experiments performed on cultured primary cells, stable cell lines and ex vivo tissue slices using a variety of membrane proteins, under different imaging conditions. In some cell types, we find no evidence for heterogeneity in mobility across the plasma membrane, but in others we find statistically significant differences with some regions of membrane showing significantly higher viscosity than others.

We use total internal reflection fluorescence microscopy combined with super-resolution tracking of multiple individual molecules, in order to create high-resolution maps of local membrane viscosity.

Revealing the dynamic adsorption and diffusion of peptide amphiphile on supported lipid bilayer by single molecule experiment and simulation

Dynamic adsorption and diffusion of peptide amphiphiles (PAs) with different numbers of alkyl tails on supported lipid bilayers (SLBs) were investigated by single molecule tracking experiment and molecule dynamic simulation. Experimental results show two distinct populations of PAs with different residence time. Residence time of adsorbed PAs increases with the increase of the alkyl tails, whereas diffusion coefficient monotonically decreases with rising the number of alkyl tails and also decreases with increasing the mobility of SLBs. All-atom molecule dynamic simulation results prove that the adsorption and diffusion of PAs on SLB surface are mainly determined by interactions between PAs and SLBs and also the intrinsic mobility of PAs in aqueous solution. The electrostatic interactions and the mobility of PAs are two dominated but contradictory coefficients for the adsorption and diffusion of PAs. By increasing the alkyl tails, the mobility of PAs decreases while the electrostatic property does not change significantly, resulting in the increase of residence time and decrease of diffusivity of PAs. Additionally, for the PAs with large number of alkyl tails (≥ 3 alkyl tails), steric hindrance of alkyl tails leads to the decrease of adsorption probability of PAs on SLB surface. These findings lay the groundwork for guiding the design of PAs with high cell affinity, potentially useful for efficient drug delivery.

Reconstructing the Intrinsic Statistical Properties of Intermittent Locomotion Through Corrections for Boundary Effects

Locomotion characteristics are often recorded within bounded spaces, a constraint which introduces geometry-specific biases and potentially complicates the inference of behavioural features from empirical observations. We describe how statistical properties of an uncorrelated random walk, namely the steady-state stopping location probability density and the empirical step probability density, are affected by enclosure in a bounded space. The random walk here is considered as a null model for an organism moving intermittently in such a space, that is, the points represent stopping locations and the step is the displacement between them. Closed-form expressions are derived for motion in one dimension and simple two-dimensional geometries, in addition to an implicit expression for arbitrary (convex) geometries. For the particular choice of no-go boundary conditions, we demonstrate that the empirical step distribution is related to the intrinsic step distribution, i.e. the one we would observe in unbounded space, via a multiplicative transformation dependent solely on the boundary geometry. This conclusion allows in practice for the compensation of boundary effects and the reconstruction of the intrinsic step distribution from empirical observations.

Identifiability analysis for stochastic differential equation models in systems biology

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

High-Precision Protein-Tracking With Interferometric Scattering Microscopy

The mobility of proteins and lipids within the cell, sculpted oftentimes by the organization of the membrane, reveals a great wealth of information on the function and interaction of these molecules as well as the membrane itself. Single particle tracking has proven to be a vital tool to study the mobility of individual molecules and unravel details of their behavior. Interferometric scattering (iSCAT) microscopy is an emerging technique well-suited for visualizing the diffusion of gold nanoparticle-labeled membrane proteins to a spatial and temporal resolution beyond the means of traditional fluorescent labels. We discuss the applicability of interferometric single particle tracking (iSPT) microscopy to investigate the minutia in the motion of a protein through measurements visualizing the mobility of the epidermal growth factor receptor in various biological scenarios on the live cell.

Gaining insight into cellular cardiac physiology using single particle tracking

Single particle tracking (SPT) is a robust technique to monitor single-molecule behaviors in living cells directly. By this approach, we can uncover the potential biological significance of particle dynamics by statistically characterizing individual molecular behaviors. SPT provides valuable information at the single-molecule level, that could be obscured by simple averaging that is inherent to conventional ensemble measurements. Here, we give a brief introduction to SPT including the commonly used optical implementations, fluorescence labeling strategies, and data analysis methods. We then focus on how SPT has been harnessed to decipher myocardial function. In this context, SPT has provided novel insight into the lateral diffusion of signal receptors and ion channels, the dynamic organization of cardiac nanodomains, subunit composition and stoichiometry of cardiac ion channels, myosin movement along actin filaments, the kinetic features of transcription factors involved in cardiac remodeling, and the intercellular communication by nanotubes. Finally, we speculate on the prospects and challenges of applying SPT to future questions regarding cellular cardiac physiology using SPT.

Diffusive transport of nanoscale objects through cell membranes: a computational perspective

Diffusion is an essential and fundamental means of transport of substances on cell membranes, and the dynamics of biomembranes plays a crucial role in the regulation of numerous cellular processes. The understanding of the complex mechanisms and the nature of particle diffusion have a bearing on establishing guidelines for the design of efficient transport materials and unique therapeutic approaches. Herein, this review article highlights the most recent advances in investigating diffusion dynamics of nanoscale objects on biological membranes, focusing on the approaches of tailored computer simulations and theoretical analysis. Due to the presence of the complicated and heterogeneous environment on native cell membranes, the diffusive transport behaviors of nanoparticles exhibit unique and variable characteristics. The general aspects and basic theories of normal diffusion and anomalous diffusion have been introduced. In addition, the influence of a series of external and internal factors on the diffusion behaviors is discussed, including particle size, membrane curvature, particle-membrane interactions or particle-inclusion, and the crowding degree of membranes. Finally, we seek to identify open problems in the existing experimental, simulation, and theoretical research studies, and to propose challenges for future development.

Dynamic heterogeneity and non-Gaussian statistics for ganglioside GM1s and acetylcholine receptors on live cell membrane

We have carried out a comparative study of the lateral motion of ganglioside GM1, which is a glycosphingolipid residing on the outer leaflet of the plasma membrane, and acetylcholine receptor (AChR), which is a well-characterized ion channel. Both the lipid molecules and the transmembrane proteins reside on the plasma membranes of live Xenopus muscle cells. From a thorough analysis of a large volume of individual molecular trajectories obtained from more than 300 live cells over a wide range of sampling rates and long durations, we find that the GM1s and AChRs share the same dynamic heterogeneity and non-Gaussian statistics. Our measurements with the ATP-depleted cells reveal that the diffusion dynamics of the GM1s and AChRs is uniformly affected by the intracellular ATP level of the living muscle cells, further demonstrating that membrane diffusion is strongly coupled to the dynamics of the underlying cortical actin network, as predicted by the dynamic picket-fence model.

Matrix mechanics and water permeation regulate extracellular vesicle transport

Cells release extracellular vesicles (EVs) to communicate over long distances, which requires EVs to traverse the extracellular matrix (ECM). However, given that the size of EVs is usually larger than the mesh size of the ECM, it is not clear how they can travel through the dense ECM. Here we show that, in contrast to synthetic nanoparticles, EVs readily transport through nanoporous ECM. Using engineered hydrogels, we demonstrate that the mechanical properties of the matrix regulate anomalous EV transport under confinement. Matrix stress relaxation allows EVs to overcome the confinement, and a higher crosslinking density facilitates a fluctuating transport motion through the polymer mesh, which leads to free diffusion and fast transport. Furthermore, water permeation through aquaporin-1 mediates the EV deformability, which further supports EV transport in hydrogels and a decellularized matrix. Our results provide evidence for the nature of EV transport within confined environments and demonstrate an unexpected dependence on matrix mechanics and water permeation.

Resolving Single-Nanoconstruct Dynamics during Targeting and Nontargeting Live-Cell Membrane Interactions

This paper describes how differences in the dynamics of targeting and nontargeting constructs can provide information on nanoparticle (NP)-cell interactions. We probed translational and rotational dynamics of functionalized Au nanostar (AuNS) nanoconstructs interacting with cells in serum-containing medium. We found that AuNS with targeting ligands had a larger dynamical footprint and faster rotational speed on cell membranes expressing human epidermal growth factor receptor 2 (HER-2) receptors compared to that of AuNS with nontargeting ligands. Targeting and nontargeting nanoconstructs displayed distinct membrane dynamics despite their similar protein adsorption profiles, which suggests that targeted interactions are preserved even in the presence of a protein corona. The high sensitivity of single-NP dynamics can be used to compare different nanoconstruct properties (such as NP size, shape, and surface chemistry) to improve their design as delivery vehicles.

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.

SIMPLE: Structured illumination based point localization estimator with enhanced precision

We present a structured illumination microscopy based point localization estimator (SIMPLE) that achieves a 2-fold increase in single molecule localization precision compared to conventional centroid estimation methods. SIMPLE advances the recently introduced MINFLUX concept by using precisely phase-shifted sinusoidal wave patterns as nanometric rulers for simultaneous particle localization based on photon count variation over a 20 μm field of view. We validate SIMPLE in silico and experimentally on a TIRF-SIM setup using a digital micro-mirror device (DMD) as a spatial light modulator.

Cadherin Extracellular Domain Clustering in the Absence of Trans-Interactions

While both cis and trans (adhesive)-interactions cooperate in the assembly of intercellular adhesions, computational simulations have predicted that two-dimensional confinement may promote cis-oligomerization, in the absence of trans-interactions. Here, single-molecule tracking of cadherin extracellular domains on supported lipid bilayers revealed the density-dependent formation of oligomers and cis-clusters in the absence of trans-interactions. Lateral oligomers were virtually eliminated by mutating a putative cis (lateral) binding interface. At low cadherin surface coverage, wild-type and mutant cadherin diffused rapidly, consistent with the motion of a lipid molecule within a cadherin-free supported bilayer and with cadherins diffusing as monomers. Although the diffusion of mutant cadherin did not change appreciably with increasing surface coverage, the average short-time diffusion coefficient of wild-type cadherin slowed significantly above a fractional surface coverage of ∼0.01 (∼1100 molecules/μm2). A detailed analysis of molecular trajectories suggested the presence of a broad size distribution of cis-cadherin oligomers. These findings verify predictions that two-dimensional confinement promotes cis-oligomerization, in the absence of trans-interactions.

Influence of Bilayer Size and Number in Multi-Bilayer DOPC Simulations at Full and Low Hydration

Biophysical studies of model cell membranes at full and low hydration are usually carried out using scattering measurements on multi-bilayer systems. Molecular simulations of lipid bilayers aimed at reproducing those experimental conditions are usually conducted using single bilayers with different amounts of water. These simulation conditions may lead to artifacts arising from size effects and self-interactions because of periodic boundary conditions. We have tested the influence of the size and number of bilayers on membrane properties using the Lipid14 force field for lipids in molecular dynamics simulations of 1,2-dioleoyl- sn-glycero-3-phosphocholine bilayers at full hydration (44 water molecules per lipid), low hydration (18 water molecules per lipid), and dehydration (9 water molecules per lipid). A number of additional simulations were conducted with the Slipids force field for comparison. We have found that the average area per lipid (APL), thickness, mass density profiles, and acyl tail order parameters are insensitive to the size and the number of bilayers for all hydration states. The Lipid14 force field can also successfully reproduce the experimentally observed decrease in APL and corresponding increase in bilayer thickness upon dehydration, reflecting the increase in ordering as the system becomes more gel-like. Additionally, decreasing hydration levels were associated with a trend away from normal lateral diffusion and toward more subdiffusive regimes across both force fields. In summary, at least for the Lipid14 force field, the use of a single bilayer with 128 phospholipid molecules provides an adequate representation of multi-bilayer systems at varying levels of hydration.

Time-Dependent Diffusion Coefficients for Chaotic Advection due to Fluctuations of Convective Rolls

The properties of chaotic advection arising from defect turbulence, that is, weak turbulence in the electroconvection of nematic liquid crystals, were experimentally investigated. Defect turbulence is a phenomenon in which fluctuations of convective rolls arise and are globally disturbed while maintaining convective rolls locally. The time-dependent diffusion coefficient, as measured from the motion of a tagged particle driven by the turbulence, was used to clarify the dependence of the type of diffusion on coarse-graining time. The results showed that, as coarse-graining time increases, the type of diffusion changes from superdiffusion → subdiffusion → normal diffusion. The change in diffusive properties over the observed timescale reflects the coexistence of local order and global disorder in the defect turbulence.

Urea-mediated anomalous diffusion in supported lipid bilayers

Diffusion in biological membranes is seldom simply Brownian motion; instead, the rate of diffusion is dependent on the time scale of observation and so is often described as anomalous. In order to help better understand this phenomenon, model systems are needed where the anomalous diffusion of the lipid bilayer can be tuned and quantified. We recently demonstrated one such model by controlling the excluded area fraction in supported lipid bilayers (SLBs) through the incorporation of lipids derivatized with polyethylene glycol. Here, we extend this work, using urea to induce anomalous diffusion in SLBs. By tuning incubation time and urea concentration, we produce bilayers that exhibit anomalous behaviour on the same scale as that observed in biological membranes.

Blob Size Controls Diffusion of Free Polymer in a Chemically Identical Brush in Semidilute Solution

The diffusion of rhodamine-labeled poly(ethylene glycol) (r-PEG) within surface-grafted poly(ethylene glycol) (s-PEG) layers in aqueous solution at 18 °C was measured by fluorescence correlation spectroscopy. The diffusion coefficient of r-PEG within s-PEG was controlled by the grafting density, σ, and scaled as σ^{–1.42±0.09}. It is proposed that a characteristic blob size associated with the grafted (brush) layer defines the region through which the r-PEG diffusion occurs. The diffusion coefficients for r-PEG in semidilute solution were found to be similar to those in the brushes.

Actin filaments partition primary cilia membranes into distinct fluid corrals

Lee et al. examine the dynamics of membrane proteins within the ciliary membrane using quantum dots and 2P Super FRAP. They show that ciliary membrane proteins diffuse rapidly within highly fluid local membrane domains delimited by actin filaments.

Physical properties of primary cilia membranes in living cells were examined using two independent, high-spatiotemporal-resolution approaches: fast tracking of single quantum dot–labeled G protein–coupled receptors and a novel two-photon super-resolution fluorescence recovery after photobleaching of protein ensemble. Both approaches demonstrated the cilium membrane to be partitioned into corralled domains spanning 274 ± 20 nm, within which the receptors are transiently confined for 0.71 ± 0.09 s. The mean membrane diffusion coefficient within the corrals, D_m1 = 2.9 ± 0.41 µm²/s, showed that the ciliary membranes were among the most fluid encountered. At longer times, the apparent membrane diffusion coefficient, D_m2 = 0.23 ± 0.05 µm²/s, showed that corral boundaries impeded receptor diffusion 13-fold. Mathematical simulations predict the probability of G protein–coupled receptors crossing corral boundaries to be 1 in 472. Remarkably, latrunculin A, cytochalasin D, and jasplakinolide treatments altered the corral permeability. Ciliary membranes are thus partitioned into highly fluid membrane nanodomains that are delimited by filamentous actin.

SPT and Imaging FCS Provide Complementary Information on the Dynamics of Plasma Membrane Molecules

The dynamics of biomolecules in the plasma membrane is of fundamental importance to understanding cellular processes. Cellular signaling often starts with extracellular ligand binding to a membrane receptor, which then transduces an intracellular signal. Ligand binding and receptor-complex activation often involve a complex rearrangement of proteins in the membrane, which results in changes in diffusion properties. Two widely used methods to characterize biomolecular diffusion are single-particle tracking (SPT) and imaging total internal reflection fluorescence correlation spectroscopy (ITIR-FCS). Here, we compare the results of recovered diffusion coefficients and mean-square displacements of the two methods by simulations of free, domain-confined, or meshwork diffusion. We introduce, to our knowledge, a new method for the determination of confinement radii from ITIR-FCS data. We further establish and demonstrate simultaneous SPT/ITIR-FCS for direct comparison within living cells. Finally, we compare the results obtained by SPT and ITIR-FCS for the receptor tyrosine kinase MET. Our results show that SPT and ITIR-FCS yield complementary information on diffusion properties of biomolecules in cell membranes.

Dynamic pattern generation in cell membranes: Current insights into membrane organization

It has been two decades since the lipid raft hypothesis was first presented. Even today, whether these nanoscale cholesterol-rich domains are present in cell membranes is not completely resolved. However, especially in the last few years, a rich body of literature has demonstrated both the presence and the importance of non-random distribution of biomolecules on the membrane, which is the focus of this review. These new developments have pushed the experimental limits of detection and have brought us closer to observing lipid domains in the plasma membrane of live cells. Characterization of biomolecules associated with lipid rafts has revealed a deep connection between biological regulation and function and membrane compositional heterogeneities. Finally, tantalizing new developments in the field have demonstrated that lipid domains might not just be associated with the plasma membrane of eukaryotes but could potentially be a ubiquitous membrane-organizing principle in several other biological systems. This article is part of a Special Issue entitled: Emergence of Complex Behavior in Biomembranes edited by Marjorie Longo.

Single-Particle Tracking of Cell Surface Proteins

Single-particle tracking has been used extensively to advance our understanding of the plasma membrane and the mechanisms controlling the movement of cell surface proteins. These studies provide fundamental insights into the regulation of membrane receptor activation and the assembly of signaling clusters. Here, we describe a method to label and track B cell receptor (BCR) and other cell surface proteins and how this method can be adapted to simultaneously track two molecular species or examine the movement of membrane proteins in relation to membrane microdomains. We recently used this method to study the role of the actin cytoskeleton in the regulation of B cell receptor dynamics at the cell surface.

A Simple and Powerful Analysis of Lateral Subdiffusion Using Single Particle Tracking

In biological membranes, many factors such as cytoskeleton, lipid composition, crowding, and molecular interactions deviate lateral diffusion from the expected random walks. These factors have different effects on diffusion but act simultaneously, so the observed diffusion is a complex mixture of diffusive behaviors (directed, Brownian, anomalous, or confined). Therefore, commonly used approaches to quantify diffusion based on averaging of the displacements such as the mean square displacement, are not adapted to the analysis of this heterogeneity. We introduce a parameter-the packing coefficient Pc, which gives an estimate of the degree of free movement that a molecule displays in a period of time independently of its global diffusivity. Applying this approach to two different situations (diffusion of a lipid probe and trapping of receptors at synapses), we show that Pc detected and localized temporary changes of diffusive behavior both in time and in space. More importantly, it allowed the detection of periods with very high confinement as well as their frequency and duration, and thus it can be used to calculate the effective k_on and k_off of scaffolding interactions such as those that immobilize receptors at synapses.

Spatiotemporal Fluctuation Analysis of Molecular Diffusion Laws in Live-Cell Membranes

A present challenge of membrane biophysics is deciphering the dynamic behavior of molecules, such as lipids and proteins, within the natural environment of a living-cell membrane. Here, a fluorescence fluctuation-based approach will be described, which makes it possible to probe the "diffusion law" of molecules directly from imaging, in the form of a mean square displacement vs time-delay plot (iMSD), with no need for interpretative models. Of note, the presented approach does not require extraction of the molecular trajectories nor the use of bright fluorophores. Conversely, it can be used at high fluorophore density and with relatively dim fluorophores, such as GFP-tagged molecules transiently expressed within cells. The ability of this approach to resolve average molecular dynamic properties well below the diffraction limit will be discussed. Overall, this novel approach is proposed as a powerful tool for the determination of kinetic and thermodynamic parameters over wide spatial and temporal scales.

The imaging FCS diffusion law in the presence of multiple diffusive modes

The cellular plasma membrane is the barrier over which cells exchange materials and communicate with their surroundings, and thus plays the central role in cellular sensing and metabolism. Therefore, the investigation of plasma membrane organization and dynamics is required for understanding of cellular functions. The plasma membrane is a heterogeneous matrix. The presence of structures such as lipid and protein domains and the cytoskeleton meshwork poses a hindrance to the free diffusion of membrane associated biomolecules. However, these domains and the cytoskeleton meshwork barriers are below the optical diffraction limit with potentially short lifetimes and are not easily detected even in super-resolution microscopy. Therefore, dynamic measurements are often used to indirectly prove the existence of domains and barriers by analyzing the mode of diffusion of probe molecules. One of these tools is the Fluorescence Correlation Spectroscopy (FCS) diffusion law. The FCS diffusion law is a plot of diffusion time (τ_d) versus observation area. For at least three different diffusive modes - free, domain confined, and meshwork hindered hop diffusion - the expected plots have been characterized, typically by its y-intercept (τ₀) when fit with a linear model, and have been verified in many cases. However, a description of τ₀ has only been given for pure diffusive modes. But in many experimental cases it is not evident that a protein will undergo only one kind of diffusion, and thus the interpretation of the τ₀ value is problematic. Here, we therefore address the question about the absolute value of τ₀ in the case of complex diffusive modes, i.e. when either one molecule is domain confined and cytoskeleton hindered or when two molecules exhibit different diffusive behavior at the same position in a sample. In addition, we investigate how τ₀ changes when the diffusive mode of a probe alters upon disruption of domains or the cytoskeleton by drug treatments. By a combination of experimental studies and simulations, we show that τ₀ is not influenced equally by the different diffusive modes as typically found in cellular environments, and that it is the relative change of τ₀ rather than its absolute value that provides information on the mode of diffusion.

Seeing the unseen: Imaging rotation in cells with designer anisotropic particles

Cellular functions are enabled by cascades of transient biological events. Imaging and tracking the dynamics of these events have proven to be a powerful means of understanding the principles of cellular processes. These studies have typically focused on translational dynamics. By contrast, investigations of rotational dynamics have been scarce, despite emerging evidence that rotational dynamics are an inherent feature of many cellular processes and may also provide valuable clues to understanding those cell functions. Such studies have been impeded by the limited availability of suitable rotational imaging probes. This has recently changed thanks to the advances in the development of anisotropic particles for rotational imaging. In this review, we will summarize current techniques for imaging rotation using particle probes that are anisotropic in shape or optical properties. We will highlight two studies that demonstrate how these techniques can be applied to explore important facets of cellular functions.

Spatiotemporal Fluctuation Analysis: A Powerful Tool for the Future Nanoscopy of Molecular Processes

The enormous wealth of information available today from optical microscopy measurements on living samples is often underexploited. We argue that spatiotemporal analysis of fluorescence fluctuations using multiple detection channels can enhance the performance of current nanoscopy methods and provide further insight into dynamic molecular processes of high biological relevance.

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.