Mechanisms Underlying Anomalous Diffusion in the Plasma Membrane

Reliable deep learning in anomalous diffusion against out-of-distribution dynamics

Anomalous diffusion plays a crucial rule in understanding molecular-level dynamics by offering valuable insights into molecular interactions, mobility states and the physical properties of systems across both biological and materials sciences. Deep-learning techniques have recently outperformed conventional statistical methods in anomalous diffusion recognition. However, deep-learning networks are typically trained by data with limited distribution, which inevitably fail to recognize unknown diffusion models and misinterpret dynamics when confronted with out-of-distribution (OOD) scenarios. In this work, we present a general framework for evaluating deep-learning-based OOD dynamics-detection methods. We further develop a baseline approach that achieves robust OOD dynamics detection as well as accurate recognition of in-distribution anomalous diffusion. We demonstrate that this method enables a reliable characterization of complex behaviors across a wide range of experimentally diverse systems, including nicotinic acetylcholine receptors in membranes, fluorescent beads in dextran solutions and silver nanoparticles undergoing active endocytosis.

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.

Lateral diffusion of ions near membrane surface

Biological membranes isolate living cells from their environment, while allowing selective molecular transport between the inner and outer realms. For example, Na+ and K+ permeability through ionic channels contributes to neural conduction. Whether the ionic currents arise directly from cations in the bulk, or from the interface, is currently unclear. There are only scant results concerning lateral diffusion of ions on aquated membrane surfaces (and strong belief that this occurs through binding to a diffusing lipid). We performed classical molecular dynamics (MD) simulations of monovalent ions, Na+, K+, and Cl-, near the surface of the zwitterionic palmitoyl-oleoyl-phosphatidylcholine (POPC) membrane. Realistic force-fields for lipids (Amber's Lipid17 and Lipid21) and water (TIP4P-Ew) are tested for the mass and charge densities and the electrostatic potential across the membrane. These calculations reveal that the chloride can bind to the choline moiety through an intervening water molecule by forming a CH⋯OH hydrogen bond, while cations bind to both the phosphatic and carbonyl oxygens of phosphatidylcholine moieties. Upon transitioning from the bulk to the interface, a cation sheds some of its hydration water, which are replaced by headgroup atoms. Notably, an interfacial cation can bind 1-4 headgroup atoms, which is a key to understanding its surface hopping mechanism. We find that cation binding to three headgroup atoms immobilizes it, while binding to four energizes it. Consequently, the lateral cation diffusion rate is only 15-25 times slower than in the bulk, and 4-5 times faster than lipid self-diffusion. K+ diffusion is notably more anomalous than Na+, switching from sub- to super-diffusion after about 2 ns.

Fractional Brownian motion with fluctuating diffusivities

Despite the success of fractional Brownian motion (fBm) in modeling systems that exhibit anomalous diffusion due to temporal correlations, recent experimental and theoretical studies highlight the necessity for a more comprehensive approach of a generalization that incorporates heterogeneities in either the tracers or the environment. This work presents a modification of Lévy's representation of fBm for the case in which the generalized diffusion coefficient is a stochastic process. We derive analytical expressions for the autocovariance function and both ensemble- and time-averaged mean squared displacements. Further, we validate the efficacy of the developed framework in two-state systems, comparing analytical asymptotic expressions with numerical simulations.

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.

Two-Dimensional Fluctuation Correlation Spectroscopy (2D-FlucCS): A Method to Determine the Origin of Relaxation Rate Dispersion

Relaxation rate dispersion, i.e., nonexponential or multicomponent kinetics, is observed in complex systems when measuring relaxation kinetics. Often, the origin of rate dispersion is associated with the heterogeneity in the system. However, both homogeneous (where all molecules experience the same rate but inherently nonexponential) and heterogeneous (where all molecules experience different rates) systems can exhibit rate dispersion. A multidimensional correlation analysis method has been demonstrated to detect and quantify rate dispersion observed in molecular rotation, diffusion, solvation, and reaction kinetics. One-dimensional (1D) autocorrelation function detects rate dispersion and measures its extent. Two-dimensional (2D) autocorrelation function measures the origin of rate dispersion and distinguishes homogeneous from heterogeneous. In a heterogeneous system, implicitly there exist subensembles of molecules experiencing different rates. A three-dimensional (3D) autocorrelation function measures subensemble exchange if present and reveals if the system possesses static or dynamic heterogeneity. This perspective discusses the principles, applications, and potential and also presents a future outlook of two-dimensional fluctuation correlation spectroscopy (2D-FlucCS). The method is applicable to any experiment or simulation where a time series of fluctuation in an observable (emission, scattering, current, etc.) around a mean value can be obtained in steady state (equilibrium or nonequilibrium), provided the system is ergodic.

Ergodic criterion of a random diffusivity model

The random diffusivity, initially proposed to explain Brownian yet non-Gaussian diffusion, has garnered significant attention due to its capacity not only for elucidating the internal physical mechanism of non-Gaussian diffusion, but also for establishing an analytical framework to characterize particle motion in complex environments. In this paper, based on the correlation function C(t_{1},t_{2})=〈D(t_{1})D(t_{2})〉 of random diffusivity D(t), we quantitatively propose a general criterion of determining the ergodic property of the Langevin equation with the arbitrary random diffusivity D(t). Due to the critical role of correlation function C(t_{1},t_{2}), we derive the criterion for the two cases with stationary diffusivity or nonstationary diffusivity, respectively. By utilizing the quantitative criterion, we can directly judge the ergodic properties of the random diffusivity model based on the correlation function C(t_{1},t_{2}) of random diffusivity D(t). Several typical diffusivities, including the common square of the Brownian motion and of the (fractional) Ornstein-Uhlenbeck process, are found to contribute to different ergodic properties, which validates our proposed criterion built on the correlation function C(t_{1},t_{2}).

GLT-1a glutamate transporter nanocluster localization is associated with astrocytic actin and neuronal Kv2 clusters at sites of neuron-astrocyte contact

Introduction: Astrocytic GLT-1 glutamate transporters ensure the fidelity of glutamic neurotransmission by spatially and temporally limiting glutamate signals. The ability to limit neuronal hyperactivity relies on the localization and diffusion of GLT-1 on the astrocytic surface, however, little is known about the underlying mechanisms. We show that two isoforms of GLT-1, GLT-1a and GLT-1b, form nanoclusters on the surface of transfected astrocytes and HEK-293 cells. Methods: We used both fixed and live cell super-resolution imaging of fluorescent protein and epitope tagged proteins in co-cultures of rat astrocytes and neurons. Immunofluorescence techniques were also used. GLT1 diffusion was assessed via single particle tracking and fluorescence recovery after photobleach (FRAP). Results: We found GLT-1a, but not GLT-1b, nanoclusters concentrated adjacent to actin filaments which was maintained after addition of glutamate. GLT-1a nanocluster concentration near actin filaments was prevented by expression of a cytosolic GLT-1a C-terminus, suggesting the C-terminus is involved in the localization adjacent to cortical actin. Using super-resolution imaging, we show that astrocytic GLT-1a and actin co-localize in net-like structures around neuronal Kv2.1 clusters at points of neuron/astrocyte contact. Conclusion: Overall, these data describe a novel relationship between GLT-1a and cortical actin filaments, which localizes GLT-1a near neuronal structures responsive to ischemic insult.

Heterogeneous anomalous transport in cellular and molecular biology

It is well established that a wide variety of phenomena in cellular and molecular biology involve anomalous transport e.g. the statistics for the motility of cells and molecules are fractional and do not conform to the archetypes of simple diffusion or ballistic transport. Recent research demonstrates that anomalous transport is in many cases heterogeneous in both time and space. Thus single anomalous exponents and single generalised diffusion coefficients are unable to satisfactorily describe many crucial phenomena in cellular and molecular biology. We consider advances in the field ofheterogeneous anomalous transport(HAT) highlighting: experimental techniques (single molecule methods, microscopy, image analysis, fluorescence correlation spectroscopy, inelastic neutron scattering, and nuclear magnetic resonance), theoretical tools for data analysis (robust statistical methods such as first passage probabilities, survival analysis, different varieties of mean square displacements, etc), analytic theory and generative theoretical models based on simulations. Special emphasis is made on high throughput analysis techniques based on machine learning and neural networks. Furthermore, we consider anomalous transport in the context of microrheology and the heterogeneous viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion systems is also considered since it plays an important role in morphogenesis and signalling. In addition, we present specific examples from cellular biology including embryonic cells, leucocytes, cancer cells, bacterial cells, bacterial biofilms, and eukaryotic microorganisms. Case studies from molecular biology include DNA, membranes, endosomal transport, endoplasmic reticula, mucins, globular proteins, and amyloids.

Simulation study on the effect of polydisperse nanoparticles on polymer diffusion in crowded environments

The diffusion of polymer chains in a crowded environment with large and small immobile, attractive nanoparticles (NPs) is studied using Langevin dynamics simulations. For orderly distributed NPs on the simple cubic lattice, our results show that the diffusion of polymer chains is dependent on the NP-NP distance or lattice distance d. At low d where NPs are placed closely, subdiffusion occurs at a sufficiently high polydispersity of NPs, PD. Both the apparent diffusion coefficient and subdiffusion exponent of polymer chains decrease with increasing PD, attributed to the adsorption of polymers on NP clusters formed by larger NPs. At large d, normal diffusion is always observed, and the diffusion coefficient increases with increasing PD. The reason is that, at high PD, the difference between single large NP adsorption and double large NP adsorption is reduced, which increases the exchange of a polymer between the two adsorption states. Finally, the impact of size polydispersity of NPs on the diffusion of polymer chains in a crowded environment with randomly distributed NPs is also investigated. The results show that the position disorder of NPs enhances the subdiffusion of the system.

Live-cell imaging uncovers the relationship between histone acetylation, transcription initiation, and nucleosome mobility

Histone acetylation and RNA polymerase II phosphorylation are associated with transcriptionally active chromatin, but their spatiotemporal relationship in live cells remains poorly understood. To address this problem, we combine Fab-based labeling of endogenous protein modifications with single-molecule tracking to quantify the dynamics of chromatin enriched with histone H3 lysine-27 acetylation (H3K27ac) and RNA polymerase II serine-5 phosphorylation (RNAP2-Ser5ph). Our analysis reveals that chromatin enriched with these two modifications is generally separate. In these separated sites, we show that the two modifications are inversely correlated with one another on the minutes time scale and that single nucleosomes within each region display distinct and opposing dynamics on the subsecond time scale. While nucleosomes diffuse ~15% faster in chromatin enriched with H3K27ac, they diffuse ~15% slower in chromatin enriched with RNAP2-Ser5ph. These results argue that high levels of H3K27ac and RNAP2-Ser5ph are not often present together at the same place and time, but rather each marks distinct transcriptionally poised or active sites, respectively.

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.

Heterogeneous nanoscopic lipid diffusion in the live cell membrane and its dependency on cholesterol

Cholesterol plays a unique role in the regulation of membrane organization and dynamics by modulating the membrane phase transition at the nanoscale. Unfortunately, due to their small sizes and dynamic nature, the effects of cholesterol-mediated membrane nanodomains on membrane dynamics remain elusive. Here, using ultrahigh-speed single-molecule tracking with advanced optical microscope techniques, we investigate the diffusive motion of single phospholipids in the live cell plasma membrane at the nanoscale and its dependency on the cholesterol concentration. We find that both saturated and unsaturated phospholipids undergo anomalous subdiffusion on the length scale of 10-100 nm. The diffusion characteristics exhibit considerable variations in space and in time, indicating that the nanoscopic lipid diffusion is highly heterogeneous. Importantly, through the statistical analysis, apparent dual-mobility subdiffusion is observed from the mixed diffusion behaviors. The measured subdiffusion agrees well with the hop diffusion model that represents a diffuser moving in a compartmentalized membrane created by the cytoskeleton meshwork. Cholesterol depletion diminishes the lipid mobility with an apparently smaller compartment size and a stronger confinement strength. Similar results are measured with temperature reduction, suggesting that the more heterogeneous and restricted diffusion is connected to the nanoscopic membrane phase transition. Our conclusion supports the model that cholesterol depletion induces the formation of gel-phase, solid-like membrane nanodomains. These nanodomains undergo restricted diffusion and act as diffusion obstacles to the membrane molecules that are excluded from the nanodomains. This work provides the experimental evidence that the nanoscopic lipid diffusion in the cell plasma membrane is heterogeneous and sensitive to the cholesterol concentration and temperature, shedding new light on the regulation mechanisms of nanoscopic membrane dynamics.

A simulation study on the subdiffusion of polymer chains in crowded environments containing nanoparticles

Polymer chains in crowded environments often show subdiffusive behavior. We adopt molecular dynamics simulations to study the conditions for the subdiffusion of polymer chains in crowded environments containing randomly distributed, immobile, attractive nanoparticles (NPs). The attraction is strong enough to adsorb polymer chains on NPs. The results show that subdiffusion occurs at a low concentration of polymer chains (c_p). A transition from subdiffusion to normal diffusion is observed when c_p exceeds the transition concentration , which increases with increasing concentration of NPs while decreases with increasing size of NPs. The high concentration and small size of NPs exert a big effect on the subdiffusion of polymer chains. The subdiffusive behavior of polymer chains can be attributed to the strong adsorption of polymer chains on the attractive NPs. For the subdiffusion case, polymer chains are adsorbed strongly on multiple NPs, and they diffuse via the NP-exchange diffusion mechanism. However for the normal diffusion case, polymer chains are either free or weakly adsorbed on one or a few NPs, and they diffuse mainly via the adsorption-and-desorption diffusion mechanism.

Objective comparison of methods to decode anomalous diffusion

Deviations from Brownian motion leading to anomalous diffusion are found in transport dynamics from quantum physics to life sciences. The characterization of anomalous diffusion from the measurement of an individual trajectory is a challenging task, which traditionally relies on calculating the trajectory mean squared displacement. However, this approach breaks down for cases of practical interest, e.g., short or noisy trajectories, heterogeneous behaviour, or non-ergodic processes. Recently, several new approaches have been proposed, mostly building on the ongoing machine-learning revolution. To perform an objective comparison of methods, we gathered the community and organized an open competition, the Anomalous Diffusion challenge (AnDi). Participating teams applied their algorithms to a commonly-defined dataset including diverse conditions. Although no single method performed best across all scenarios, machine-learning-based approaches achieved superior performance for all tasks. The discussion of the challenge results provides practical advice for users and a benchmark for developers.

A deep learning-based approach to model anomalous diffusion of membrane proteins: the case of the nicotinic acetylcholine receptor

We present a concatenated deep-learning multiple neural network system for the analysis of single-molecule trajectories. We apply this machine learning-based analysis to characterize the translational diffusion of the nicotinic acetylcholine receptor at the plasma membrane, experimentally interrogated using superresolution optical microscopy. The receptor protein displays a heterogeneous diffusion behavior that goes beyond the ensemble level, with individual trajectories exhibiting more than one diffusive state, requiring the optimization of the neural networks through a hyperparameter analysis for different numbers of steps and durations, especially for short trajectories (<50 steps) where the accuracy of the models is most sensitive to localization errors. We next use the statistical models to test for Brownian, continuous-time random walk and fractional Brownian motion, and introduce and implement an additional, two-state model combining Brownian walks and obstructed diffusion mechanisms, enabling us to partition the two-state trajectories into segments, each of which is independently subjected to multiple analysis. The concatenated multi-network system evaluates and selects those physical models that most accurately describe the receptor's translational diffusion. We show that the two-state Brownian-obstructed diffusion model can account for the experimentally observed anomalous diffusion (mostly subdiffusive) of the population and the heterogeneous single-molecule behavior, accurately describing the majority (72.5 to 88.7% for α-bungarotoxin-labeled receptor and between 73.5 and 90.3% for antibody-labeled molecules) of the experimentally observed trajectories, with only ~15% of the trajectories fitting to the fractional Brownian motion model.

Aging power spectrum of membrane protein transport and other subordinated random walks

Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion processes in complex systems are usually not stationary, the traditional Wiener-Khinchin theorem for the analysis of power spectral densities is invalid. Here, we employ a recently developed tool named aging Wiener-Khinchin theorem to derive the power spectral density of fractional Brownian motion coexisting with a scale-free continuous time random walk, the two most typical anomalous diffusion processes. Using this analysis, we characterize the motion of voltage-gated sodium channels on the surface of hippocampal neurons. Our results show aging where the power spectral density can either increase or decrease with observation time depending on the specific parameters of both underlying processes.

Variable-order fractional master equation and clustering of particles: non-uniform lysosome distribution

In this paper, we formulate the space-dependent variable-order fractional master equation to model clustering of particles, organelles, inside living cells. We find its solution in the long-time limit describing non-uniform distribution due to a space-dependent fractional exponent. In the continuous space limit, the solution of this fractional master equation is found to be exactly the same as the space-dependent variable-order fractional diffusion equation. In addition, we show that the clustering of lysosomes, an essential organelle for healthy functioning of mammalian cells, exhibit space-dependent fractional exponents. Furthermore, we demonstrate that the non-uniform distribution of lysosomes in living cells is accurately described by the asymptotic solution of the space-dependent variable-order fractional master equation. Finally, Monte Carlo simulations of the fractional master equation validate our analytical solution. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Local Analysis of Heterogeneous Intracellular Transport: Slow and Fast Moving Endosomes

Trajectories of endosomes inside living eukaryotic cells are highly heterogeneous in space and time and diffuse anomalously due to a combination of viscoelasticity, caging, aggregation and active transport. Some of the trajectories display switching between persistent and anti-persistent motion, while others jiggle around in one position for the whole measurement time. By splitting the ensemble of endosome trajectories into slow moving subdiffusive and fast moving superdiffusive endosomes, we analyzed them separately. The mean squared displacements and velocity auto-correlation functions confirm the effectiveness of the splitting methods. Applying the local analysis, we show that both ensembles are characterized by a spectrum of local anomalous exponents and local generalized diffusion coefficients. Slow and fast endosomes have exponential distributions of local anomalous exponents and power law distributions of generalized diffusion coefficients. This suggests that heterogeneous fractional Brownian motion is an appropriate model for both fast and slow moving endosomes. This article is part of a Special Issue entitled: "Recent Advances In Single-Particle Tracking: Experiment and Analysis" edited by Janusz Szwabiński and Aleksander Weron.

Characterising the diffusion of biological nanoparticles on fluid and cross-linked membranes

Tracing the motion of macromolecules, viruses, and nanoparticles adsorbed onto cell membranes is currently the most direct way of probing the complex dynamic interactions behind vital biological processes, including cell signalling, trafficking, and viral infection. The resulting trajectories are usually consistent with some type of anomalous diffusion, but the molecular origins behind the observed anomalous behaviour are usually not obvious. Here we use coarse-grained molecular dynamics simulations to help identify the physical mechanisms that can give rise to experimentally observed trajectories of nanoscopic objects moving on biological membranes. We find that diffusion on membranes of high fluidities typically results in normal diffusion of the adsorbed nanoparticle, irrespective of the concentration of receptors, receptor clustering, or multivalent interactions between the particle and membrane receptors. Gel-like membranes on the other hand result in anomalous diffusion of the particle, which becomes more pronounced at higher receptor concentrations. This anomalous diffusion is characterised by local particle trapping in the regions of high receptor concentrations and fast hopping between such regions. The normal diffusion is recovered in the limit where the gel membrane is saturated with receptors. We conclude that hindered receptor diffusivity can be a common reason behind the observed anomalous diffusion of viruses, vesicles, and nanoparticles adsorbed on cell and model membranes. Our results enable direct comparison with experiments and offer a new route for interpreting motility experiments on cell membranes.

Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x)=D_{0}|x|^{α}. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble- and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y∼|x|^{1/(2/(2-α))}/t^{H} coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P_{HDP-FBM}(y)=e^{-y^{2}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.

Fluorescence strategies for mapping cell membrane dynamics and structures

Fluorescence spectroscopy has been a cornerstone of research in membrane dynamics and organization. Technological advances in fluorescence spectroscopy went hand in hand with discovery of various physicochemical properties of membranes at nanometric spatial and microsecond timescales. In this perspective, we discuss the various challenges associated with quantification of physicochemical properties of membranes and how various modes of fluorescence spectroscopy have overcome these challenges to shed light on the structure and organization of membranes. Finally, we discuss newer measurement strategies and data analysis tools to investigate the structure, dynamics, and organization of membranes.

To Hop or not to Hop: Exceptions in the FCS Diffusion Law

Diffusion obstacles in membranes have not been directly visualized because of fast membrane dynamics and the occurrence of subresolution molecular complexes. To understand the obstacle characteristics, mobility-based methods are often used as an indirect way of assessing the membrane structure. Molecular movement in biological plasma membranes is often characterized by anomalous diffusion, but the exact underlying mechanisms are still elusive. Imaging total internal reflection fluorescence correlation spectroscopy (ITIR-FCS) is a well-established mobility-based method that provides spatially resolved diffusion coefficient maps and is combined with FCS diffusion law analysis to examine subresolution membrane organization. In recent years, although FCS diffusion law analysis has been instrumental in providing new insights into the membrane structure below the optical diffraction limit, there are certain exceptions and anomalies that require further clarification. To this end, we correlate the membrane structural features imaged by atomic force microscopy (AFM) with the dynamics measured using ITIR-FCS. We perform ITIR-FCS measurements on supported lipid bilayers (SLBs) of various lipid compositions to characterize the anomalous diffusion of lipid molecules in distinct obstacle configurations, along with the high-resolution imaging of the membrane structures with AFM. Furthermore, we validate our experimental results by performing simulations on image grids with experimentally determined obstacle configurations. This study demonstrates that FCS diffusion law analysis is a powerful tool to determine membrane heterogeneities implied from dynamics measurements. Our results corroborate the commonly accepted interpretations of imaging FCS diffusion law analysis, and we show that exceptions happen when domains reach the percolation threshold in a biphasic membrane and a network of domains behaves rather like a meshwork, resulting in hop diffusion.

Diffusion of DNA-Binding Species in the Nucleus: A Transient Anomalous Subdiffusion Model

Single-particle tracking experiments have measured escape times of DNA-binding species diffusing in living cells: CRISPR-Cas9, TetR, and LacI. The observed distribution is a truncated power law. Working backward from the experimental results, the observed distribution appears inconsistent with a Gaussian distribution of binding energies. Working forward, the observed distribution leads to transient anomalous subdiffusion, in which diffusion is anomalous at short times and normal at long times, here only mildly anomalous. Monte Carlo simulations are used to characterize the time-dependent diffusion coefficient D(t) in terms of the anomalous exponent α, the crossover time tcross, and the limits D(0) and D(∞) and to relate these quantities to the escape time distribution. The simplest interpretations identify the escape time as the actual binding time to DNA or the period of one-dimensional diffusion on DNA in the standard model combining one-dimensional and three-dimensional search, but a more complicated interpretation may be required. The model has several implications for cell biophysics. 1) The initial anomalous regime represents the search of the DNA-binding species for its target DNA sequence. 2) Non-target DNA sites have a significant effect on search kinetics. False positives in bioinformatic searches of the genome are potentially rate-determining in vivo. For simple binding, the search would be speeded if false-positive sequences were eliminated from the genome. 3) Both binding and obstruction affect diffusion. Obstruction ought to be measured directly, using as the primary probe the DNA-binding species with the binding site inactivated and eGFP as a calibration standard among laboratories and cell types. 4) Overexpression of the DNA-binding species reduces anomalous subdiffusion because the deepest binding sites are occupied and unavailable. 5) The model provides a coarse-grained phenomenological description of diffusion of a DNA-binding species, useful in larger-scale modeling of kinetics, FCS, and FRAP.

Live cell single molecule tracking and localization microscopy of bioorthogonally labeled plasma membrane proteins

Tracking the localization and mobility of individual proteins in live cells is key for understanding how they mediate their function. Such information can be obtained from single molecule imaging techniques including as Single Particle Tracking (SPT) and Single Molecule Localization Microscopy (SMLM). Genetic code expansion (GCE) combined with bioorthogonal chemistry offers an elegant approach for direct labeling of proteins with fluorescent dyes, holding great potential for improving protein labeling in single molecule applications. Here we calibrated conditions for performing SPT and live-SMLM of bioorthogonally labeled plasma membrane proteins in live mammalian cells. Using SPT, the diffusion of bioorthogonally labeled EGF receptor and the prototypical Shaker voltage-activated potassium channel (Kv) was measured and characterized. Applying live-SMLM to bioorthogonally labeled Shaker Kv channels enabled visualizing the plasma membrane distribution of the channel over time with ∼30 nm accuracy. Finally, by competitive labeling with two Fl-dyes, SPT and live-SMLM were performed in a single cell and both the density and dynamics of the EGF receptor were measured at single molecule resolution in subregions of the cell. We conclude that GCE and bioorthogonal chemistry is a highly suitable, flexible approach for protein labeling in quantitative single molecule applications that outperforms current protein live-cell labeling approaches.

Increased Confinement and Polydispersity of STIM1 and Orai1 after Ca2+ Store Depletion

STIM1 (a Ca²⁺ sensor in the endoplasmic reticulum (ER) membrane) and Orai1 (a pore-forming subunit of the Ca²⁺-release-activated calcium channel in the plasma membrane) diffuse in the ER membrane and plasma membrane, respectively. Upon depletion of Ca²⁺ stores in the ER, STIM1 translocates to the ER-plasma membrane junction and binds Orai1 to trigger store-operated Ca²⁺ entry. However, the motion of STIM1 and Orai1 during this process and its roles to Ca²⁺ entry is poorly understood. Here, we report real-time tracking of single STIM1 and Orai1 particles in the ER membrane and plasma membrane in living cells before and after Ca²⁺ store depletion. We found that the motion of single STIM1 and Orai1 particles exhibits anomalous diffusion both before and after store depletion, and their mobility-measured by the radius of gyration of the trajectories, mean-square displacement, and generalized diffusion coefficient-decreases drastically after store depletion. We also found that the measured displacement distribution is non-Gaussian, and the non-Gaussian parameter drastically increases after store depletion. Detailed analyses and simulations revealed that single STIM1 and Orai1 particles are confined in the compartmentalized membrane both before and after store depletion, and the changes in the motion after store depletion are explained by increased confinement and polydispersity of STIM1-Orai1 complexes formed at the ER-plasma membrane junctions. Further simulations showed that this increase in the confinement and polydispersity after store depletion localizes a rapid increase of Ca²⁺ influx, which can facilitate the rapid activation of local Ca²⁺ signaling pathways and the efficient replenishing of Ca²⁺ store in the ER in store-operated Ca²⁺ entry.

A novel computational framework for D(t) from Fluorescence Recovery after Photobleaching data reveals various anomalous diffusion types in live cell membranes

Diffusion of proteins and lipids in lipid membranes plays a pivotal role in almost all aspects of cellular biology, including motility, exo-/endocytosis and signal transduction. For this reason, gaining a detailed understanding of membrane structure and function has long been a major area of cell biology research. To better elucidate this structure-function relationship, various tools have been developed for diffusion measurements, including Fluorescence Recovery After Photobleaching (FRAP). Because of the complexity of cellular microenvironments, biological diffusion is often correlated over time and described by a time-dependent diffusion coefficient, D(t), although the underlying mechanisms are not fully understood. Since D(t) provides important information regarding cellular structures, such as the existence of subresolution barriers to diffusion, many efforts have been made to quantify D(t) by FRAP assuming a single power law, D(t) = Γt ^α - 1 where Γ and α are transport coefficient and anomalous exponent. However, straightforward approaches to quantify a general form of D(t) are lacking. In this study, we develop a novel mathematical and computational framework to compute the mean square displacement of diffusing molecules and diffusion coefficient D(t) from each individual time point of confocal FRAP data without the single power law assumption. Additionally, we developed an auxiliary equation for D(t) which can readily distinguish normal diffusion or single power law anomalous diffusion from other types of anomalous diffusion directly from FRAP data. Importantly, by applying this approach to FRAP data from a variety of membrane markers, we demonstrate the single power law anomalous diffusion assumption is not sufficient to describe various types of D(t) of membrane proteins. Lastly, we discuss how our new approaches can be applied to other fluorescence microscopy tools such as Fluorescence Correlation Spectroscopy (FCS) and Single Particle Tracking (SPT).

Membrane Diffusion Occurs by Continuous-Time Random Walk Sustained by Vesicular Trafficking

Diffusion in cellular membranes is regulated by processes that occur over a range of spatial and temporal scales. These processes include membrane fluidity, interprotein and interlipid interactions, interactions with membrane microdomains, interactions with the underlying cytoskeleton, and cellular processes that result in net membrane movement. The complex, non-Brownian diffusion that results from these processes has been difficult to characterize, and moreover, the impact of factors such as membrane recycling on membrane diffusion remains largely unexplored. We have used a careful statistical analysis of single-particle tracking data of the single-pass plasma membrane protein CD93 to show that the diffusion of this protein is well described by a continuous-time random walk in parallel with an aging process mediated by membrane corrals. The overall result is an evolution in the diffusion of CD93: proteins initially diffuse freely on the cell surface but over time become increasingly trapped within diffusion-limiting membrane corrals. Stable populations of freely diffusing and corralled CD93 are maintained by an endocytic/exocytic process in which corralled CD93 is selectively endocytosed, whereas freely diffusing CD93 is replenished by exocytosis of newly synthesized and recycled CD93. This trafficking not only maintained CD93 diffusivity but also maintained the heterogeneous distribution of CD93 in the plasma membrane. These results provide insight into the nature of the biological and biophysical processes that can lead to significantly non-Brownian diffusion of membrane proteins and demonstrate that ongoing membrane recycling is critical to maintaining steady-state diffusion and distribution of proteins in the plasma membrane.

Influence of Oligonucleotide Grafting Density on Surface-Mediated DNA Transport and Hybridization

Adsorption of soluble DNA to surfaces decorated with complementary DNA plays an important role in many bionanotechnology applications, and previous studies have reported complex dependencies of the surface density of immobilized DNA on hybridization. While these effects have been speculatively ascribed to steric or electrostatic effects, the influence of surface-mediated molecular transport (i.e., intermittent "hopping diffusion") has not been fully appreciated. Here, single-molecule tracking and Förster resonance energy transfer (FRET) were employed to characterize the mobility and the hybridization efficiency of adsorbed ssDNA oligonucleotides ("target") at solid-liquid interfaces exhibiting surface-immobilized ssDNA ("probe") over a wide range of surface grafting densities. Two distinct regimes were observed, with qualitatively different transport and hybridization behaviors. At dilute grafting density, only 1-3% of target molecules were observed to associate with probes (i.e., to hybridize). Adsorbing target molecules often searched unsuccessfully and "flew", via desorption-mediated diffusion, to secondary locations before hybridizing. In contrast, at high probe grafting density, approximately 20% of target DNA hybridized to immobilized probes, and almost always in the vicinity of initial adsorption. Moreover, following a dehybridization event, target molecules rehybridized at high probe density, but rehybridization was infrequent in the dilute density regime. Interestingly, the intermittent interfacial transport of mobile target molecules was suppressed by the presence of immobilized probe DNA, presumably due to an increased probability of readsorption following each "hop". Together, these findings suggested that many salient effects of grafting density on surface-mediated DNA hybridization can be directly related to the mechanisms of surface-mediated intermittent diffusion.

Temporal dependence of shifts in mu opioid receptor mobility at the cell surface after agonist binding observed by single-particle tracking

Agonist binding to the mu opioid receptor (MOR) results in conformational changes that allow recruitment of G-proteins, activation of downstream effectors and eventual desensitization and internalization, all of which could affect receptor mobility. The present study employed single particle tracking (SPT) of quantum dot labeled FLAG-tagged MORs to examine shifts in MOR mobility after agonist binding. FLAG-MORs on the plasma membrane were in both mobile and immobile states under basal conditions. Activation of FLAG-MORs with DAMGO caused an acute increase in the fraction of mobile MORs, and free portions of mobile tracks were partially dependent on interactions with G-proteins. In contrast, 10-minute exposure to DAMGO or morphine increased the fraction of immobile FLAG-MORs. While the decrease in mobility with prolonged DAMGO exposure corresponded to an increase in colocalization with clathrin, the increase in colocalization was present in both mobile and immobile FLAG-MORs. Thus, no single mobility state of the receptor accounted for colocalization with clathrin. These findings demonstrate that SPT can be used to track agonist-dependent changes in MOR mobility over time, but that the mobility states observed likely arise from a diverse set of interactions and will be most informative when examined in concert with particular downstream effectors.

Assessing metastatic potential of breast cancer cells based on EGFR dynamics

Derailed transmembrane receptor trafficking could be a hallmark of tumorigenesis and increased tumor invasiveness, but receptor dynamics have not been used to differentiate metastatic cancer cells from less invasive ones. Using single-particle tracking techniques, we  developed a phenotyping asssay named Transmembrane Receptor Dynamics (TReD), studied the dynamics of epidermal growth factor receptor (EGFR) in seven breast epithelial cell lines and developed a phenotyping assay named Transmembrane Receptor Dynamics (TReD). Here we show a clear evidence that increased EGFR diffusivity and enlarged EGFR confinement size in the plasma membrane (PM) are correlated with the enhanced metastatic potential in these cell lines. By comparing the TReD results with the gene expression profiles, we found a clear negative correlation between the EGFR diffusivities and the breast cancer luminal differentiation scores (r = -0.75). Upon the induction of epithelial-mesenchymal transition (EMT), EGFR diffusivity significantly increased for the non-tumorigenic MCF10A (99%) and the non-invasive MCF7 (56%) cells, but not for the highly metastatic MDA-MB-231 cell. We believe that the reorganization of actin filaments during EMT modified the PM structures, causing the receptor dynamics to change. TReD can thus serve as a new biophysical marker to probe the metastatic potential of cancer cells and even to monitor the transition of metastasis.

Identifying diffusive motions in single-particle trajectories on the plasma membrane via fractional time-series models

In this paper we show that an autoregressive fractionally integrated moving average time-series model can identify two types of motion of membrane proteins on the surface of mammalian cells. Specifically we analyze the motion of the voltage-gated sodium channel Nav1.6 and beta-2 adrenergic receptors. We find that the autoregressive (AR) part models well the confined dynamics whereas the fractionally integrated moving average (FIMA) model describes the nonconfined periods of the trajectories. Since the Ornstein-Uhlenbeck process is a continuous counterpart of the AR model, we are also able to calculate its physical parameters and show their biological relevance. The fitted FIMA and AR parameters show marked differences in the dynamics of the two studied molecules.

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

A fractional heteroscedastic time series model explains inhomogeneous membrane receptor diffusion and interprets ergodicity when the diffusion parameter fluctuates in time.

Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing

We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.

Cholesterol modulates acetylcholine receptor diffusion by tuning confinement sojourns and nanocluster stability

Translational motion of neurotransmitter receptors is key for determining receptor number at the synapse and hence, synaptic efficacy. We combine live-cell STORM superresolution microscopy of nicotinic acetylcholine receptor (nAChR) with single-particle tracking, mean-squared displacement (MSD), turning angle, ergodicity, and clustering analyses to characterize the lateral motion of individual molecules and their collective behaviour. nAChR diffusion is highly heterogeneous: subdiffusive, Brownian and, less frequently, superdiffusive. At the single-track level, free walks are transiently interrupted by ms-long confinement sojourns occurring in nanodomains of ~36 nm radius. Cholesterol modulates the time and the area spent in confinement. Turning angle analysis reveals anticorrelated steps with time-lag dependence, in good agreement with the permeable fence model. At the ensemble level, nanocluster assembly occurs in second-long bursts separated by periods of cluster disassembly. Thus, millisecond-long confinement sojourns and second-long reversible nanoclustering with similar cholesterol sensitivities affect all trajectories; the proportion of the two regimes determines the resulting macroscopic motional mode and breadth of heterogeneity in the ensemble population.

Fluctuations of random walks in critical random environments

Percolation networks have been widely used in the description of porous media but are now found to be relevant to understand the motion of particles in cellular membranes or the nucleus of biological cells. Random walks on the infinite cluster at criticality of a percolation network are asymptotically ergodic. On any finite size cluster of the network stationarity is reached at finite times, depending on the cluster's size. Despite of this we here demonstrate by combination of analytical calculations and simulations that at criticality the disorder and cluster size average of the ensemble of clusters leads to a non-vanishing variance of the time averaged mean squared displacement, regardless of the measurement time. Fluctuations of this relevant experimental quantity due to the disorder average of such ensembles are thus persistent and non-negligible. The relevance of our results for single particle tracking analysis in complex and biological systems is discussed.

Compartmentalization of the plasma membrane

The compartmentalization of the plasma membrane is essential for cells to perform specialized biochemical functions, in particular those responsible for intracellular and intercellular signaling pathways. Study of membrane compartmentalization requires state-of-the-art imaging tools that can reveal dynamics of individual molecules with high spatial and temporal resolution. In addition, quantitative analyses are employed to identify transient changes in molecule dynamics. In this review, membrane compartments are classified as stable domains, transient compartments, or nanodomains where proteins aggregate. Interestingly, in most cases, the cortical cytoskeleton plays important roles. Recent studies of the membrane-cytoskeleton interface are providing new insights about membrane organization involving a scale-free self-similar fractal structure and cytoskeleton active processes coupled to membrane dynamics.

Transient subdiffusion from an Ising environment

We introduce a model in which a particle performs a continuous-time random walk (CTRW) coupled to an environment with Ising dynamics. The particle shows locally varying diffusivity determined by the geometrical properties of the underlying Ising environment, that is, the diffusivity depends on the size of the connected area of spins pointing in the same direction. The model shows anomalous diffusion when the Ising environment is at critical temperature. We show that any finite scale introduced by a temperature different from the critical one, or a finite size of the environment, cause subdiffusion only during a transient time. The characteristic time, at which the system returns to normal diffusion after the subdiffusive plateau depends on the limiting scale and on how close the temperature is to criticality. The system also displays apparent ergodicity breaking at intermediate time, while ergodicity breaking at longer time occurs only under the idealized infinite environment at the critical temperature.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Elucidating distinct ion channel populations on the surface of hippocampal neurons via single-particle tracking recurrence analysis

Protein and lipid nanodomains are prevalent on the surface of mammalian cells. In particular, it has been recently recognized that ion channels assemble into surface nanoclusters in the soma of cultured neurons. However, the interactions of these molecules with surface nanodomains display a considerable degree of heterogeneity. Here, we investigate this heterogeneity and develop statistical tools based on the recurrence of individual trajectories to identify subpopulations within ion channels in the neuronal surface. We specifically study the dynamics of the K^{+} channel Kv1.4 and the Na^{+} channel Nav1.6 on the surface of cultured hippocampal neurons at the single-molecule level. We find that both these molecules are expressed in two different forms with distinct kinetics with regards to surface interactions, emphasizing the complex proteomic landscape of the neuronal surface. Further, the tools presented in this work provide new methods for the analysis of membrane nanodomains, transient confinement, and identification of populations within single-particle trajectories.

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.

Multiscale Modeling of Diffusion in a Crowded Environment

We present a multiscale approach to model diffusion in a crowded environment and its effect on the reaction rates. Diffusion in biological systems is often modeled by a discrete space jump process in order to capture the inherent noise of biological systems, which becomes important in the low copy number regime. To model diffusion in the crowded cell environment efficiently, we compute the jump rates in this mesoscopic model from local first exit times, which account for the microscopic positions of the crowding molecules, while the diffusing molecules jump on a coarser Cartesian grid. We then extract a macroscopic description from the resulting jump rates, where the excluded volume effect is modeled by a diffusion equation with space-dependent diffusion coefficient. The crowding molecules can be of arbitrary shape and size, and numerical experiments demonstrate that those factors together with the size of the diffusing molecule play a crucial role on the magnitude of the decrease in diffusive motion. When correcting the reaction rates for the altered diffusion we can show that molecular crowding either enhances or inhibits chemical reactions depending on local fluctuations of the obstacle density.

Ergodicity breaking on the neuronal surface emerges from random switching between diffusive states

Stochastic motion on the surface of living cells is critical to promote molecular encounters that are necessary for multiple cellular processes. Often the complexity of the cell membranes leads to anomalous diffusion, which under certain conditions it is accompanied by non-ergodic dynamics. Here, we unravel two manifestations of ergodicity breaking in the dynamics of membrane proteins in the somatic surface of hippocampal neurons. Three different tagged molecules are studied on the surface of the soma: the voltage-gated potassium and sodium channels Kv1.4 and Nav1.6 and the glycoprotein CD4. In these three molecules ergodicity breaking is unveiled by the confidence interval of the mean square displacement and by the dynamical functional estimator. Ergodicity breaking is found to take place due to transient confinement effects since the molecules alternate between free diffusion and confined motion.