Wanted: a positive control for anomalous subdiffusion

Synthetic Stochastic Motion Platform for Testing Single Particle Tracking Microscopes

We describe the design and implementation of a control system for testing the performance of single particle tracking microscopes with the method of synthetic motion. Single particle tracking (SPT) has become a common and powerful tool in the study of biomolecular transport in cellular biology, providing the ability to track individual biological macromolecules in their native environment. Existing methods for testing SPT techniques rely on physical simulations and there is a clear need for experimental-based schemes for both comparing different approaches and for characterizing the accuracy and precision of techniques on particular experimental setups. Synthetic motion, that is, using an actuator such as a nanopositioning stage to drive a particle along a known ground-truth trajectory, is a means for achieving these ends. However, the resolution, accuracy, and flexibility of this method is limited by the actuator static and dynamic characteristics. In this work we apply system identification and model inverse feedforward control to increase actuator bandwidth and address some common actuator nonlinearities, develop a set of dimensionless numbers that describe system limitations, and provide a set of guidelines for the practical use of synthetic motion in SPT.

Protein-coated nanoparticles exhibit Lévy flights on a suspended lipid bilayer

Lateral diffusion of nano-objects on lipid membranes is a crucial process in cell biology. Recent studies indicate that nanoparticle lateral diffusion is affected by the presence of membrane proteins and deviates from Brownian motion. Gold nanoparticles (Au NPs) stabilized by short thiol ligands were dispersed near a free-standing bilayer formed in a 3D microfluidic chip. Using dark-field microscopy, the position of single NPs at the bilayer surface was tracked over time. Numerical analysis of the NP trajectories shows that NP diffusion on the bilayer surface corresponds to Brownian motion. The addition of bovine serum albumin (BSA) protein to the solution led to the formation of a protein corona on the NP surface. We found that protein-coated NPs show anomalous superdiffusion and that the distribution of their relative displacement obeys Lévy flight statistics. This superdiffusive motion is attributed to a drastic reduction in adhesive energies between the NPs and the bilayer in the presence of the protein corona. This hypothesis was confirmed by numerical simulations mimicking the random walk of a single particle near a weakly adhesive surface. These results may be generalized to other classes of nano-objects that experience adsorption-desorption behaviour with a weakly adhesive surface.

Normal and anomalous diffusion in the disordered wind-tree model

Ehrenfests' wind-tree model (EWTM) refers to a two-dimensional system where noninteracting point tracer particles move through a random arrangement of overlapping or nonoverlapping square-shaped scatterers. Here, extensive event-driven molecular dynamics simulations of the EWTM at different reduced scatterer densities ρ are presented. For nonoverlapping scatterers, the asymptotic motion of the tracer particles is diffusive. We compare their diffusion coefficient D, as obtained from the simulation, with that predicted by kinetic theory where D^{-1} is expanded up to the second order in the scatterer density. While at low density quantitative agreement between theory and simulation is found, we show that beyond the low-density regime deviations to the theory are associated with the emergence of a maximum in the non-Gaussian parameter at intermediate times. For the case of overlapping scatterers, in agreement with a theoretical prediction, the asymptotic motion of the tracer particles is subdiffusive, i.e., the mean-squared displacement at long times t grows like t^{1-2ρ/3}. We propose a model of the van Hove correlation function that describes the density dependence of the tracer particles' asymptotic subdiffusive transport on a quantitative level.

Real-Time Feedback-Driven Single-Particle Tracking: A Survey and Perspective

Real-time feedback-driven single-particle tracking (RT-FD-SPT) is a class of techniques in the field of single-particle tracking that uses feedback control to keep a particle of interest in a detection volume. These methods provide high spatiotemporal resolution on particle dynamics and allow for concurrent spectroscopic measurements. This review article begins with a survey of existing techniques and of applications where RT-FD-SPT has played an important role. Each of the core components of RT-FD-SPT are systematically discussed in order to develop an understanding of the trade-offs that must be made in algorithm design and to create a clear picture of the important differences, advantages, and drawbacks of existing approaches. These components are feedback tracking and control, ranging from simple proportional-integral-derivative control to advanced nonlinear techniques, estimation to determine particle location from the measured data, including both online and offline algorithms, and techniques for calibrating and characterizing different RT-FD-SPT methods. Then a collection of metrics for RT-FD-SPT is introduced to help guide experimentalists in selecting a method for their particular application and to help reveal where there are gaps in the techniques that represent opportunities for further development. Finally, this review is concluded with a discussion on future perspectives in the field.

Effects of Macromolecular Crowding on Nanoparticle Diffusion: New Insights from Mössbauer Spectroscopy

In this work, we used Mössbauer spectroscopy as a new approach for experimental quantification of the self-diffusion coefficient (D_Mössbauer) and hydrodynamic (HD) size of iron-containing nanoparticles (NPs) in complex crowded solutions, mimicking cell cytoplasm. As a probe, we used 9 nm cobalt ferrite NPs (CFNs) dispersed in solutions of bovine serum albumin (BSA) with a volume fraction (φ_BSA) of 0-0.2. Our results show that the broadening of Mössbauer spectra is highly sensitive to the diffusion of CFNs, while when φ_BSA = 0.2, the CFN-normalized diffusivity is reduced by 86% compared to that of a protein-free solution. CFN colloids were also studied by dynamic light scattering (DLS). Comparison of the experimental data shows that DLS significantly underestimates the diffusion coefficient of CFNs and, consequently, overestimates the HD size of CFNs at φ_BSA > 0, which cannot be attributed to the formation of the BSA monolayer on the surface of CFNs.

Feedforward Control for Single Particle Tracking Synthetic Motion

Single particle tracking (SPT) is a method to study the transport of biomolecules with nanometer resolution. Unfortunately, recent reports show that systematic errors in position localization and uncertainty in model parameter estimates limits the utility of these techniques in studying biological processes. There is a need for an experimental method with a known ground-truth that tests the total SPT system (sample, microscope, algorithm) on both localization and estimation of model parameters. Synthetic motion is a known ground-truth method that moves a particle along a trajectory. This trajectory is a realization of a Markovian stochastic process that represents models of biomolecular transport. Here we describe a platform for creating synthetic motion using common equipment and well-known, simple methods that can be easily adopted by the biophysics community. In this paper we describe the synthetic motion system and calibration to achieve nanometer accuracy and precision. Steady state input-output characteristics are analyzed with both line scans and grid scans. The resulting relationship is described by an affine transformation, which is inverted and used as a prefilter. Model inverse feed forward control is used to increase the system bandwidth. The system model was identified from frequency response function measurements using an integrated stepped-sine with coherent demodulation built into the FPGA controller. Zero magnitude error tracking controller method was used to invert non-minimum phase zeros to achieve a stable discrete time feed forward filter.

Line-FRAP, A Versatile Method to Measure Diffusion Rates In Vitro and In Vivo

The crowded cellular milieu affect molecular diffusion through hard (occluded space) and soft (weak, non-specific) interactions. Multiple methods have been developed to measure diffusion coefficients at physiological protein concentrations within cells, each with its limitations. Here, we show that Line-FRAP, combined with rigours data analysis, is able to determine diffusion coefficients in a variety of environments, from in vitro to in vivo. The use of Line mode greatly improves time resolution of FRAP data acquisition, from 20-100 Hz in the classical mode to 800 Hz in the line mode. This improves data analysis, as intensity and radius of the bleach at the first post-bleach frame is critical. We evaluated the method on different proteins labelled chemically or fused to YFP in a wide range of environments. The diffusion coefficients measured in HeLa and in E. coli were ~2.5-fold and 15-fold slower than in buffer, and were comparable to previously published data. Increasing the osmotic pressure on E. coli further decreases diffusion, to the point at which proteins virtually stop moving. The method presented here, which requires a confocal microscope equipped with dual scanners, can be applied to study a large range of molecules with different sizes, and provides robust results in a wide range of environments and protein concentrations for fast diffusing molecules.

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.

Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement

The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them.

Nanoparticle Tracking to Probe Transport in Porous Media

From the granular and fractured subsurface environment to highly engineered polymer membranes used in pharmaceutical purification, porous materials are ubiquitous in nature and industrial applications. In particular, porous media are used extensively in processes including water treatment, pharmaceutical sterilization, food/beverage processing, and heterogeneous catalysis, where hindered mass transport is either essential to the process or a necessary but undesirable limitation. Unfortunately, there are currently no universal models capable of predicting mass transport based on a description of the porous material because real porous materials are complex and because many coupled dynamic mechanisms (e.g., adsorption, steric effects, hydrodynamic effects, electrostatic interactions, etc.) give rise to the observed macroscopic transport phenomena.While classical techniques, like nuclear magnetic resonance and dynamic light scattering, provide useful information about mass transport in porous media at the ensemble level, they provide limited insight into the microscopic mechanisms that give rise to complex phenomena such as anomalous diffusion, hindered pore-space accessibility, and unexpected retention under flow, among many others. To address this issue, we have developed refractive index matching imaging systems, combined with single-particle tracking methods, allowing the direct visualization of single-particle motion within a variety of porous materials.In this Account, we summarize our recent efforts to advance the understanding of nanoparticle transport in porous media using single-particle tracking methods in both fundamental and applied scenarios. First, we describe the basic principles for two-dimensional and three-dimensional single-particle tracking in porous materials. Then, we provide concrete examples of nanoparticle transport in porous materials from two perspectives: (1) understanding fundamental elementary particle transport processes in porous media, including pore accessibility and cavity escape, which limit transport in porous media, and (2) facilitating applications in industrial processes, e.g., by understanding the mechanisms of particle fouling and remobilization in filtration membranes. Finally, we provide an outlook of opportunities associated with investigating other types of mass transport in confined environments using single-particle tracking methods, including electrophoretic and self-propelled motion.

Accelerating and retarding anomalous diffusion: A Bernstein function approach

We have discovered here a duality relation between infinitely divisible subordinators which can produce both retarding and accelerating anomalous diffusion in the framework of the special Bernstein function approach. As a consequence, we show that conjugate pairs of Bernstein functions taken as Laplace exponents can produce in a natural way both retarding and accelerating anomalous diffusion (either subdiffusion or superdiffusion). This provides a unified way to control the dynamics of complex biological processes leading to transient anomalous diffusion in single-particle tracking experiments. Moreover, this permits one to explain better the relaxation diagram positioning two different power laws of relaxation, including the celebrated Havriliak-Negami law.

The diffusion of doxorubicin drug molecules in silica nanoslits is non-Gaussian, intermittent and anticorrelated

The motion of the confined doxorubicin drug molecule exhibits an interesting combination of anomalous diffusion features.

Short-time fractal analysis of biological autoluminescence

Biological systems manifest continuous weak autoluminescence, which is present even in the absence of external stimuli. Since this autoluminescence arises from internal metabolic and physiological processes, several works suggested that it could carry information in the time series of the detected photon counts. However, there is little experimental work which would show any difference of this signal from random Poisson noise and some works were prone to artifacts due to lacking or improper reference signals. Here we apply rigorous statistical methods and advanced reference signals to test the hypothesis whether time series of autoluminescence from germinating mung beans display any intrinsic correlations. Utilizing the fractional Brownian bridge that employs short samples of time series in the method kernel, we suggest that the detected autoluminescence signal from mung beans is not totally random, but it seems to involve a process with a negative memory. Our results contribute to the development of the rigorous methodology of signal analysis of photonic biosignals.

Anomalous transport in the soft-sphere Lorentz model

The sensitivity of anomalous transport in crowded media to the form of the inter-particle interactions is investigated through computer simulations. We extend the highly simplified Lorentz model towards realistic natural systems by modeling the interactions between the tracer and the obstacles with a smooth potential. We find that the anomalous transport at the critical point happens to be governed by the same universal exponent as for hard exclusion interactions, although the mechanism of how narrow channels are probed is rather different. The scaling behavior of simulations close to the critical point confirm this exponent. Our result indicates that the simple Lorentz model may be applicable to describing the fundamental properties of long-range transport in real crowded environments.

Dynamics of proteins confined in non-ionic bicontinuous microemulsions: a FCS study

In our present work we present an approach which allows one to confine proteins in structurally nearly identical bicontinuous microemulsions with systematically decreasing water domain size. It is shown that sub-diffusive behaviour occurs already at water domain sizes below 13 nm. However, above 13 nm normal diffusion is seen. Moreover, we compare protein diffusion in microemulsions to the transport of a much smaller fluorescent dye.

Controlling Anomalous Diffusion in Lipid Membranes

Diffusion in cell membranes is not just simple two-dimensional Brownian motion but typically depends on the timescale of the observation. The physical origins of this anomalous subdiffusion are unresolved, and model systems capable of quantitative and reproducible control of membrane diffusion have been recognized as a key experimental bottleneck. Here, we control anomalous diffusion using supported lipid bilayers containing lipids derivatized with polyethylene glycol (PEG) headgroups. Bilayers with specific excluded area fractions are formed by control of PEG lipid mole fraction. These bilayers exhibit a switch in diffusive behavior, becoming anomalous as bilayer continuity is disrupted. Using a combination of single-molecule fluorescence and interferometric imaging, we measure the anomalous behavior in this model over four orders of magnitude in time. Diffusion in these bilayers is well described by a power-law dependence of the mean-square displacement with observation time. Anomaleity in this system can be tailored by simply controlling the mole fraction of PEG lipid, producing bilayers with diffusion parameters similar to those observed for anomalous diffusion in biological membranes.

Simulation Best Practices for Lipid Membranes [Article v1.0]

We establish a reliable and robust standardization of settings for practical molecular dynamics (MD) simulations of pure and mixed (single- and multi-component) lipid bilayer membranes. In lipid membranes research, particle-based molecular simulations are a powerful tool alongside continuum theory, lipidomics, and model, in vitro, and in vivo experiments. Molecular simulations can provide precise and reproducible spatiotemporal (atomic- and femtosecond-level) information about membrane structure, mechanics, thermodynamics, kinetics, and dynamics. Yet the simulation of lipid membranes can be a daunting task, given the uniqueness of lipid membranes relative to conventional liquid-liquid and solid-liquid interfaces, the immense and complex thermodynamic and statistical mechanical theory, the diversity of multiscale lipid models, limitations of modern computing power, the difficulty and ambiguity of simulation controls, finite size effects, competitive continuum simulation alternatives, and the desired application, including vesicle experiments and biological membranes. These issues can complicate an essential understanding of the field of lipid membranes, and create major bottlenecks to simulation advancement. In this article, we clarify these issues and present a consistent, thorough, and user-friendly framework for the design of state-of-the-art lipid membrane MD simulations. We hope to allow early-career researchers to quickly overcome common obstacles in the field of lipid membranes and reach maximal impact in their simulations.

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.

Generalized Langevin equation formulation for anomalous diffusion in the Ising model at the critical temperature

We consider the two- (2D) and three-dimensional (3D) Ising models on a square lattice at the critical temperature T_{c}, under Monte Carlo spin flip dynamics. The bulk magnetization and the magnetization of a tagged line in the 2D Ising model, and the bulk magnetization and the magnetization of a tagged plane in the 3D Ising model, exhibit anomalous diffusion. Specifically, their mean-square displacements increase as power laws in time, collectively denoted as ∼t^{c}, where c is the anomalous exponent. We argue that the anomalous diffusion in all these quantities for the Ising model stems from time-dependent restoring forces, decaying as power laws in time-also with exponent c -in striking similarity to anomalous diffusion in polymeric systems. Prompted by our previous work that has established a memory-kernel based generalized Langevin equation (GLE) formulation for polymeric systems, we show that a closely analogous GLE formulation holds for the Ising model as well. We obtain the memory kernels from spin-spin correlation functions, and the formulation allows us to consistently explain anomalous diffusion as well as anomalous response of the Ising model to an externally applied magnetic field in a consistent manner.

Diffusion of small particles in polymer films

The motion of small probe molecules in a two-dimensional system containing frozen polymer chains was studied by means of Monte Carlo simulations. The model macromolecules were coarse-grained and restricted to vertices of a triangular lattice. The cooperative motion algorithm was used to generate representative configurations of macromolecular systems of different polymer concentrations. The remaining unoccupied lattice sites of the system were filled with small molecules. The structure of the polymer film, especially near the percolation threshold, was determined. The dynamic lattice liquid algorithm was then employed for studies of the dynamics of small objects in the polymer matrix. The influence of chain length and polymer concentration on the mobility and the character of motion of small molecules were studied. Short- and long-time dynamic behaviors of solvent molecules were also described. Conditions of anomalous diffusions' appearance in such systems are discussed. The influence of the structure of the matrix of obstacles on the molecular transport was discussed.

Mixed Macromolecular Crowding: A Protein and Solvent Perspective

In the living cell, biomolecules perform their respective functions in the presence of not only one type of macromolecules but rather in the presence of various macromolecules with different shapes and sizes. In this study, we have investigated the effects of five single macromolecular crowding agents, Dextran 6, Dextran 40, Dextran 70, Ficoll 70, and PEG 8000 and their binary mixtures on the modulation in the domain separation of human serum albumin using a Förster resonance energy transfer-based approach and the translational mobility of a small fluorescent probe fluorescein isothiocyanate (FITC) using fluorescence correlation spectroscopy (FCS). Our observations suggest that mixed crowding induces greater cooperativity in the domain movement as compared to the components of the mixtures. Thermodynamic analyses of the same provide evidence of crossovers from enthalpy-based interactions to effects dominated by hard-sphere potential. When compared with those obtained for individual crowders, both domain movements and FITC diffusion studies show significant deviations from ideality, with an ideal solution being considered to be that arising from the sum of the contributions of those obtained in the presence of individual crowding agents. Considering the fact that domain movements are local (on the order of a few angstroms) in nature while translational movements span much larger lengthscales, our results imply that the observed deviation from simple additivity exists at several possible levels or lengthscales in such mixtures. Moreover, the nature and the type of deviation not only depend on the identities of the components of the crowder mixtures but are also influenced by the particular face of the serum protein (either the domain I-II or the domain II-III face) that the crowders interact with, thus providing further insights into the possible existence of microheterogeneities in such solutions.

Moonlighting Proteins and Their Role in the Control of Signaling Microenvironments, as Exemplified by cGMP and Phytosulfokine Receptor 1 (PSKR1)

Signal generating and processing complexes and changes in concentrations of messenger molecules such as calcium ions and cyclic nucleotides develop gradients that have critical roles in relaying messages within cells. Cytoplasmic contents are densely packed, and in plant cells this is compounded by the restricted cytoplasmic space. To function in such crowded spaces, scaffold proteins have evolved to keep key enzymes in the correct place to ensure ordered spatial and temporal and stimulus-specific message generation. Hence, throughout the cytoplasm there are gradients of messenger molecules that influence signaling processes. However, it is only recently becoming apparent that specific complexes involving receptor molecules can generate multiple signal gradients and enriched microenvironments around the cytoplasmic domains of the receptor that regulate downstream signaling. Such gradients or signal circuits can involve moonlighting proteins, so called because they can enable fine-tune signal cascades via cryptic additional functions that are just being defined. This perspective focuses on how enigmatic activity of moonlighting proteins potentially contributes to regional intracellular microenvironments. For instance, the proteins associated with moonlighting proteins that generate cyclic nucleotides may be regulated by cyclic nucleotide binding directly or indirectly. In this perspective, we discuss how generation of cyclic nucleotide-enriched microenvironments can promote and regulate signaling events. As an example, we use the phytosulfokine receptor (PSKR1), discuss the function of its domains and their mutual interactions and argue that this complex architecture and function enhances tuning of signals in microenvironments.

An efficient algorithm for extracting the magnitude of the measurement error for fractional dynamics

Modern live-imaging fluorescent microscopy techniques following the stochastic motion of labeled tracer particles, i.e. single particle tracking (SPT) experiments, have uncovered significant deviations from the laws of Brownian motion in a variety of biological systems. Accurately characterizing the anomalous diffusion for SPT experiments has become a central issue in biophysics. However, measurement errors raise difficulty in the analysis of single trajectories. In this paper, we introduce a novel surface calibration method based on a fractionally integrated moving average (FIMA) process as an effective tool for extracting both the magnitude of the measurement error and the anomalous exponent for autocorrelated processes of various origins. This method is developed using a toy model - fractional Brownian motion disturbed by independent Gaussian white noise - and is illustrated on both simulated and experimental biological data. We also compare this new method with the mean-squared displacement (MSD) technique, extended to capture the measurement noise in the toy model, which shows inferior results. The introduced procedure is expected to allow for more accurate analysis of fractional anomalous diffusion trajectories with measurement errors across different experimental fields and without the need for any calibration measurements.

Measurement of Rapid Protein Diffusion in the Cytoplasm by Photo-Converted Intensity Profile Expansion

The fluorescence microscopy methods presently used to characterize protein motion in cells infer protein motion from indirect observables, rather than measuring protein motion directly. Operationalizing these methods requires expertise that can constitute a barrier to their broad utilization. Here, we have developed PIPE (photo-converted intensity profile expansion) to directly measure the motion of tagged proteins and quantify it using an effective diffusion coefficient. PIPE works by pulsing photo-convertible fluorescent proteins, generating a peaked fluorescence signal at the pulsed region, and analyzing the spatial expansion of the signal. We demonstrate PIPE's success in measuring accurate diffusion coefficients in silico and in vitro and compare effective diffusion coefficients of native cellular proteins and free fluorophores in vivo. We apply PIPE to measure diffusion anomality in the cell and use it to distinguish free fluorophores from native cellular proteins. PIPE's direct measurement and ease of use make it appealing for cell biologists.

Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement

The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.

The Dark Matter of Biology

The inside of the cell is full of important, yet invisible species of molecules and proteins that interact weakly but couple together to have huge and important effects in many biological processes. Such “dark matter” inside cells remains mostly hidden, because our tools were developed to investigate strongly interacting species and folded proteins. Example dark-matter species include intrinsically disordered proteins, posttranslational states, ion species, and rare, transient, and weak interactions undetectable by biochemical assays. The dark matter of biology is likely to have multiple, vital roles to regulate signaling, rates of reactions, water structure and viscosity, crowding, and other cellular activities. We need to create new tools to image, detect, and understand these dark-matter species if we are to truly understand fundamental physical principles of biology.

Protein diffusion in a bicontinuous microemulsion: inducing sub-diffusion by tuning the water domain size

We study the diffusion of an enhanced green fluorescent protein (GFP+) in bicontinuous sugar-surfactant based microemulsions. The size of the water domains in such systems is controlled by changes of the oil-to-water ratio. Hence, microemulsions allow to produce confinement effects in a controlled way. At high water content the protein is found to exhibit Fickian diffusion. Decreasing the water domain size leads to a slowing down of the protein diffusion and sub-diffusive behavior is obtained on the scale observed by fluorescence correlation spectroscopy. Further decrease of the water domain size finally nearly fixes the GFP+ in these domains and forces it to increasingly follow the breathing mode of the microemulsion matrix.

Mean-squared-displacement statistical test for fractional Brownian motion

Anomalous diffusion in crowded fluids, e.g., in cytoplasm of living cells, is a frequent phenomenon. A common tool by which the anomalous diffusion of a single particle can be classified is the time-averaged mean square displacement (TAMSD). A classical mechanism leading to the anomalous diffusion is the fractional Brownian motion (FBM). A validation of such process for single-particle tracking data is of great interest for experimentalists. In this paper we propose a rigorous statistical test for FBM based on TAMSD. To this end we analyze the distribution of the TAMSD statistic, which is given by the generalized chi-squared distribution. Next, we study the power of the test by means of Monte Carlo simulations. We show that the test is very sensitive for changes of the Hurst parameter. Moreover, it can easily distinguish between two models of subdiffusion: FBM and continuous-time random walk.

Comparison of different models of motion in a crowded environment: a Monte Carlo study

In this paper we investigate the motion of molecules in crowded environments for two dramatically different types of molecular transport. The first type is realized by the dynamic lattice liquid model, which is based on a cooperative movement concept and thus, the motion of molecules is highly correlated. The second one corresponds to a so-called motion of a single agent where the motion of molecules is considered as a random walk without any correlation with other moving elements. The crowded environments are modeled as a two-dimensional triangular lattice with fixed impenetrable obstacles. Our simulation results indicate that the type of transport has an impact on the dynamics of the system, the percolation threshold, critical exponents, and on molecules' trajectories.

Analytical form of the autocorrelation function for the fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) can provide information about diffusion coefficients and rate constants of chemical reactions in small systems of interacting molecules. However, the interpretation of FCS experiments depends crucially on the model of the autocorrelation function for the fluorescence intensity fluctuations. In this theoretical work, we consider a system of fluorescent molecules that diffuse and interact with massive particles, e.g. surfactant micelles. Using the general formalism of FCS, we derive a new analytical approximation of the autocorrelation function for systems in which both diffusion and a binary reaction occur. This approximation provides a smooth interpolation between the limit of fast reaction (much faster than diffusion), and the opposite limit of slow reaction. Our studies of noncovalent interactions of micelles with dyes by FCS provided an experimental case to which the approximate autocorrelation function was successfully applied [X. Zhang, A. Poniewierski, A. Jelińska, A. Zagożdżon, A. Wisniewska, S. Hou and R. Hołyst, Soft Matter, 2016, 12, 8186-8194].

Classification and Segmentation of Nanoparticle Diffusion Trajectories in Cellular Micro Environments

Darkfield and confocal laser scanning microscopy both allow for a simultaneous observation of live cells and single nanoparticles. Accordingly, a characterization of nanoparticle uptake and intracellular mobility appears possible within living cells. Single particle tracking allows to measure the size of a diffusing particle close to a cell. However, within the more complex system of a cell’s cytoplasm normal, confined or anomalous diffusion together with directed motion may occur. In this work we present a method to automatically classify and segment single trajectories into their respective motion types. Single trajectories were found to contain more than one motion type. We have trained a random forest with 9 different features. The average error over all motion types for synthetic trajectories was 7.2%. The software was successfully applied to trajectories of positive controls for normal- and constrained diffusion. Trajectories captured by nanoparticle tracking analysis served as positive control for normal diffusion. Nanoparticles inserted into a diblock copolymer membrane was used to generate constrained diffusion. Finally we segmented trajectories of diffusing (nano-)particles in V79 cells captured with both darkfield- and confocal laser scanning microscopy. The software called “TraJClassifier” is freely available as ImageJ/Fiji plugin via https://git.io/v6uz2.

Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

Subdiffusion in Membrane Permeation of Small Molecules

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

Characterizing anomalous diffusion in crowded polymer solutions and gels over five decades in time with variable-lengthscale fluorescence correlation spectroscopy

The diffusion of macromolecules in cells and in complex fluids is often found to deviate from simple Fickian diffusion. One explanation offered for this behavior is that molecular crowding renders diffusion anomalous, where the mean-squared displacement of the particles scales as 〈r(2)〉∝t(α) with α < 1. Unfortunately, methods such as fluorescence correlation spectroscopy (FCS) or fluorescence recovery after photobleaching (FRAP) probe diffusion only over a narrow range of lengthscales and cannot directly test the dependence of the mean-squared displacement (MSD) on time. Here we show that variable-lengthscale FCS (VLS-FCS), where the volume of observation is varied over several orders of magnitude, combined with a numerical inversion procedure of the correlation data, allows retrieving the MSD for up to five decades in time, bridging the gap between diffusion experiments performed at different lengthscales. In addition, we show that VLS-FCS provides a way to assess whether the propagator associated with the diffusion is Gaussian or non-Gaussian. We used VLS-FCS to investigate two systems where anomalous diffusion had been previously reported. In the case of dense cross-linked agarose gels, the measured MSD confirmed that the diffusion of small beads was anomalous at short lengthscales, with a cross-over to simple diffusion around ≈1 μm, consistent with a caged diffusion process. On the other hand, for solutions crowded with marginally entangled dextran molecules, we uncovered an apparent discrepancy between the MSD, found to be linear, and the propagators at short lengthscales, found to be non-Gaussian. These contradicting features call to mind the "anomalous, yet Brownian" diffusion observed in several biological systems, and the recently proposed "diffusing diffusivity" model.

Automatic detection of diffusion modes within biological membranes using back-propagation neural network

Background

Single particle tracking (SPT) is nowadays one of the most popular technique to probe spatio-temporal dynamics of proteins diffusing within the plasma membrane. Indeed membrane components of eukaryotic cells are very dynamic molecules and can diffuse according to different motion modes. Trajectories are often reconstructed frame-by-frame and dynamic properties often evaluated using mean square displacement (MSD) analysis. However, to get statistically significant results in tracking experiments, analysis of a large number of trajectories is required and new methods facilitating this analysis are still needed.

Results

In this study we developed a new algorithm based on back-propagation neural network (BPNN) and MSD analysis using a sliding window. The neural network was trained and cross validated with short synthetic trajectories. For simulated and experimental data, the algorithm was shown to accurately discriminate between Brownian, confined and directed diffusion modes within one trajectory, the 3 main of diffusion encountered for proteins diffusing within biological membranes. It does not require a minimum number of observed particle displacements within the trajectory to infer the presence of multiple motion states. The size of the sliding window was small enough to measure local behavior and to detect switches between different diffusion modes for segments as short as 20 frames. It also provides quantitative information from each segment of these trajectories. Besides its ability to detect switches between 3 modes of diffusion, this algorithm is able to analyze simultaneously hundreds of trajectories with a short computational time.

Conclusion

This new algorithm, implemented in powerful and handy software, provides a new conceptual and versatile tool, to accurately analyze the dynamic behavior of membrane components.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-016-1064-z) contains supplementary material, which is available to authorized users.

Modeling of band-3 protein diffusion in the normal and defective red blood cell membrane

We employ a two-component red blood cell (RBC) membrane model to simulate lateral diffusion of band-3 proteins in the normal RBC and in the RBC with defective membrane proteins. The defects reduce the connectivity between the lipid bilayer and the membrane skeleton (vertical connectivity), or the connectivity of the membrane skeleton itself (horizontal connectivity), and are associated with the blood disorders of hereditary spherocytosis (HS) and hereditary elliptocytosis (HE) respectively. Initially, we demonstrate that the cytoskeleton limits band-3 lateral mobility by measuring the band-3 macroscopic diffusion coefficients in the normal RBC membrane and in a lipid bilayer without the cytoskeleton. Then, we study band-3 diffusion in the defective RBC membrane and quantify the relation between band-3 diffusion coefficients and percentage of protein defects in HE RBCs. In addition, we illustrate that at low spectrin network connectivity (horizontal connectivity) band-3 subdiffusion can be approximated as anomalous diffusion, while at high horizontal connectivity band-3 diffusion is characterized as confined diffusion. Our simulations show that the band-3 anomalous diffusion exponent depends on the percentage of protein defects in the membrane cytoskeleton. We also confirm that the introduction of attraction between the lipid bilayer and the spectrin network reduces band-3 diffusion, but we show that this reduction is lower than predicted by the percolation theory. Furthermore, we predict that the attractive force between the spectrin filament and the lipid bilayer is at least 20 times smaller than the binding forces at band-3 and glycophorin C, the two major membrane binding sites. Finally, we explore diffusion of band-3 particles in the RBC membrane with defects related to vertical connectivity. We demonstrate that in this case band-3 diffusion can be approximated as confined diffusion for all attraction levels between the spectrin network and the lipid bilayer. By comparing the diffusion coefficients measured in horizontal vs. vertical defects, we conclude that band-3 mobility is primarily controlled by the horizontal connectivity.

Ergodicity testing for anomalous diffusion: small sample statistics

The analysis of trajectories recorded in experiments often requires calculating time averages instead of ensemble averages. According to the Boltzmann hypothesis, they are equivalent only under the assumption of ergodicity. In this paper, we implement tools that allow to study ergodic properties. This analysis is conducted in two classes of anomalous diffusion processes: fractional Brownian motion and subordinated Ornstein-Uhlenbeck process. We show that only first of them is ergodic. We demonstrate this by applying rigorous statistical methods: mean square displacement, confidence intervals, and dynamical functional test. Our methodology is universal and can be implemented for analysis of many experimental data not only if a large sample is available but also when there are only few trajectories recorded.

Splitting of the Universality Class of Anomalous Transport in Crowded Media

We investigate the emergence of subdiffusive transport by obstruction in continuum models for molecular crowding. While the underlying percolation transition for the accessible space displays universal behavior, the dynamic properties depend in a subtle nonuniversal way on the transport through narrow channels. At the same time, the different universality classes are robust with respect to introducing correlations in the obstacle matrix as we demonstrate for quenched hard-sphere liquids as underlying structures. Our results confirm that the microscopic dynamics can dominate the relaxational behavior even at long times, in striking contrast to glassy dynamics.

Confined diffusion of transmembrane proteins and lipids induced by the same actin meshwork lining the plasma membrane

Ultraspeed single-molecule tracking with <25-μs resolution and electron tomography show that transmembrane proteins and phospholipids in the plasma membrane hop among submicrometer compartments of the same size, probably delimited by the anchored-transmembrane-protein pickets lining the actin-based membrane-skeleton fence, once every 1–58 ms.

The mechanisms by which the diffusion rate in the plasma membrane (PM) is regulated remain unresolved, despite their importance in spatially regulating the reaction rates in the PM. Proposed models include entrapment in nanoscale noncontiguous domains found in PtK2 cells, slow diffusion due to crowding, and actin-induced compartmentalization. Here, by applying single-particle tracking at high time resolutions, mainly to the PtK2-cell PM, we found confined diffusion plus hop movements (termed “hop diffusion”) for both a nonraft phospholipid and a transmembrane protein, transferrin receptor, and equal compartment sizes for these two molecules in all five of the cell lines used here (actual sizes were cell dependent), even after treatment with actin-modulating drugs. The cross-section size and the cytoplasmic domain size both affected the hop frequency. Electron tomography identified the actin-based membrane skeleton (MSK) located within 8.8 nm from the PM cytoplasmic surface of PtK2 cells and demonstrated that the MSK mesh size was the same as the compartment size for PM molecular diffusion. The extracellular matrix and extracellular domains of membrane proteins were not involved in hop diffusion. These results support a model of anchored TM-protein pickets lining actin-based MSK as a major mechanism for regulating diffusion.

Lipid Diffusion in Supported Lipid Bilayers: A Comparison between Line-Scanning Fluorescence Correlation Spectroscopy and Single-Particle Tracking

Diffusion in lipid membranes is an essential component of many cellular process and fluorescence a method of choice to study membrane dynamics. The goal of this work was to directly compare two common fluorescence methods, line-scanning fluorescence correlation spectroscopy and single-particle tracking, to observe the diffusion of a fluorescent lipophilic dye, DiD, in a complex five-component mitochondria-like solid-supported lipid bilayer. We measured diffusion coefficients of DFCS ~ 3 um2 * s-1 and DSPT ~ 2 um2 * s-1, respectively. These comparable, yet statistically different values are used to highlight the main message of the paper, namely that the two considered methods give access to distinctly different dynamic ranges: D sup or approximatively 1um2 * s-1 for FCS and D inf or approximatively 5 um2 s-1 for SPT (with standard imaging conditions). In the context of membrane diffusion, this means that FCS allows studying lipid diffusion in fluid membranes, as well as the diffusion of loosely-bound proteins hovering above the membrane. SPT, on the other hand, is ideal to study the motions of membrane-inserted proteins, especially those presenting different conformations, but only allows studying lipid diffusion in relatively viscous membranes, such as supported lipid bilayers and cell membranes.

Subdiffusive motion of bacteriophage in mucosal surfaces increases the frequency of bacterial encounters

Bacteriophages (phages) defend mucosal surfaces against bacterial infections. However, their complex interactions with their bacterial hosts and with the mucus-covered epithelium remain mostly unexplored. Our previous work demonstrated that T4 phage with Hoc proteins exposed on their capsid adhered to mucin glycoproteins and protected mucus-producing tissue culture cells in vitro. On this basis, we proposed our bacteriophage adherence to mucus (BAM) model of immunity. Here, to test this model, we developed a microfluidic device (chip) that emulates a mucosal surface experiencing constant fluid flow and mucin secretion dynamics. Using mucus-producing human cells and Escherichia coli in the chip, we observed similar accumulation and persistence of mucus-adherent T4 phage and nonadherent T4∆hoc phage in the mucus. Nevertheless, T4 phage reduced bacterial colonization of the epithelium >4,000-fold compared with T4∆hoc phage. This suggests that phage adherence to mucus increases encounters with bacterial hosts by some other mechanism. Phages are traditionally thought to be completely dependent on normal diffusion, driven by random Brownian motion, for host contact. We demonstrated that T4 phage particles displayed subdiffusive motion in mucus, whereas T4∆hoc particles displayed normal diffusion. Experiments and modeling indicate that subdiffusive motion increases phage-host encounters when bacterial concentration is low. By concentrating phages in an optimal mucus zone, subdiffusion increases their host encounters and antimicrobial action. Our revised BAM model proposes that the fundamental mechanism of mucosal immunity is subdiffusion resulting from adherence to mucus. These findings suggest intriguing possibilities for engineering phages to manipulate and personalize the mucosal microbiome.

Nuclear physics (of the cell, not the atom)

The nucleus is physically distinct from the cytoplasm in ways that suggest new ideas and approaches for interrogating the operation of this organelle. Chemical bond formation and breakage underlie the lives of cells, but as this special issue of Molecular Biology of the Cell attests, the nonchemical aspects of cell nuclei present a new frontier to biologists and biophysicists.

Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors - An alternative approach

Accurately characterizing the anomalous diffusion of a tracer particle has become a central issue in biophysics. However, measurement errors raise difficulty in the characterization of single trajectories, which is usually performed through the time-averaged mean square displacement (TAMSD). In this paper, we study a fractionally integrated moving average (FIMA) process as an appropriate model for anomalous diffusion data with measurement errors. We compare FIMA and traditional TAMSD estimators for the anomalous diffusion exponent. The ability of the FIMA framework to characterize dynamics in a wide range of anomalous exponents and noise levels through the simulation of a toy model (fractional Brownian motion disturbed by Gaussian white noise) is discussed. Comparison to the TAMSD technique, shows that FIMA estimation is superior in many scenarios. This is expected to enable new measurement regimes for single particle tracking (SPT) experiments even in the presence of high measurement errors.

Biased diffusion in three-dimensional comb-like structures

In this paper, we study biased diffusion of point Brownian particles in a three-dimensional comb-like structure formed by a main cylindrical tube with identical periodic cylindrical dead ends. It is assumed that the dead ends are thin cylinders whose radius is much smaller than both the radius of the main tube and the distance between neighboring dead ends. It is also assumed that in the main tube, the particle, in addition to its regular diffusion, moves with a uniform constant drift velocity. For such a system, we develop a formalism that allows us to derive analytical expressions for the Laplace transforms of the first two moments of the particle displacement along the main tube axis. Inverting these Laplace transforms numerically, one can find the time dependences of the two moments for arbitrary values of both the drift velocity and the dead-end length, including the limiting case of infinitely long dead ends, where the unbiased diffusion becomes anomalous at sufficiently long times. The expressions for the Laplace transforms are used to find the effective drift velocity and diffusivity of the particle as functions of its drift velocity in the main tube and the tube geometric parameters. As might be expected from common-sense arguments, the effective drift velocity monotonically decreases from the initial drift velocity to zero as the dead-end length increases from zero to infinity. The effective diffusivity is a more complex, non-monotonic function of the dead-end length. As this length increases from zero to infinity, the effective diffusivity first decreases, reaches a minimum, and then increases approaching a plateau value which is proportional to the square of the particle drift velocity in the main tube.

Aging scaled Brownian motion

Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.

Hindered nanoparticle diffusion and void accessibility in a three-dimensional porous medium

The inherent pore-scale heterogeneity of many natural and synthetic porous materials can make it difficult to model and predict porous transport because the underlying microscopic processes are often poorly understood. Here we present the results of single-particle tracking experiments in which we followed the pore-scale diffusion of individual nanoparticles, deep within a three-dimensional porous material of moderate porosity. We observed significant hydrodynamic damping of particle motion at subpore length scales, resulting in heterogeneous and spatially dependent mobility. The accessibility of the void space was strongly dependent on particle size, and related to the heterogeneous hydrodynamics. Our results suggest that pore-scale diffusion is more heterogeneous and volume accessibility more limited than previously expected. The method demonstrated here will enable studies of a broad new class of materials including porous polymers of technological interest.