Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study

Multifractal spectral features enhance classification of anomalous diffusion

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

Record ages of non-Markovian scale-invariant random walks

How long is needed for an observable to exceed its previous highest value and establish a new record? This time, known as the age of a record plays a crucial role in quantifying record statistics. Until now, general methods for determining record age statistics have been limited to observations of either independent random variables or successive positions of a Markovian (memoryless) random walk. Here we develop a theoretical framework to determine record age statistics in the presence of memory effects for continuous non-smooth processes that are asymptotically scale-invariant. Our theoretical predictions are confirmed by numerical simulations and experimental realisations of diverse representative non-Markovian random walk models and real time series with memory effects, in fields as diverse as genomics, climatology, hydrology, geology and computer science. Our results reveal the crucial role of the number of records already achieved in time series and change our view on analysing record statistics.

What's past is prologue: FRAP keeps delivering 50 years later

Fluorescence recovery after photobleaching (FRAP) has emerged as one of the most widely utilized techniques to quantify binding and diffusion kinetics of biomolecules in biophysics. Since its inception in the mid-1970s, FRAP has been used to address an enormous array of questions including the characteristic features of lipid rafts, how cells regulate the viscosity of their cytoplasm, and the dynamics of biomolecules inside condensates formed by liquid-liquid phase separation. In this perspective, I briefly summarize the history of the field and discuss why FRAP has proven to be so incredibly versatile and popular. Next, I provide an overview of the extensive body of knowledge that has emerged on best practices for quantitative FRAP data analysis, followed by some recent examples of biological lessons learned using this powerful approach. Finally, I touch on new directions and opportunities for biophysicists to contribute to the continued development of this still-relevant research tool.

Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories

Single-particle traces of the diffusive motion of molecules, cells, or animals are by now routinely measured, similar to stochastic records of stock prices or weather data. Deciphering the stochastic mechanism behind the recorded dynamics is vital in understanding the observed systems. Typically, the task is to decipher the exact type of diffusion and/or to determine the system parameters. The tools used in this endeavor are currently being revolutionized by modern machine-learning techniques. In this Perspective we provide an overview of recently introduced methods in machine-learning for diffusive time series, most notably, those successfully competing in the anomalous diffusion challenge. As such methods are often criticized for their lack of interpretability, we focus on means to include uncertainty estimates and feature-based approaches, both improving interpretability and providing concrete insight into the learning process of the machine. We expand the discussion by examining predictions on different out-of-distribution data. We also comment on expected future developments.

Expanding the applicability of multiphoton fluorescence recovery after photobleaching by incorporating shear stress in laminar flow

Significance

Multi-photon fluorescence recovery after photobleaching (MPFRAP) is a nonlinear microscopy technique used to measure the diffusion coefficient of fluorescently tagged molecules in solution. Previous MPFRAP fitting models calculate the diffusion coefficient in systems with diffusion or diffusion in laminar flow.

Aim

We propose an MPFRAP fitting model that accounts for shear stress in laminar flow, making it a more applicable technique for in vitro and in vivo studies involving diffusion.

Approach

Fluorescence recovery curves are generated using high-throughput molecular dynamics simulations and then fit to all three models (diffusion, diffusion and flow, and diffusion and shear flow) to define the limits within which accurate diffusion coefficients are produced. Diffusion is simulated as a random walk with a variable horizontal bias to account for shear flow.

Results

Contour maps of the accuracy of the fitted diffusion coefficient as a function of scaled velocity and scaled shear rate show the parameter space within which each model produces accurate diffusion coefficients; the shear-flow model covers a larger area than the previous models.

Conclusion

The shear-flow model allows MPFRAP to be a viable optical tool for studying more biophysical systems than previous models.

Normal and anomalous diffusion in a bouncing ball over an irregular surface

The problem of a bouncing ball on a nonplanar surface is investigated. We discovered that surface undulation adds a horizontal component to the impact force, which acquires a random character. Some aspects of Brownian motion are found in the horizontal distribution of the particle. On the x axis, normal and superdiffusion are observed. For the probability density's functional form, a scaling hypothesis is presented.

Bayesian deep learning for error estimation in the analysis of anomalous diffusion

Modern single-particle-tracking techniques produce extensive time-series of diffusive motion in a wide variety of systems, from single-molecule motion in living-cells to movement ecology. The quest is to decipher the physical mechanisms encoded in the data and thus to better understand the probed systems. We here augment recently proposed machine-learning techniques for decoding anomalous-diffusion data to include an uncertainty estimate in addition to the predicted output. To avoid the Black-Box-Problem a Bayesian-Deep-Learning technique named Stochastic-Weight-Averaging-Gaussian is used to train models for both the classification of the diffusion model and the regression of the anomalous diffusion exponent of single-particle-trajectories. Evaluating their performance, we find that these models can achieve a well-calibrated error estimate while maintaining high prediction accuracies. In the analysis of the output uncertainty predictions we relate these to properties of the underlying diffusion models, thus providing insights into the learning process of the machine and the relevance of the output.

Inferences from FRAP data are model dependent: A subdiffusive analysis

Fluorescence recovery after photobleaching (FRAP) is a widely used biological experiment to study the kinetics of molecules that react and move randomly. Since the development of FRAP in the 1970s, many reaction-diffusion models have been used to interpret FRAP data. However, intracellular molecules are widely observed to move by anomalous subdiffusion instead of normal diffusion. In this article, we extend a popular reaction-diffusion model of FRAP to the case of subdiffusion modeled by a fractional diffusion equation. By analyzing this reaction-subdiffusion model, we show that FRAP data are consistent with both diffusive and subdiffusive motion in many scenarios. We illustrate this general result by fitting our model to FRAP data from glucocorticoid receptors in a cell nucleus. We further show that the assumed model of molecular motion (normal diffusion or subdiffusion) strongly impacts the biological parameter values inferred from a given experimentally observed FRAP curve. We additionally analyze our model in three simplified parameter regimes and discuss parameter identifiability for varying subdiffusion exponents.

A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes and general space domains. The stability results are proved for all algorithms and the accuracy of obtained approximations is estimated by solving well-known test problems. A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem. Results of computational experiments are presented and analyzed.

The TFIIH subunits p44/p62 act as a damage sensor during nucleotide excision repair

Nucleotide excision repair (NER) in eukaryotes is orchestrated by the core form of the general transcription factor TFIIH, containing the helicases XPB, XPD and five ‘structural’ subunits, p62, p44, p34, p52 and p8. Recent cryo-EM structures show that p62 makes extensive contacts with p44 and in part occupies XPD’s DNA binding site. While p44 is known to regulate the helicase activity of XPD during NER, p62 is thought to be purely structural. Here, using helicase and adenosine triphosphatase assays we show that a complex containing p44 and p62 enhances XPD’s affinity for dsDNA 3-fold over p44 alone. Remarkably, the relative affinity is further increased to 60-fold by dsDNA damage. Direct binding studies show this preference derives from p44/p62’s high affinity (20 nM) for damaged ssDNA. Single molecule imaging of p44/p62 complexes without XPD reveals they bind to and randomly diffuse on DNA, however, in the presence of UV-induced DNA lesions these complexes stall. Combined with the analysis of a recent cryo-EM structure, we suggest that p44/p62 acts as a novel DNA-binding entity that enhances damage recognition in TFIIH. This revises our understanding of TFIIH and prompts investigation into the core subunits for an active role during DNA repair and/or transcription.

Log-Normal Superstatistics for Brownian Particles in a Heterogeneous Environment

Superstatistical approaches have played a crucial role in the investigations of mixtures of Gaussian processes. Such approaches look to describe non-Gaussian diffusion emergence in single-particle tracking experiments realized in soft and biological matter. Currently, relevant progress in superstatistics of Gaussian diffusion processes has been investigated by applying χ2-gamma and χ2-gamma inverse superstatistics to systems of particles in a heterogeneous environment whose diffusivities are randomly distributed; such situations imply Brownian yet non-Gaussian diffusion. In this paper, we present how the log-normal superstatistics of diffusivities modify the density distribution function for two types of mixture of Brownian processes. Firstly, we investigate the time evolution of the ensemble of Brownian particles with random diffusivity through the analytical and simulated points of view. Furthermore, we analyzed approximations of the overall probability distribution for log-normal superstatistics of Brownian motion. Secondly, we propose two models for a mixture of scaled Brownian motion and to analyze the log-normal superstatistics associated with them, which admits an anomalous diffusion process. The results found in this work contribute to advances of non-Gaussian diffusion processes and superstatistical theory.

How Do Surface Properties of Nanoparticles Influence Their Diffusion in the Extracellular Matrix? A Model Study in Matrigel Using Polymer-Grafted Nanoparticles

Diffusion of nanomedicines inside the extracellular matrix (ECM) has been identified as a key factor to achieve homogeneous distribution and therefore therapeutic efficacy. Here, we sought to determine the impact of nanoparticles' (NPs) surface properties on their ability to diffuse in the ECM. As model nano-objects, we used a library of gold nanoparticles grafted with a versatile polymethacrylate corona, which enabled the surface properties to be modified. To accurately recreate the features of the native ECM, diffusion studies were carried out in a tumor-derived gel (Matrigel). We developed two methods to evaluate the diffusion ability of NPs inside this model gel: an easy-to-implement one based on optical monitoring and another one using small-angle X-ray scattering (SAXS) measurements. Both enabled the determination of the diffusion coefficients of NPs and comparison of the influence of their various surface properties, while the SAXS technique also allowed to monitor the NPs' structure as they diffused inside the gel. Positive charges and hydrophobicity were found to particularly hinder diffusion, and the different results suggested on the whole the presence of NPs-matrix interactions, therefore underlying the importance of the ECM model. The accuracy of the tumor-derived gels used in this study was evidenced by in vivo experiments involving intratumoral injections of NPs on mice, which showed that diffusion patterns in the peripheral tumor tissues were quite similar to the ones obtained within the chosen ECM model.

Generalization of the Kubo relation for confined motion and ergodicity breakdown

A time-dependent generalized Kubo relation is derived by introducing the notion of a diffusion function for a particle confined in a harmonic potential. The relation reduces to the standard Kubo relation as a special case but holds for anomalous diffusion, nonergodic processes, and bounded motion. We analyze in detail the behaviors of the diffusion and memory functions and report a generalized Stokes-Einstein relation concerning anomalous diffusion. Furthermore, we demonstrate that when a high finite-frequency cutoff is imposed on the noise spectral density, a breakdown in ergodicity accompanied by the appearance of nonstationarity in the velocity autocorrelation function occurs in forced systems. This breakdown is taken as explicit evidence for either decay-spring-memory or recovering-force effects leading to nonexponential relaxation kinematics.

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.

Molecular transport in systems containing binding obstacles

We studied the movement of particles in crowded environments by means of extensive Monte Carlo simulations. The dynamic lattice liquid model was employed for this purpose. It is based on the cooperative movement concept and allows the study of systems at high densities. The cooperative model of molecular transport is assumed: the motion of all moving particles is highly correlated. The model is supposed to mimic lateral motion in a membrane and therefore the system is two-dimensional with moving objects and traps placed on a triangular lattice. In our study the interaction (binding) of traps with moving particles was assumed. The conditions in which subdiffusive motion appeared in the system were analysed. The influence of the strength of binding on the dynamic percolation threshold was also shown. The differences in the dynamics compared to systems with impenetrable obstacles and with systems without correlation in motion were presented and discussed. It was shown that in the case of correlated motion the influence of deep traps is similar to that of impenetrable obstacles. If the traps are shallow a recovery to normal diffusion was observed for longer time periods.

Mechanical Interaction between Cells Facilitates Molecular Transport

In vivo, eukaryotic cells are embedded in a matrix environment, where they grow and develop. Generally, this extracellular matrix (ECM) is an anisotropic fibrous structure, through which macromolecules and biochemical signaling molecules at the nanometer scale diffuse. The ECM is continuously remodeled by cells, via mechanical interactions, which lead to a potential link between biomechanical and biochemical cell-cell interactions. Here, it is studied how cell-induced forces applied on the ECM impact the biochemical transport of molecules between distant cells. It is experimentally observed that cells remodel the ECM by increasing fiber alignment and density of the matrix between them over time. Using random walk simulations on a 3D lattice, elongated fixed obstacles are implemented that mimic the fibrous ECM structure. Both diffusion of a tracer molecule and the mean first-passage time a molecule secreted from one cell takes to reach another cell are measured. The model predicts that cell-induced remodeling can lead to a dramatic speedup in the transport of molecules between cells. Fiber alignment and densification cause reduction of the transport dimensionality from a 3D to a much more rapid 1D process. Thus, a novel mechanism of mechano-biochemical feedback in the regulation of long-range cell-cell communication is suggested.

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).

Nonrenewal resetting of scaled Brownian motion

We investigate an intermittent stochastic process in which diffusive motion with a time-dependent diffusion coefficient, D(t)∼t^{α-1}, α>0 (scaled Brownian motion), is stochastically reset to its initial position and starts anew. The resetting follows a renewal process with either an exponential or a power-law distribution of the waiting times between successive renewals. The resetting events, however, do not affect the time dependence of the diffusion coefficient, so that the whole process appears to be a nonrenewal one. We discuss the mean squared displacement of a particle and the probability density function of its positions in this process. We show that scaled Brownian motion with resetting demonstrates rich behavior whose properties essentially depend on the interplay of the parameters of the resetting process and the particle's displacement infree motion. The motion of particles can remain almost unaffected by resetting but can also get slowed down or even be completely suppressed. Especially interesting are the nonstationary situations in which the mean squared displacement stagnates but the distribution of positions does not tend to any steady state. This behavior is compared to the situation [discussed in the companion paper; A. S. Bodrova et al., Phys. Rev. E 100, 012120 (2019)10.1103/PhysRevE.100.012120] in which the memory of the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. We show that the properties of the probability densities in such processes (erasing or retaining the memory on the diffusion coefficient) are vastly different.

Multistep Track Segmentation and Motion Classification for Transient Mobility Analysis

Molecular interactions are often transient and might change within the window of observation, leading to changes in molecule movement. Therefore, accurate motion analysis often requires transient motion classification. Here we present an accurate and computationally efficient transient mobility analysis framework, termed "divide-and-conquer moment scaling spectrum" (DC-MSS). DC-MSS works in a multistep fashion: 1) it utilizes a local movement descriptor throughout a track to divide it into initial segments of putatively different motion classes; 2) it classifies these segments via moment scaling spectrum (MSS) analysis of molecule displacements; and 3) it uses the MSS analysis results to refine the track segmentation. This strategy uncouples the initial identification of motion switches from motion classification, allowing DC-MSS to circumvent the sensitivity-accuracy tradeoff of classic rolling window approaches for transient motion analysis, while at the same time harnessing the classification power of MSS analysis. Testing of DC-MSS demonstrates that it detects switches among free diffusion, confined diffusion, directed diffusion, and immobility with great sensitivity. To illustrate the utility of DC-MSS, we have applied it to single-particle tracks of the transmembrane protein CD44 on the surface of macrophages, revealing actin cortex-dependent transient mobility changes.

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.

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.

Numerical Simulation and FRAP Experiments Show That the Plasma Membrane Binding Protein PH-EFA6 Does Not Exhibit Anomalous Subdiffusion in Cells

The fluorescence recovery after photobleaching (FRAP) technique has been used for decades to measure movements of molecules in two-dimension (2D). Data obtained by FRAP experiments in cell plasma membranes are assumed to be described through a means of two parameters, a diffusion coefficient, D (as defined in a pure Brownian model) and a mobile fraction, M. Nevertheless, it has also been shown that recoveries can be nicely fit using anomalous subdiffusion. Fluorescence recovery after photobleaching (FRAP) at variable radii has been developed using the Brownian diffusion model to access geometrical characteristics of the surrounding landscape of the molecule. Here, we performed numerical simulations of continuous time random walk (CTRW) anomalous subdiffusion and interpreted them in the context of variable radii FRAP. These simulations were compared to experimental data obtained at variable radii on living cells using the pleckstrin homology (PH) domain of the membrane binding protein EFA6 (exchange factor for ARF6, a small G protein). This protein domain is an excellent candidate to explore the structure of the interface between cytosol and plasma membrane in cells. By direct comparison of our numerical simulations to the experiments, we show that this protein does not exhibit anomalous diffusion in baby hamster kidney (BHK) cells. The non Brownian PH-EFA6 dynamics observed here are more related to spatial heterogeneities such as cytoskeleton fence effects.

Bioengineering of a Full-Thickness Skin Equivalent in a 96-Well Insert Format for Substance Permeation Studies and Organ-On-A-Chip Applications

The human skin is involved in protecting the inner body from constant exposure to outer environmental stimuli. There is an evident need to screen for toxicity and the efficacy of drugs and cosmetics applied to the skin. To date, animal studies are still the standard method for substance testing, although they are currently controversially discussed Therefore, the multi-organ chip is an attractive alternative to replace animal testing. The two-organ chip is designed to hold 96-well cell culture inserts (CCIs). Small-sized skin equivalents are needed for this. In this study, full-thickness skin equivalents (ftSEs) were generated successfully inside 96-well CCIs. These skin equivalents developed with in vivo-like histological architecture, with normal differentiation marker expressions and proliferation rates. The 96-well CCI-based ftSEs were successfully integrated into the two-organ chip. The permeation of fluorescein sodium salt through the ftSEs was monitored during the culture. The results show a decreasing value for the permeation over time, which seems a promising method to track the development of the ftSEs. Additionally, the permeation was implemented in a computational fluid dynamics simulation, as a tool to predict results in long-term experiments. The advantage of these ftSEs is the reduced need for cells and substances, which makes them more suitable for high throughput assays.

Fractional cable equation for general geometry: A model of axons with swellings and anomalous diffusion

Different experimental studies have reported anomalous diffusion in brain tissues and notably this anomalous diffusion is expressed through fractional derivatives. Axons are important to understand neurodegenerative diseases such as multiple sclerosis, Alzheimer's disease, and Parkinson's disease. Indeed, abnormal accumulation of proteins and organelles in axons is a hallmark of these diseases. The diffusion in the axons can become anomalous as a result of this abnormality. In this case the voltage propagation in axons is affected. Another hallmark of different neurodegenerative diseases is given by discrete swellings along the axon. In order to model the voltage propagation in axons with anomalous diffusion and swellings, in this paper we propose a fractional cable equation for a general geometry. This generalized equation depends on fractional parameters and geometric quantities such as the curvature and torsion of the cable. For a cable with a constant radius we show that the voltage decreases when the fractional effect increases. In cables with swellings we find that when the fractional effect or the swelling radius increases, the voltage decreases. Similar behavior is obtained when the number of swellings and the fractional effect increase. Moreover, we find that when the radius swelling (or the number of swellings) and the fractional effect increase at the same time, the voltage dramatically decreases.

Tetramethylpyrazine-Loaded Hydrogels: Preparation, Penetration Through a Subcutaneous-Mucous-Membrane Model, and a Molecular Dynamics Simulation

Tetramethylpyrazine (TMP) was extracted from Ligusticum chuanxiong hort. The compound is known to have a variety of medicinal functions; in particular, it is used for the treatment of cerebral ischemic diseases. TMP-loaded hydrogels offer an excellent preparation with the capacity to bypass the blood-brain barrier, allowing treatment of the brain through intranasal administration. We prepared TMP-loaded hydrogels using carbomer 940 and evaluated the release of TMP from the hydrogel. We determined the release rate using Franz-type diffusion cell experiments with a subcutaneous-mucous-membrane model and also by a molecular dynamics (MD) simulation. In general, the former method was more complicated than the latter was. The dynamic behavior of TMP release from the hydrogel was revealed by analysis of the mean square displacement of the trajectory in the MD simulation. The coefficient of TMP diffusion from the hydrogel was calculated at different temperatures (277, 298, and 310 K) by using MD software. The results showed that the coefficient of diffusion increased with an increase in temperature. This trend was observed both experimentally and in the MD simulation. Therefore, the MD simulation was a complementary method to verify the experimental data.

The particle in the spider's web: transport through biological hydrogels

Biological hydrogels such as mucus, extracellular matrix, biofilms, and the nuclear pore have diverse functions and compositions, but all act as selectively permeable barriers to the diffusion of particles. Each barrier has a crosslinked polymeric mesh that blocks penetration of large particles such as pathogens, nanotherapeutics, or macromolecules. These polymeric meshes also employ interactive filtering, in which affinity between solutes and the gel matrix controls permeability. Interactive filtering affects the transport of particles of all sizes including peptides, antibiotics, and nanoparticles and in many cases this filtering can be described in terms of the effects of charge and hydrophobicity. The concepts described in this review can guide strategies to exploit or overcome gel barriers, particularly for applications in diagnostics, pharmacology, biomaterials, and drug delivery.

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.

Rad4 recognition-at-a-distance: Physical basis of conformation-specific anomalous diffusion of DNA repair proteins

Since Robert Brown's first observations of random walks by pollen particles suspended in solution, the concept of diffusion has been subject to countless theoretical and experimental studies in diverse fields from finance and social sciences, to physics and biology. Diffusive transport of macromolecules in cells is intimately linked to essential cellular functions including nutrient uptake, signal transduction, gene expression, as well as DNA replication and repair. Advancement in experimental techniques has allowed precise measurements of these diffusion processes. Mathematical and physical descriptions and computer simulations have been applied to model complicated biological systems in which anomalous diffusion, in addition to simple Brownian motion, was observed. The purpose of this review is to provide an overview of the major physical models of anomalous diffusion and corresponding experimental evidence on the target search problem faced by DNA-binding proteins, with an emphasis on DNA repair proteins and the role of anomalous diffusion in DNA target recognition.

Simulation of Molecular Transport in Systems Containing Mobile Obstacles

In this paper, we investigate the movement of molecules in crowded environments with obstacles undergoing Brownian motion by means of extensive Monte Carlo simulations. Our investigations were performed using the dynamic lattice liquid model, which was based on the cooperative movement concept and allowed to mimic systems at high densities where the motion of all elements (obstacles as well as moving particles) were highly correlated. The crowded environments are modeled on a two-dimensional triangular lattice containing obstacles (particles whose mobility was significantly reduced) moving by a Brownian motion. The subdiffusive motion of both elements in the system was analyzed. It was shown that the percolation transition does not exist in such systems in spite of the cooperative character of the particles' motion. The reduction of the obstacle mobility leads to the longer caging of liquid particles by mobile obstacles.

Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion

It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases.

Anomalous diffusion and dynamics of fluorescence recovery after photobleaching in the random-comb model

We address the problem of diffusion on a comb whose teeth display varying lengths. Specifically, the length ℓ of each tooth is drawn from a probability distribution displaying power law behavior at large ℓ,P(ℓ)∼ℓ^{-(1+α)} (α>0). To start with, we focus on the computation of the anomalous diffusion coefficient for the subdiffusive motion along the backbone. This quantity is subsequently used as an input to compute concentration recovery curves mimicking fluorescence recovery after photobleaching experiments in comblike geometries such as spiny dendrites. Our method is based on the mean-field description provided by the well-tested continuous time random-walk approach for the random-comb model, and the obtained analytical result for the diffusion coefficient is confirmed by numerical simulations of a random walk with finite steps in time and space along the backbone and the teeth. We subsequently incorporate retardation effects arising from binding-unbinding kinetics into our model and obtain a scaling law characterizing the corresponding change in the diffusion coefficient. Finally, we show that recovery curves obtained with the help of the analytical expression for the anomalous diffusion coefficient cannot be fitted perfectly by a model based on scaled Brownian motion, i.e., a standard diffusion equation with a time-dependent diffusion coefficient. However, differences between the exact curves and such fits are small, thereby providing justification for the practical use of models relying on scaled Brownian motion as a fitting procedure for recovery curves arising from particle diffusion in comblike systems.

Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes

We investigate the diffusive and ergodic properties of massive and confined particles in a model disordered medium, in which the local diffusivity fluctuates in time while its mean has a power law dependence on the diffusion time.

A review of progress in single particle tracking: from methods to biophysical insights

Optical microscopy has for centuries been a key tool to study living cells with minimum invasiveness. The advent of single molecule techniques over the past two decades has revolutionized the field of cell biology by providing a more quantitative picture of the complex and highly dynamic organization of living systems. Amongst these techniques, single particle tracking (SPT) has emerged as a powerful approach to study a variety of dynamic processes in life sciences. SPT provides access to single molecule behavior in the natural context of living cells, thereby allowing a complete statistical characterization of the system under study. In this review we describe the foundations of SPT together with novel optical implementations that nowadays allow the investigation of single molecule dynamic events with increasingly high spatiotemporal resolution using molecular densities closer to physiological expression levels. We outline some of the algorithms for the faithful reconstruction of SPT trajectories as well as data analysis, and highlight biological examples where the technique has provided novel insights into the role of diffusion regulating cellular function. The last part of the review concentrates on different theoretical models that describe anomalous transport behavior and ergodicity breaking observed from SPT studies in living cells.

Fluorescence recovery after photobleaching in material and life sciences: putting theory into practice

Fluorescence recovery after photobleaching (FRAP) is a versatile tool for determining diffusion and interaction/binding properties in biological and material sciences. An understanding of the mechanisms controlling the diffusion requires a deep understanding of structure–interaction–diffusion relationships. In cell biology, for instance, this applies to the movement of proteins and lipids in the plasma membrane, cytoplasm and nucleus. In industrial applications related to pharmaceutics, foods, textiles, hygiene products and cosmetics, the diffusion of solutes and solvent molecules contributes strongly to the properties and functionality of the final product. All these systems are heterogeneous, and accurate quantification of the mass transport processes at the local level is therefore essential to the understanding of the properties of soft (bio)materials. FRAP is a commonly used fluorescence microscopy-based technique to determine local molecular transport at the micrometer scale. A brief high-intensity laser pulse is locally applied to the sample, causing substantial photobleaching of the fluorescent molecules within the illuminated area. This causes a local concentration gradient of fluorescent molecules, leading to diffusional influx of intact fluorophores from the local surroundings into the bleached area. Quantitative information on the molecular transport can be extracted from the time evolution of the fluorescence recovery in the bleached area using a suitable model. A multitude of FRAP models has been developed over the years, each based on specific assumptions. This makes it challenging for the non-specialist to decide which model is best suited for a particular application. Furthermore, there are many subtleties in performing accurate FRAP experiments. For these reasons, this review aims to provide an extensive tutorial covering the essential theoretical and practical aspects so as to enable accurate quantitative FRAP experiments for molecular transport measurements in soft (bio)materials.

Universal Approach to FRAP Analysis of Arbitrary Bleaching Patterns

The original approach to calculating diffusion coefficients of a fluorescent probe from Fluorescence Recovery After Photobleaching (FRAP) measurements assumes bleaching with a circular laser beam of a Gaussian intensity profile. This method was used without imaging the bleached cell. An empirical equation for calculating diffusion coefficients from a rectangular bleaching geometry, created in a confocal image, was later published, however a single method allowing the calculation of diffusion coefficients for arbitrary geometry does not exist. Our simulation approach allows computation of diffusion coefficients regardless of bleaching geometry used in the FRAP experiment. It accepts a multiple-frame TIFF file, representing the experiment as input, and simulates the (pure) diffusion of the fluorescent probes (2D random walk) starting with the first post-bleach frame of the actual data. It then fits the simulated data to the real data and extracts the diffusion coefficient. We validate our approach using a well characterized diffusing molecule (DiIC18) against well-established analytical procedures. We show that the algorithm is able to calculate the absolute value of diffusion coefficients for arbitrary bleaching geometries, including exaggeratedly large ones. It is provided freely as an ImageJ plugin, and should facilitate quantitative FRAP measurements for users equipped with standard fluorescence microscopy setups.

Validation of Normalizations, Scaling, and Photofading Corrections for FRAP Data Analysis

Fluorescence Recovery After Photobleaching (FRAP) has been a versatile tool to study transport and reaction kinetics in live cells. Since the fluorescence data generated by fluorescence microscopy are in a relative scale, a wide variety of scalings and normalizations are used in quantitative FRAP analysis. Scaling and normalization are often required to account for inherent properties of diffusing biomolecules of interest or photochemical properties of the fluorescent tag such as mobile fraction or photofading during image acquisition. In some cases, scaling and normalization are also used for computational simplicity. However, to our best knowledge, the validity of those various forms of scaling and normalization has not been studied in a rigorous manner. In this study, we investigate the validity of various scalings and normalizations that have appeared in the literature to calculate mobile fractions and correct for photofading and assess their consistency with FRAP equations. As a test case, we consider linear or affine scaling of normal or anomalous diffusion FRAP equations in combination with scaling for immobile fractions. We also consider exponential scaling of either FRAP equations or FRAP data to correct for photofading. Using a combination of theoretical and experimental approaches, we show that compatible scaling schemes should be applied in the correct sequential order; otherwise, erroneous results may be obtained. We propose a hierarchical workflow to carry out FRAP data analysis and discuss the broader implications of our findings for FRAP data analysis using a variety of kinetic models.

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.

Subdiffusion in an external potential: Anomalous effects hiding behind normal behavior

We propose a model of subdiffusion in which an external force is acting on a particle at all times not only at the moment of jump. The implication of this assumption is the dependence of the random trapping time on the force with the dramatic change of particles behavior compared to the standard continuous time random walk model in the long time limit. Constant force leads to the transition from non-ergodic subdiffusion to ergodic diffusive behavior. However, we show this behavior remains anomalous in a sense that the diffusion coefficient depends on the external force and on the anomalous exponent. For quadratic potential we find that the system remains non-ergodic. The anomalous exponent in this case defines not only the speed of convergence but also the stationary distribution which is different from standard Boltzmann equilibrium.

Investigating the cellular distribution and interactions of HIV-1 nucleocapsid protein by quantitative fluorescence microscopy

The nucleocapsid protein (NCp7) of the Human immunodeficiency virus type 1 (HIV-1) is a small basic protein containing two zinc fingers. About 2000 NCp7 molecules coat the genomic RNA in the HIV-1 virion. After infection of a target cell, the viral core enters into the cytoplasm, where NCp7 chaperones the reverse transcription of the genomic RNA into the proviral DNA. As a consequence of their much lower affinity for double-stranded DNA as compared to single-stranded RNAs, NCp7 molecules are thought to be released in the cytoplasm and the nucleus of infected cells in the late steps of reverse transcription. Yet, little is known on the cellular distribution of the released NCp7 molecules and on their possible interactions with cell components. Hence, the aim of this study was to identify potential cellular partners of NCp7 and to monitor its intracellular distribution and dynamics by means of confocal fluorescence microscopy, fluorescence lifetime imaging microscopy, fluorescence recovery after photobleaching, fluorescence correlation and cross-correlation spectroscopy, and raster imaging correlation spectroscopy. HeLa cells transfected with eGFP-labeled NCp7 were used as a model system. We found that NCp7-eGFP localizes mainly in the cytoplasm and the nucleoli, where it binds to cellular RNAs, and notably to ribosomal RNAs which are the most abundant. The binding of NCp7 to ribosomes was further substantiated by the intracellular co-diffusion of NCp7 with the ribosomal protein 26, a component of the large ribosomal subunit. Finally, gradient centrifugation experiments demonstrate a direct association of NCp7 with purified 80S ribosomes. Thus, our data suggest that NCp7 molecules released in newly infected cells may primarily bind to ribosomes, where they may exert a new potential role in HIV-1 infection.

In complex fluids the Gaussian Diffusion Approximation is generally invalid

The Gaussian Diffusion Approximation proposes that the distribution of displacements Δx for a particle diffusing through a complex fluid has a Gaussian dependence on Δx. We demonstrate from experimental evidence and computer simulations that this approximation is generally incorrect in complex fluids. As a result, experimental studies that have used the Gaussian Diffusion Approximation to interpret their findings will need to be reconsidered.

Quantifying non-ergodic dynamics of force-free granular gases

We demonstrate how non-ergodicity arises in simple mechanistic systems such as force free, dissipative granular gases. This behaviour results from the strong non-stationarity of the process mirrored in the continuous decay of the gas temperature.

Interactions and diffusion in fine-stranded β-lactoglobulin gels determined via FRAP and binding

The effects of electrostatic interactions and obstruction by the microstructure on probe diffusion were determined in positively charged hydrogels. Probe diffusion in fine-stranded gels and solutions of β-lactoglobulin at pH 3.5 was determined using fluorescence recovery after photobleaching (FRAP) and binding, which is widely used in biophysics. The microstructures of the β-lactoglobulin gels were characterized using transmission electron microscopy. The effects of probe size and charge (negatively charged Na2-fluorescein (376Da) and weakly anionic 70kDa FITC-dextran), probe concentration (50 to 200 ppm), and β-lactoglobulin concentration (9% to 12% w/w) on the diffusion properties and the electrostatic interaction between the negatively charged probes and the positively charged gels or solutions were evaluated. The results show that the diffusion of negatively charged Na2-fluorescein is strongly influenced by electrostatic interactions in the positively charged β-lactoglobulin systems. A linear relationship between the pseudo-on binding rate constant and the β-lactoglobulin concentration for three different probe concentrations was found. This validates an important assumption of existing biophysical FRAP and binding models, namely that the pseudo-on binding rate constant equals the product of the molecular binding rate constant and the concentration of the free binding sites. Indicators were established to clarify whether FRAP data should be analyzed using a binding-diffusion model or an obstruction-diffusion model.

Simulation of diffusion in a crowded environment

We performed extensive and systematic simulation studies of two-dimensional fluid motion in a complex crowded environment. In contrast to other studies we focused on cooperative phenomena that occurred if the motion of particles takes place in a dense crowded system, which can be considered as a crude model of a cellular membrane. Our main goal was to answer the following question: how do the fluid molecules move in an environment with a complex structure, taking into account the fact that motions of fluid molecules are highly correlated. The dynamic lattice liquid (DLL) model, which can work at the highest fluid density, was employed. Within the frame of the DLL model we considered cooperative motion of fluid particles in an environment that contained static obstacles. The dynamic properties of the system as a function of the concentration of obstacles were studied. The subdiffusive motion of particles was found in the crowded system. The influence of hydrodynamics on the motion was investigated via analysis of the displacement in closed cooperative loops. The simulation and the analysis emphasize the influence of the movement correlation between moving particles and obstacles.