Fluorescence recovery after photobleaching: the case of anomalous diffusion

Concentration-dependent anomalous diffusion of crystal violet dye in agar gel: application of the continuous time random walk model

The transport of materials is of fundamental importance, with studies on diffusion being at the forefront. Diffusion in a simple matrix is typically considered Fickian. However, anomalous diffusion in various media is a dominant process. It is well documented that anomalous diffusion results from medium heterogeneity, extreme events, phase transitions, medium surface dynamics, and other factors. The present work demonstrated that the diffusion of a cationic dye (crystal violet, CV) in an agar gel medium was anomalous, as shown by spatial and time-series data of dye movement. We estimated the classical diffusion coefficients using the Einstein-Smoluchowski solution to Fick's law, but these did not yield consistent Gaussianity, stationarity, or non-seasonality. However, anomalous diffusion was confirmed by modelling the experimental data. The exponent values (α) ranged from 0.468 ± 0.027 to 0.883 ± 0.107, indicating anomalous sub-diffusion that depended on the concentration of the CV dye in the same medium. We then evaluated time- and ensemble-averaged mean squared displacement using the dye-spreading data via image processing. The discrepancy in the distribution functions over a long experimental period highlighted the non-ergodic nature of the stochastic process. The dimensionless ergodicity-breaking parameter was evaluated, confirming the non-ergodic nature of the process. We found that the continuous time random walk (CTRW) model was well-suited for describing the anomalous diffusion in this system. We attributed the sub-diffusion of the dye to violations of the assumptions of Brownian motion of the particles and the nature of the diffusing medium.

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.

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.

Standard and fractional Ornstein-Uhlenbeck process on a growing domain

We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing or contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers both the case of Brownian diffusion and the case of a subdiffusive continuous-time random walk (CTRW). We find a high sensitivity of the random walk properties to the details of the domain growth rate, which gives rise to a variety of regimes with extremely different behaviors. At the origin of this rich phenomenology is the fact that the walkers still move while they wait to jump, since they are dragged by the deterministic drift arising from the domain growth. Thus, the increasingly long waiting times associated with the aging of the subdiffusive CTRW imply that, in the time interval between two consecutive jumps, the walkers might travel over much longer distances than in the normal diffusive case. This gives rise to seemingly counterintuitive effects. For example, on a static domain, both Brownian diffusion and subdiffusive CTRWs yield a stationary particle distribution with finite width when a harmonic potential is at play, thus indicating a confinement of the diffusing particle. However, for a sufficiently fast growing or contracting domain, this qualitative behavior breaks down, and differences between the Brownian case and the subdiffusive case are found. In the case of Brownian particles, a sufficiently fast exponential domain growth is needed to break the confinement induced by the harmonic force; in contrast, for subdiffusive particles such a breakdown may already take place for a sufficiently fast power-law domain growth. Our analytic and numerical results for both types of diffusion are fully confirmed by random walk simulations.

Frapbot: An open-source application for FRAP data

We introduce Frapbot, a free-of-charge open source software web application written in R, which provides manual and automated analyses of fluorescence recovery after photobleaching (FRAP) datasets. For automated operation, starting from data tables containing columns of time-dependent intensity values for various regions of interests within the images, a pattern recognition algorithm recognizes the relevant columns and identifies the presence or absence of prebleach values and the time point of photobleaching. Raw data, residuals, normalization, and boxplots indicating the distribution of half times of recovery (t_1/2 ) of all uploaded files are visualized instantly in a batch-wise manner using a variety of user-definable fitting options. The fitted results are provided as .zip file, which contains .csv formatted output tables. Alternatively, the user can manually control any of the options described earlier. © 2017 International Society for Advancement of Cytometry.

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.

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.

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.

Dissecting chromatin interactions in living cells from protein mobility maps

The genome of eukaryotes is organized into a dynamic nucleoprotein complex referred to as chromatin, which can adopt different functional states. Both the DNA and the protein component of chromatin are subject to various post-translational modifications that define the cell's gene expression program. Their readout and establishment occurs in a spatio-temporally coordinated manner that is controlled by numerous chromatin-interacting proteins. Binding to chromatin in living cells can be measured by a spatially resolved analysis of protein mobility using fluorescence microscopy based approaches. Recent advancements in the acquisition of protein mobility data using fluorescence bleaching and correlation methods provide data sets on diffusion coefficients, binding kinetics, and cellular concentrations on different time and length scales. The combination of different techniques is needed to dissect the complex interplay of diffusive translocations, binding events, and mobility constraints of the chromatin environment. While bleaching techniques have their strength in the characterization of particles that are immobile on the second/minute time scale, a correlation analysis is advantageous to characterize transient binding events with millisecond residence time. The application and synergy effects of the different approaches to obtain protein mobility and interaction maps in the nucleus are illustrated for the analysis of heterochromatin protein 1.

Reaction-subdiffusion model of morphogen gradient formation

We study gradient formation of subdiffusive morphogens. The morphogens are produced at a source point at a constant rate. From there they move subdiffusively and are also subject to degradation at a rate that may depend on location and on time. Our analysis is based on a reaction-subdiffusion equation obtained from a continuous time random-walk model with a long-tailed waiting time distribution that also incorporates an evanescence process. Spatially uniform degradation at a constant rate leads to an exponentially decreasing stationary concentration profile hardly distinguishable from that obtained with normal diffusion. On the other hand, with location-dependent degradation we find a rich gamut of profiles, some qualitatively quite different from those occurring with normal diffusion. We conclude that long-time morphogen concentration profiles are very sensitive to the spatial dependence of the reactivity and may also serve as a sensitive measure of the occurrence of anomalous diffusion.

An improved strip FRAP method for estimating diffusion coefficients: correcting for the degree of photobleaching

Fluorescence recovery after photobleaching is a widely established method for the estimation of diffusion coefficients, strip bleaching with an associated recovery curve analysis being one of the simplest techniques. However, its implementation requires near 100% bleaching in the region of interest with negligible fluorescence loss outside, both constraints being hard to achieve concomitantly for fast diffusing molecules. We demonstrate that when these requirements are not met there is an error in the estimation of the diffusion coefficient D, either an under- or overestimation depending on which assumption is violated the most. We propose a simple modification to the recovery curve analysis incorporating the concept of the relative bleached mass m giving a revised recovery time parametrization tau=m(2)w(2)/4piD for a strip of width w. This modified model removes the requirement of 100% bleaching in the region of interest and allows for limited diffusion of the fluorophore during bleaching. We validate our method by estimating the (volume) diffusion coefficient of FITC-labelled IgG in 60% glycerol solution, D= 4.09 +/- 0.21 microm(2) s(-1), and the (surface) diffusion coefficient of a green-fluorescent protein-tagged class I MHC protein expressed at the surface of a human B cell line, D= 0.32 +/- 0.03 microm(2) s(-1) for a population of cells.

Determination of protein mobility in mitochondrial membranes of living cells

Molecular mobility in membranes of intracellular organelles is poorly understood, due to the lack of experimental tools applicable for a great diversity of shapes and sizes such organelles can acquire. Determinations of diffusion within the plasma membrane or cytosol are based mostly on the assumption of an infinite flat space, not valid for curved membranes of smaller organelles. Here we extend the application of FRAP to mitochondria of living cells by application of numerical analysis to data collected from a small region inside a single organelle. The spatiotemporal pattern of light pulses generated by the laser scanning microscope during the measurement is reconstructed in silico and consequently the values of diffusion parameters best suited to the particular organelle are found. The mobility of the outer membrane proteins hFis and Tom7, as well as oxidative phosphorylation complexes COX and F(1)F(0) ATPase located in the inner membrane is analyzed in detail. Several alternative models of diffusivity applied to these proteins provide insight into the mechanisms determining the rate of motion in each of the membranes. Tom7 and hFis move along the mitochondrial axis in the outer membrane with similar diffusion coefficients (D=0.7μm(2)/s and 0.6μm(2)/s respectively) and equal immobile fraction (7%). The notably slower motion of the inner membrane proteins is best represented by a dual-component model with approximately equal partitioning of the fractions (F(1)F(0) ATPase: 0.4μm(2)/s and 0.0005μm(2)/s; COX: 0.3μm(2)/s and 0.007μm(2)/s). The mobility patterns specific for the membranes of this organelle are unambiguously distinguishable from those of the plasma membrane or artificial lipid environments: The parameters of mitochondrial proteins indicate a distinct set of factors responsible for their diffusion characteristics.

Fractional chemotaxis diffusion equations

We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macromolecular crowding or other obstacles.

FRAP and kinetic modeling in the analysis of nuclear protein dynamics: what do we really know?

The binding of nuclear proteins to chromatin in live cells has been analyzed by kinetic modeling procedures applied to experimental data from fluorescence recovery after photobleaching (FRAP). The kinetic models have yielded a number of important biological predictions about transcription, but concerns have arisen about the accuracy of these predictions. First, different studies using different kinetic models have arrived at very different predictions for the same or similar proteins. Second, some of these divergent predictions have been shown to arise from technical issues rather than biological differences. For confidence and accuracy, gold standards for the measurement of in vivo binding must be established by extensive cross validation using both different experimental methods and different kinetic modeling procedures.

Analysis of reaction-diffusion systems with anomalous subdiffusion

Reaction-diffusion equations are the cornerstone of modeling biochemical systems with spatial gradients, which are relevant to biological processes such as signal transduction. Implicit in the formulation of these equations is the assumption of Fick's law, which states that the local diffusive flux of species i is proportional to its concentration gradient; however, in the context of complex fluids such as cytoplasm and cell membranes, the use of Fick's law is based on empiricism, whereas evidence has been mounting that such media foster anomalous subdiffusion (with mean-squared displacement increasing less than linearly with time) over certain length scales. Particularly when modeling diffusion-controlled reactions and other systems where the spatial domain is considered semi-infinite, assuming Fickian diffusion might not be appropriate. In this article, two simple, conceptually extreme models of anomalous subdiffusion are used in the framework of Green's functions to demonstrate the solution of four reaction-diffusion problems that are well known in the biophysical context of signal transduction: fluorescence recovery after photobleaching, the Smolochowski limit for diffusion-controlled reactions in solution, the spatial range of a diffusing molecule with finite lifetime, and the collision coupling mechanism of diffusion-controlled reactions in two dimensions. In each case, there are only subtle differences between the two subdiffusion models, suggesting how measurements of mean-squared displacement versus time might generally inform models of reactive systems with partial diffusion control.

Fluorescence correlation spectroscopy: the case of subdiffusion

The theory of fluorescence correlation spectroscopy is revisited here for the case of subdiffusing molecules. Subdiffusion is assumed to stem from a continuous-time random walk process with a fat-tailed distribution of waiting times and can therefore be formulated in terms of a fractional diffusion equation (FDE). The FDE plays the central role in developing the fluorescence correlation spectroscopy expressions, analogous to the role played by the simple diffusion equation for regular systems. Due to the nonstationary nature of the continuous-time random walk/FDE, some interesting properties emerge that are amenable to experimental verification and may help in discriminating among subdiffusion mechanisms. In particular, the current approach predicts 1), a strong dependence of correlation functions on the initial time (aging); 2), sensitivity of correlation functions to the averaging procedure, ensemble versus time averaging (ergodicity breaking); and 3), that the basic mean-squared displacement observable depends on how the mean is taken.