Anomalous Diffusion in Inverted Variable-Lengthscale Fluorescence Correlation Spectroscopy

Using fluorescence correlation spectroscopy (FCS) to distinguish between different types of diffusion processes is often a perilous undertaking because the analysis of the resulting autocorrelation data is model dependant. Two recently introduced strategies, however, can help move toward a model-independent interpretation of FCS experiments: 1) the obtention of correlation data at different length scales and 2) their inversion to retrieve the mean-squared displacement associated with the process under study. We use computer simulations to examine the signature of several biologically relevant diffusion processes (simple diffusion, continuous-time random walk, caged diffusion, obstructed diffusion, two-state diffusion, and diffusing diffusivity) in variable-length-scale FCS. We show that, when used in concert, length-scale variation and data inversion permit us to identify non-Gaussian processes and, regardless of Gaussianity, to retrieve their mean-squared displacement over several orders of magnitude in time. This makes unbiased discrimination between different classes of diffusion models possible.     

Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study

Anomalous subdiffusion is hindered diffusion in which the mean-square displacement of a diffusing particle is proportional to some power of time less than one. Anomalous subdiffusion has been observed for a variety of lipids and proteins in the plasma membranes of a variety of cells. Fluorescence photobleaching recovery experiments with anomalous subdiffusion are simulated to see how to analyze the data. It is useful to fit the recovery curve with both the usual recovery equation and the anomalous one, and to judge the goodness of fit on log-log plots. The simulations show that the simplest approximate treatment of anomalous subdiffusion usually gives good results. Three models of anomalous subdiffusion are considered: obstruction, fractional Brownian motion, and the continuous-time random walk. The models differ significantly in their behavior at short times and in their noise level. For obstructed diffusion the approach to the percolation threshold is marked by a large increase in noise, a broadening of the distribution of diffusion coefficients and anomalous subdiffusion exponents, and the expected abrupt decrease in the mobile fraction. The extreme fluctuations in the recovery curves at and near the percolation threshold result from extreme fluctuations in the geometry of the percolation cluster.     

Autocorrelation function of finite-length data in fluorescence correlation spectroscopy

The experimental autocorrelation function of fluorescence correlation spectroscopy calculated from finite-length data is a biased estimator of the theoretical correlation function. This study presents a new theoretical framework that explicitly accounts for the data length to allow for unbiased analysis of experimental autocorrelation functions. To validate our theory, we applied it to experiments and simulations of diffusion and characterized the accuracy and precision of the resulting parameter estimates. Because measurements in living cells are often affected by instabilities of the fluorescence signal, autocorrelation functions are typically calculated on segmented data to improve their robustness. Our reformulated theory extends the range of usable segment times down to timescales approaching the diffusion time. This flexibility confers unique advantages for live-cell data that contain intensity variations and instabilities. We describe several applications of short segmentation to analyze data contaminated with unwanted fluctuations, drifts, or spikes in the intensity that are not suited for conventional fluorescence correlation analysis. These results demonstrate the potential of our theoretical framework to significantly expand the experimental systems accessible to fluorescence correlation spectroscopy.     

Measuring Mobility in Chromatin by Intensity-Sorted FCS

The architectural organization of chromatin can play an important role in genome regulation by affecting the mobility of molecules within its surroundings via binding interactions and molecular crowding. The diffusion of molecules at specific locations in the nucleus can be studied by fluorescence correlation spectroscopy (FCS), a well-established technique based on the analysis of fluorescence intensity fluctuations detected in a confocal observation volume. However, detecting subtle variations of mobility between different chromatin regions remains challenging with currently available FCS methods. Here, we introduce a method that samples multiple positions by slowly scanning the FCS observation volume across the nucleus. Analyzing the data in short time segments, we preserve the high temporal resolution of single-point FCS while probing different nuclear regions in the same cell. Using the intensity level of the probe (or a DNA marker) as a reference, we efficiently sort the FCS segments into different populations and obtain average correlation functions that are associated to different chromatin regions. This sorting and averaging strategy renders the method statistically robust while preserving the observation of intranuclear variations of mobility. Using this approach, we quantified diffusion of monomeric GFP in high versus low chromatin density regions. We found that GFP mobility was reduced in heterochromatin, especially within perinucleolar heterochromatin. Moreover, we found that modulation of chromatin compaction by ATP depletion, or treatment with solutions of different osmolarity, differentially affected the ratio of diffusion in both regions. Then, we used the approach to probe the mobility of estrogen receptor-α in the vicinity of an integrated multicopy prolactin gene array. Finally, we discussed the coupling of this method with stimulated emission depletion FCS for performing FCS at subdiffraction spatial scales.     

Separation of the rotational contribution in fluorescence correlation experiments

The theory of fluorescence correlation spectroscopy is reexamined with the aim of separating the contribution of rotational diffusion. Under constant excitation, fluorescence correlation experiments are characterized by three polarizations: one of the incident beam and two of the two photon detectors. A set of experiments of different polarizations is proposed for study. From the results of the experiments the isotropic factor of the fluorescence intensity correlation functions can be determined, which is independent of the rotational motion of the sample molecule. This function can be used to represent each fluorescence intensity correlation function as the product of the isotropic and the rotational factors. The theory is illustrated by an experiment in which rotational diffusion of porcine pancreatic lipase labeled with Texas Red was observed Texas Red is a label that allows precise fluorescence correlation experiments even in the nanosecond time range.     

Brownian dynamics simulations of fluorescence fluctuation spectroscopy

We have developed a program for the simulation of the fluorescence fluctuations as detected from highly diluted samples of (bio)molecules. The model is applied to translational diffusion and takes into account the hydrodynamic interactions. The solution concentration is kept constant by assuming periodic boundary conditions and spans here the range 0.5< C < 10 nM. We show that the fluorescence correlation functions can be accurately computed on systems of limited size (a few molecules per simulation box) by simulating for a total time approximately 100-300 times the diffusion relaxation time of the fluorescence autocorrelation function. The model is applied also to the simulation of the scanning fluorescence correlation spectroscopy (FCS) and of the photon counting histograms for the confocal collection configuration. Scanning FCS simulations of highly diluted samples (C approximately equals 0.5 nM) show anticorrelation effects in the autocorrelation functions of the fluorescence signal that are less evident for higher concentrations. We suggest here that this effect may be due to the non-uniform occupancy of the scanning area by the fluorophores.     

Peak intensity analysis as a method for estimation of fluorescent probe binding to artificial and natural nanoparticles: Tetramethylrhodamine uptake by isolated mitochondria

A modified version of fluorescence correlation spectroscopy (FCS) closely related to the photon counting histogram (PCH) method, which is used in the case of a mixture of molecules with similar diffusion coefficients, was applied here for analyzing the binding of the potential-sensitive dye tetramethylrhodamine ethyl ester, TMRE, to isolated mitochondria both in energized and deenergized states. Fluorescence time traces of suspensions of TMRE-doped mitochondria representing sequences of peaks of different intensity appeared to be similar to those of fluorescent beads and TMRE-doped latex particles. The experimental data were obtained under stirring conditions which increased the number of events by about three orders of magnitude thus substantially enhancing the resolution of the method. The statistics of the brightness of identical fluorescent particles reflecting their random walk through the confocal volume was described by a simple analytical equation which enabled us to perform the peak intensity analysis (PIA) of TMRE-doped mitochondria. The validity of PIA was tested with fluorescent beads of different sizes and TMRE-doped latex particles. Mitochondrial energization in the presence of TMRE led to the increase in the number and the intensity of the peaks in fluorescence time traces, the PIA of which allowed us to determine mitochondrial membrane potential and additionally a number of mitochondrial particles per ml of the suspension. The value of the membrane potential on a single mitochondrion was estimated to be about 180 mV in agreement with the data related to mitochondrial suspensions. Importantly, the PIA method required less than 1 microgram of mitochondrial protein per measurement.     

Fractional Poisson–Nernst–Planck Model for Ion Channels I: Basic Formulations and Algorithms

In this work, we propose a fractional Poisson–Nernst–Planck model to describe ion permeation in gated ion channels. Due to the intrinsic conformational changes, crowdedness in narrow channel pores, binding and trapping introduced by functioning units of channel proteins, ionic transport in the channel exhibits a power-law-like anomalous diffusion dynamics. We start from continuous-time random walk model for a single ion and use a long-tailed density distribution function for the particle jump waiting time, to derive the fractional Fokker–Planck equation. Then, it is generalized to the macroscopic fractional Poisson–Nernst–Planck model for ionic concentrations. Necessary computational algorithms are designed to implement numerical simulations for the proposed model, and the dynamics of gating current is investigated. Numerical simulations show that the fractional PNP model provides a more qualitatively reasonable match to the profile of gating currents from experimental observations. Meanwhile, the proposed model motivates new challenges in terms of mathematical modeling and computations.     

Temporal evolution of the electron diffusion coefficient in electrolyte-filled mesoporous nanocrystalline TiO2 films

Electron transport in electrolyte-filled mesoporous TiO₂-based solar cells is described quantitatively from the perspective of the continuous-time random walk model. An analytical expression is derived for the time-dependent diffusion coefficient of electrons, which transforms at a characteristic (Fermi) time from strongly time-dependent values (dispersive transport) at short times to relatively time-independent values (nondispersive transport) at long times. At short times, the diffusion coefficient displays a power-law behavior with time. The timescale for the diffusion coefficient to reach its steady-state value is substantially longer than the Fermi time. The Fermi time and the steepness of the distribution of waiting times associated with trap sites have a strong influence on both the steady-state diffusion coefficient of electrons and on the dispersiveness of electron transport. At short timescales, ionic drag, associated with the ambipolar effect, slows electron transport through the TiO₂ matrix, whereas at steady state, transport is trap limited. Decreasing the electron density lowers the steady-state limit of the diffusion coefficient and increases the timescale over which transport is dispersive.     

Probe diffusion in aqueous poly(vinyl alcohol) solutions studied by fluorescence correlation spectroscopy

We report fluorescence correlation spectroscopy measurements of the translational diffusion coefficient of various probe particles in dilute and semidilute aqueous poly(vinyl alcohol) solutions. The range of sizes of the particles (fluorescent molecules, proteins, and polymers) was chosen to explore various length scales of the polymer solutions as defined by the polymer-polymer correlation length. For particles larger than the correlation length, we find that the diffusion coefficient, D, decreases exponentially with the polymer concentration. This can be explained by an exponential increase in the solution viscosity, consistent with the Stokes-Einstein equation. For probes on the order of the correlation length, the decrease of the diffusion coefficient cannot be accounted for by the Stokes-Einstein equation, but can be fit by a stretched exponential, D approximately exp(-alphacn), where we find n = 0.73-0.84 and alpha is related to the probe size. These results are in accord with a diffusion model of Langevin and Rondelez (Polymer 1978, 19, 1875), where these values of n indicate a good solvent quality.     

Detection and correction of blinking bias in image correlation transport measurements of quantum dot tagged macromolecules

Semiconductor nanocrystals or quantum dots (QDs) are becoming widely used as fluorescent labels for biological applications. Here we demonstrate that fluorescence fluctuation analysis of their diffusional mobility using temporal image correlation spectroscopy is highly susceptible to systematic errors caused by fluorescence blinking of the nanoparticles. Temporal correlation analysis of fluorescence microscopy image time series of streptavidin-functionalized (CdSe)ZnS QDs freely diffusing in two dimensions shows that the correlation functions are fit well to a commonly used diffusion decay model, but the transport coefficients can have significant systematic errors in the measurements due to blinking. Image correlation measurements of the diffusing QD samples measured at different laser excitation powers and analysis of computer simulated image time series verified that the effect we observe is caused by fluorescence intermittency. We show that reciprocal space image correlation analysis can be used for mobility measurements in the presence of blinking emission because it separates the contributions of fluctuations due to photophysics from those due to transport. We also demonstrate application of the image correlation methods for measurement of the diffusion coefficient of glycosyl phosphatidylinositol-anchored proteins tagged with QDs as imaged on living fibroblasts.     

Easy Measurement of Diffusion Coefficients of EGFP-tagged Plasma Membrane Proteins Using k-Space Image Correlation Spectroscopy

Lateral diffusion and compartmentalization of plasma membrane proteins are tightly regulated in cells and thus, studying these processes will reveal new insights to plasma membrane protein function and regulation. Recently, k-Space Image Correlation Spectroscopy (kICS)¹ was developed to enable routine measurements of diffusion coefficients directly from images of fluorescently tagged plasma membrane proteins, that avoided systematic biases introduced by probe photophysics. Although the theoretical basis for the analysis is complex, the method can be implemented by nonexperts using a freely available code to measure diffusion coefficients of proteins. kICS calculates a time correlation function from a fluorescence microscopy image stack after Fourier transformation of each image to reciprocal (k-) space. Subsequently, circular averaging, natural logarithm transform and linear fits to the correlation function yields the diffusion coefficient. This paper provides a step-by-step guide to the image analysis and measurement of diffusion coefficients via kICS.First, a high frame rate image sequence of a fluorescently labeled plasma membrane protein is acquired using a fluorescence microscope. Then, a region of interest (ROI) avoiding intracellular organelles, moving vesicles or protruding membrane regions is selected. The ROI stack is imported into a freely available code and several defined parameters (see Method section) are set for kICS analysis. The program then generates a "slope of slopes" plot from the k-space time correlation functions, and the diffusion coefficient is calculated from the slope of the plot. Below is a step-by-step kICS procedure to measure the diffusion coefficient of a membrane protein using the renal water channel aquaporin-3 tagged with EGFP as a canonical example.     

Photon and fluorescence correlation spectroscopy and light scattering of eye-lens proteins at moderate concentrations

The bovine eye-lens protein, alpha L-crystallin, has been studied with photon correlation spectroscopy to obtain the mutual diffusion coefficient, Dm, with fluorescence correlation spectroscopy to determine the tracer diffusion coefficient, DT, and with light scattering to get the isothermal osmotic compressibility (delta pi/delta c) P,T. The concentration dependence of Dm, DT, and (delta pi/delta c) P,T up to a volume fraction phi of the protein of 2.5 x 10(-2) has been interpreted on the basis of four different interaction potentials: (a) an extended hard-sphere potential; (b) a shielded Coulomb potential; (c) a shielded Coulomb interaction where the effect of counterions is included; (d) a simple mixed potential. The three parameters Dm, DT, and (delta pi/delta c) P,T have also been combined in the generalized Stokes-Einstein equation, Dm = [(delta pi/delta c)P,T . (1--phi) . (DT)]/(kappa B . T). Our results indicate that, in the case that photon correlation spectroscopy gives the mutual diffusion coefficient Dm, the applicability of the Stokes-Einstein equation can be questioned; or that, when one assumes the Stokes-Einstein equation to be valid, there is significant discrepancy between the result of photon correlation spectroscopy and Dm.     

Propagators and time-dependent diffusion coefficients for anomalous diffusion

Complex diffusive dynamics are often observed when one is investigating the mobility of macromolecules in living cells and other complex environments, yet the underlying physical or chemical causes of anomalous diffusion are often not fully understood and are thus a topic of ongoing research interest. Theoretical models capturing anomalous dynamics are widely used to analyze mobility data from fluorescence correlation spectroscopy and other experimental measurements, yet there is significant confusion regarding these models because published versions are not entirely consistent and in some cases do not appear to satisfy the diffusion equation. Further confusion is introduced through variations in how fitting parameters are reported. A clear definition of fitting parameters and their physical significance is essential for accurate interpretation of experimental data and comparison of results from different studies acquired under varied experimental conditions. This article aims to clarify the physical meaning of the time-dependent diffusion coefficients associated with commonly used fitting models to facilitate their use for investigating the underlying causes of anomalous diffusion. We discuss a propagator for anomalous diffusion that captures the power law dependence of the mean-square displacement and can be shown to rigorously satisfy the extended diffusion equation provided one correctly defines the time-dependent diffusion coefficient. We also clarify explicitly the relation between the time-dependent diffusion coefficient and fitting parameters in fluorescence correlation spectroscopy.     

Lateral diffusion and aggregation. A Monte Carlo study.

Aggregation in a lipid bilayer is modeled as cluster-cluster aggregation on a square lattice. In the model, clusters carry out a random walk on the lattice, with a diffusion coefficient inversely proportional to mass. On contact, they adhere with a prescribed probability, rigidly and irreversibly. Monte Carlo calculations show that, as expected, rotational diffusion of the aggregating species is highly sensitive to the initial stages of aggregation. Lateral diffusion of an inert tracer obstructed by the aggregate is a sensitive probe of the later stages of aggregation. Cluster-cluster aggregates are much more effective barriers to lateral diffusion of an inert tracer than the same area fraction of random point obstacles is, but random point obstacles are more effective barriers than the same area fraction of compact obstacles. The effectiveness of aggregates as obstacles is discussed in terms of particle-particle correlation functions and fractal dimensions. Results are applicable to aggregation of membrane proteins, and at least qualitatively to aggregation of gel-phase lipid during lateral phase separation.     

荧光相关谱测量技术研究

荧光相关谱(fluorescence correlation spectroscopy,FCS)是对处于热平衡态条件下的荧光分子发出的荧光强度涨落进行时间相关处理的一种单分子检测方法,能够直接测量分子在溶液里的扩散系数和浓度.影响FCS测量扩散系数准确性的因素有分子量子效率,测量时间,样本折射率和温度偏差等.用FCS分别测量溶有荧光染料罗丹明6G(rhodamine 6G,Rh.6G)和青色素Cy5甘油水溶液的粘滞系数,实验结果表明:荧光分子的量子效率是影响测量准确性的重要因素,要求其每秒发射的光子数目(photon counts per second,cps)至少达到1 000(photons/s).     

Anomalous diffusion of fluorescent probes inside living cell nuclei investigated by spatially-resolved fluorescence correlation spectroscopy

We have investigated spatial variations of the diffusion behavior of the green fluorescent protein mutant EGFP (F64L/S65T) and of the EGFP-beta-galactosidase fusion protein in living cells with fluorescence correlation spectroscopy. Our fluorescence correlation spectroscopy device, in connection with a precision x-y translation stage, provides submicron spatial resolution and a detection volume smaller than a femtoliter. The fluorescence fluctuations in cell lines expressing EGFP are caused by molecular diffusion as well as a possible internal and a pH-dependent external protonation process of the EGFP chromophore. The latter processes result in two apparent nonfluorescent states that have to be taken into account when evaluating the fluorescence correlation spectroscopy data. The diffusional contribution deviates from ideal behavior and depends on the position in the cell. The fluorescence correlation spectroscopy data can either be evaluated as a two component model with one fraction of the molecules undergoing free Brownian motion with a diffusion coefficient approximately five times smaller than in aqueous solution, and another fraction diffusing one or two orders of magnitude slower. This latter component is especially noticeable in the nuclei. Alternatively, we can fit the data to an anomalous diffusion model where the time dependence of the diffusion serves as a measure for the degree of obstruction, which is large especially in nuclei. Possible mechanisms for this long tail behavior include corralling, immobile obstacles, and binding with a broad distribution of binding affinities. The results are consistent with recent numerical models of the chromosome territory structure in the cell nucleus.     

Accurate Determination of Membrane Dynamics with Line-Scan FCS

Here we present an efficient implementation of line-scan fluorescence correlation spectroscopy (i.e., one-dimensional spatio-temporal image correlation spectroscopy) using a commercial laser scanning microscope, which allows the accurate measurement of diffusion coefficients and concentrations in biological lipid membranes within seconds. Line-scan fluorescence correlation spectroscopy is a calibration-free technique. Therefore, it is insensitive to optical artifacts, saturation, or incorrect positioning of the laser focus. In addition, it is virtually unaffected by photobleaching. Correction schemes for residual inhomogeneities and depletion of fluorophores due to photobleaching extend the applicability of line-scan fluorescence correlation spectroscopy to more demanding systems. This technique enabled us to measure accurate diffusion coefficients and partition coefficients of fluorescent lipids in phase-separating supported bilayers of three commonly used raft-mimicking compositions. Furthermore, we probed the temperature dependence of the diffusion coefficient in several model membranes, and in human embryonic kidney cell membranes not affected by temperature-induced optical aberrations.     

Fluorescence correlation spectroscopy and nonideal solutions

The information that may be obtained from a fluorescence correlation spectroscopic study of a nonideal solution is considered. If all of the macromolecules in a two-component solution are fluorescently labeled, the mutual diffusion coefficient will be measured. If only a few of the macromolecules in a solution are fluorescently labeled, the tracer diffusion coefficient will be obtained. Two nonideal systems that probably may usefully be studied with fluorescence correlation spectroscopy are proposed. The application of fluorescence correlation spectroscopy to studies of lateral diffusion in biological membranes is discussed; the form of the contribution to the fluorescence correlation spectrum of bulk motion within a membrane is noted.