Multiple diffusion mechanisms due to nanostructuring in crowded environments

One of the key questions regarding intracellular diffusion is how the environment affects molecular mobility. Mostly, intracellular diffusion has been described as hindered, and the physical reasons for this behavior are: immobile barriers, molecular crowding, and binding interactions with immobile or mobile molecules. Using results from multi-photon fluorescence correlation spectroscopy, we describe how immobile barriers and crowding agents affect translational mobility. To study the hindrance produced by immobile barriers, we used sol-gels (silica nanostructures) that consist of a continuous solid phase and aqueous phase in which fluorescently tagged molecules diffuse. In the case of molecular crowding, translational mobility was assessed in increasing concentrations of 500 kDa dextran solutions. Diffusion of fluorescent tracers in both sol-gels and dextran solutions shows clear evidence of anomalous subdiffusion. In addition, data from the autocorrelation function were analyzed using the maximum entropy method as adapted to fluorescence correlation spectroscopy data and compared with the standard model that incorporates anomalous diffusion. The maximum entropy method revealed evidence of different diffusion mechanisms that had not been revealed using the anomalous diffusion model. These mechanisms likely correspond to nanostructuring in crowded environments and to the relative dimensions of the crowding agent with respect to the tracer molecule. Analysis with the maximum entropy method also revealed information about the degree of heterogeneity in the environment as reported by the behavior of diffusive molecules.     

Biased random walk models for chemotaxis and related diffusion approximations

Summary Stochastic models of biased random walk are discussed, which describe the behavior of chemosensitive cells like bacteria or leukocytes in the gradient of a chemotactic factor. In particular the turning frequency and turn angle distributions are derived from certain biological hypotheses on the background of related experimental observations. Under suitable assumptions it is shown that solutions of the underlying differential-integral equation approximately satisfy the well-known Patlak-Keller-Segel diffusion equation, whose coefficients can be expressed in terms of the microscopic parameters. By an appropriate energy functional a precise error estimation of the diffusion approximation is given within the framework of singular perturbation theory.     

Fluorescence correlation spectroscopy diffusion laws to probe the submicron cell membrane organization

To probe the complexity of the cell membrane organization and dynamics, it is important to obtain simple physical observables from experiments on live cells. Here we show that fluorescence correlation spectroscopy (FCS) measurements at different spatial scales enable distinguishing between different submicron confinement models. By plotting the diffusion time versus the transverse area of the confocal volume, we introduce the so-called FCS diffusion law, which is the key concept throughout this article. First, we report experimental FCS diffusion laws for two membrane constituents, which are respectively a putative raft marker and a cytoskeleton-hindered transmembrane protein. We find that these two constituents exhibit very distinct behaviors. To understand these results, we propose different models, which account for the diffusion of molecules either in a membrane comprising isolated microdomains or in a meshwork. By simulating FCS experiments for these two types of organization, we obtain FCS diffusion laws in agreement with our experimental observations. We also demonstrate that simple observables derived from these FCS diffusion laws are strongly related to confinement parameters such as the partition of molecules in microdomains and the average confinement time of molecules in a microdomain or a single mesh of a meshwork.     

Quantitative Microscopy: Protein Dynamics and Membrane Organisation

The mobility of membrane proteins is a critical determinant of their interaction capabilities and protein functions. The heterogeneity of cell membranes imparts different types of motion onto proteins; immobility, random Brownian motion, anomalous sub-diffusion, 'hop' or confined diffusion, or directed flow. Quantifying the motion of proteins therefore enables insights into the lateral organisation of cell membranes, particularly membrane microdomains with high viscosity such as lipid rafts. In this review, we examine the hypotheses and findings of three main techniques for analysing protein dynamics: fluorescence recovery after photobleaching, single particle tracking and fluorescence correlation spectroscopy. These techniques, and the physical models employed in data analysis, have become increasingly sophisticated and provide unprecedented details of the biophysical properties of protein dynamics and membrane domains in cell membranes. Yet despite these advances, there remain significant unknowns in the relationships between cholesterol-dependent lipid microdomains, protein-protein interactions, and the effect of the underlying cytoskeleton. New multi-dimensional microscopy approaches may afford greater temporal and spatial resolution resulting in more accurate quantification of protein and membrane dynamics in live cells.     

Size effects on diffusion processes within agarose gels

To investigate diffusion processes in agarose gel, nanoparticles with sizes in the range between 1 and 140 nm have been tested by means of fluorescence correlation spectroscopy. Understanding the diffusion properties in agarose gels is interesting, because such gels are good models for microbial biofilms and cells cytoplasm. The fluorescence correlation spectroscopy technique is very useful for such investigations due to its high sensitivity and selectivity, its excellent spatial resolution compared to the pore size of the gel, and its ability to probe a wide range of sizes of diffusing nanoparticles. The largest hydrodynamic radius (R(c)) of trapped particles that displayed local mobility was estimated to be 70 nm for a 1.5% agarose gel. The results showed that diffusion of particles in agarose gel is anomalous, with a diverging fractal dimension of diffusion when the large particles become entrapped in the pores of the gel. The latter situation occurs when the reduced size (R(A)/R(c)) of the diffusing particle, A, is >0.4. Variations of the fractal exponent of diffusion (d(w)) with the reduced particle size were in agreement with three-dimensional Monte Carlo simulations in porous media. Nonetheless, a systematic offset of d(w) was observed in real systems and was attributed to weak nonelastic interactions between the diffusing particles and polymer fibers, which was not considered in the Monte Carlo simulations.     

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.     

Resolution of end-to-end diffusion coefficients and distance distributions of flexible molecules using fluorescent donor-acceptor and donor-quencher pairs.

We used time-dependent fluorescence energy transfer, time-dependent collisional quenching, and global analysis of the data resulting from these through-space and contact interactions to recover the end-to-end distance distributions and diffusion coefficients of flexible fluorescent molecules. The fluorescence decays of covalently linked tryptamine-acceptor and tryptamine-quencher pairs were measured by the frequency-domain method. These data were fit using numerical solutions of the differential equation, which predicts the time- and distance-dependent population of the excited state donors in the presence of energy transfer or collisional quenching, followed by transformation to the frequency domain for nonlinear least-squares comparison with the experimental data. We found that the energy transfer data for the donor-acceptor pair alone were adequate to recover the starting distribution and the end-to-end diffusion coefficient; however, the resolution is dramatically improved by the use of both the through-space and contact interactions.     

Membrane lateral diffusion and capture of CFTR within transient confinement zones

The cystic fibrosis transmembrane conductance regulator (CFTR) channel interacts with scaffolding and other proteins that are expected to restrict its lateral movement, yet previous studies have reported predominantly free diffusion. We examined the lateral mobility of CFTR channels on live baby hamster kidney cells using three complementary methods. Channels bearing an extracellular biotinylation target sequence were labeled with streptavidin conjugated with fluorescent dyes (Alexa Fluor 488 or 568) or quantum dots (qDot605). Fluorescence recovery after photobleaching and image correlation spectroscopy of the dye-labeled channels revealed a significant immobile population ( approximately 50%), which was confirmed by direct single particle tracking (SPT) of qDot605-labeled CFTR. Adding 10 histidine residues at the C-terminus of CFTR to mask the postsynaptic density 95, Discs large, ZO-1 (PDZ) binding motif abolished its association with EBP50/NHERF1, reduced the immobile fraction, and increased mobility. Other interactions that are not normally detected on this timescale became apparent when binding of PDZ domain proteins was disrupted. SPT revealed that CFTR(His-10) channels diffuse randomly, become immobilized for periods lasting up to 1 min, and in some instances are recaptured at the same location. The impact of transient confinement on the measured diffusion using the three fluorescence techniques were assessed using computer simulations of the biological experiments. Finally, the impact of endosomal CFTR on mobility measurements was assessed by fluorescence correlation spectroscopy. These results reveal unexpected features of CFTR dynamics which may influence its ion channel activity.     

Resolving the kinetics of lipid, protein and peptide diffusion in membranes

Recent developments in the understanding of molecular diffusion phenomena in membranes are reviewed. Both model bilayers and biological membranes are considered in respect of lateral diffusion, rotational diffusion and transverse diffusion (flip-flop). For model systems, particular attention is paid to recent data obtained using surface-specific techniques such as sum frequency generation vibrational spectroscopy on supported lipid bilayers, and fluorescence correlation spectroscopy on giant unilamellar vesicles, both of which have yielded new insights into the intrinsic rates of diffusion and the energetic barriers to processes such as lipid flip-flop. Advances in single-molecule and many-molecule fluorescence methodologies have enabled the observation of processes such as anomalous diffusion for some membrane species in biological membranes. These are discussed in terms of new models for the role of membrane interactions with the cytoskeleton, the effects of molecular crowding in membranes, and the formation of lipid rafts. The diffusion of peptides, proteins and lipids is considered, particularly in relation to the means by which antimicrobial peptide activity may be rationalized in terms of membrane poration and lipid flip-flop.     

Analysis of fluorophore diffusion by continuous distributions of diffusion coefficients: application to photobleaching measurements of multicomponent and anomalous diffusion.

Fluorescence recovery after photobleaching (FRAP) is widely used to measure fluorophore diffusion in artificial solutions and cellular compartments. Two new strategies to analyze FRAP data were investigated theoretically and applied to complex systems with anomalous diffusion or multiple diffusing species: 1) continuous distributions of diffusion coefficients, alpha(D), and 2) time-dependent diffusion coefficients, D(t). A regression procedure utilizing the maximum entropy method was developed to resolve alpha(D) from fluorescence recovery curves, F(t). The recovery of multi-component alpha(D) from simulated F(t) with random noise was demonstrated and limitations of the method were defined. Single narrow Gaussian alpha(D) were recovered for FRAP measurements of thin films of fluorescein and size-fractionated FITC-dextrans and Ficolls, and multi-component alpha(D) were recovered for defined fluorophore mixtures. Single Gaussian alpha(D) were also recovered for solute diffusion in viscous media containing high dextran concentrations. To identify anomalous diffusion from FRAP data, a theory was developed to compute F(t) and alpha(D) for anomalous diffusion models defined by arbitrary nonlinear mean-squared displacement <x2> versus time relations. Several characteristic alpha(D) profiles for anomalous diffusion were found, including broad alpha(D) for subdiffusion, and alpha(D) with negative amplitudes for superdiffusion. A method to deduce apparent D(t) from F(t) was also developed and shown to provide useful complementary information to alpha(D). alpha(D) and D(t) were determined from photobleaching measurements of systems with apparent anomalous subdiffusion (nonuniform solution layer) and superdiffusion (moving fluid layer). The results establish a practical strategy to characterize complex diffusive phenomena from photobleaching recovery measurements.     

A method to investigate the diffusion properties of nuclear calcium

Modeling biophysical processes in general requires knowledge about underlying biological parameters. The quality of simulation results is strongly influenced by the accuracy of these parameters, hence the identification of parameter values that the model includes is a major part of simulating biophysical processes. In many cases, secondary data can be gathered by experimental setups, which are exploitable by mathematical inverse modeling techniques. Here we describe a method for parameter identification of diffusion properties of calcium in the nuclei of rat hippocampal neurons. The method is based on a Gauss-Newton method for solving a least-squares minimization problem and was formulated in such a way that it is ideally implementable in the simulation platform uG. Making use of independently published space- and time-dependent calcium imaging data, generated from laser-assisted calcium uncaging experiments, here we could identify the diffusion properties of nuclear calcium and were able to validate a previously published model that describes nuclear calcium dynamics as a diffusion process.     

Simultaneous diffusion and brightness measurements and brightness profile visualization from single fluorescence fluctuation traces of GFP in living cells

Fluorescence correlation spectroscopy (FCS) and photon-counting histogram (PCH) analysis use the same experimental fluorescence intensity fluctuations, but each analytical method focuses on a different property of the signal. The time-dependent decay of the correlation of fluorescence fluctuations is measured in FCS yielding, for instance, molecular diffusion coefficients. The amplitude distribution of these fluctuations is calculated by PCH analysis yielding information about the molecular brightness of fluorescent species. Analysis of both FCS and PCH results in the molecular concentration of the sample. Using a previously described global analysis procedure we report here precise, simultaneous measurements of diffusion constants and brightness values from single fluorescence fluctuation traces of green-fluorescent protein (GFP, S65T) in the cytoplasm of Dictyostelium cells. The use of a polynomial profile in PCH analysis, describing the detected three-dimensional shape of the confocal volume, enabled us to obtain well fitting results for GFP in cells. We could visualize the polynomial profile and show its deviation from a Gaussian profile.     

Out of the Randomness: Correlating Noise in Biological Systems

The study of the dynamics of biological systems requires one to follow relaxation processes in time with micron-size spatial resolution. This need has led to the development of different fluorescence correlation techniques with high spatial resolution and a tremendous (from nanoseconds to seconds) temporal dynamic range. Spatiotemporal information can be obtained even on complex dynamic processes whose time evolution is not forecast by simple Brownian diffusion. Our discussion of the most recent applications of image correlation spectroscopy to the study of anomalous sub- or superdiffusion suggests that this field still requires the development of multidimensional image analyses based on analytical models or numerical simulations. We focus in particular on the framework of spatiotemporal image correlation spectroscopy and examine the critical steps in getting information on anomalous diffusive processes from the correlation maps. We point out how a dual space-time correlative analysis, in both the direct and the Fourier space, can provide quantitative information on superdiffusional processes when these are analyzed through an empirical model based on intermittent active dynamics. We believe that this dual space-time analysis, potentially amenable to mathematical treatment and to the exact fit of experimental data, could be extended to include the rich phenomenology of subdiffusive processes, thereby quantifying relevant parameters for the various motivating biological problems of interest.     

Resolving the kinetics of lipid,protein and peptide diffusion in membranes

Abstract

Recent developments in the understanding of molecular diffusion phenomena in membranes are reviewed. Both model bilayers and biological membranes are considered in respect of lateral diffusion, rotational diffusion and transverse diffusion (flip-flop). For model systems, particular attention is paid to recent data obtained using surface-specific techniques such as sum frequency generation vibrational spectroscopy on supported lipid bilayers, and fluorescence correlation spectroscopy on giant unilamellar vesicles, both of which have yielded new insights into the intrinsic rates of diffusion and the energetic barriers to processes such as lipid flip-flop. Advances in single-molecule and many-molecule fluorescence methodologies have enabled the observation of processes such as anomalous diffusion for some membrane species in biological membranes. These are discussed in terms of new models for the role of membrane interactions with the cytoskeleton, the effects of molecular crowding in membranes, and the formation of lipid rafts. The diffusion of peptides, proteins and lipids is considered, particularly in relation to the means by which antimicrobial peptide activity may be rationalized in terms of membrane poration and lipid flip-flop.     

Modeling of slow glutamate diffusion and AMPA receptor activation in the cerebellar glomerulus

Synaptic conductances are influenced markedly by the geometry of the space surrounding the synapse since the transient glutamate concentration in the synaptic cleft is determined by this geometry. Our paper is an attempt to understand the reasons for slow glutamate diffusion in the cerebellar glomerulus, a structure situated around the enlarged mossy fiber terminal in the cerebellum and surrounded by a glial sheath. For this purpose, analytical expressions for glutamate diffusion in the glomerulus were considered in models with two-, three-, and fractional two-three-dimensional (2D-3D) geometry with an absorbing boundary. The time course of average glutamate concentration in the synaptic cleft of the mossy fiber-granule cell connection was calculated for both direct release of glutamate from the same synaptic unit, and for cumulative spillover of glutamate from neighboring release sites. Several kinetic schemes were examined, and the parameters of the diffusion models were estimated by identifying theoretical activation of AMPA receptors with direct release and spillover components of published experimental AMPA receptor-mediated EPSCs. For model selection, the correspondence of simulated paired-pulse ratio and EPSC increase after prevention of desensitization to experimental values were also taken into consideration. Our results suggest at least a 7- to 10-fold lower apparent diffusion coefficient of glutamate in the porous medium of the glomerulus than in water. The modeling of glutamate diffusion in the 2D-3D geometry gives the best fit of experimental EPSCs. We show that it could be only partly explained by normal diffusion of glutamate in the complex geometry of the glomerulus. We assume that anomalous diffusion of glutamate occurs in the glomerulus. A good match of experimental estimations and theoretical parameters, obtained in the simulations that use an approximation of anomalous diffusion by a solution for fractional Brownian motion, confirms our assumption.     

Fluorescence pattern photobleaching recovery for samples with multi-component diffusion

The translational mobility of proteins and lipids in phospholipid bilayers is often not well described as ideal self diffusion. One of the best methods for characterizing such non-ideal diffusion is to use fluorescence pattern photobleaching recovery. In this method, the spatial gradient of the monitoring and bleaching intensity is created by using epi-fluorescence and an expanded Gaussian-shaped laser beam which passes though a Ronchi ruling placed at the back image plane of a microscope. A difficulty arises when the fluorescence recovery from the exchange of slowly diffusing molecules between illuminated and non-illuminated stripes temporally overlaps with the recovery from the exchange of more rapidly diffusing molecules through the gradient produced by the broad Gaussian shape of the illumination. In the work presented here, a general theory is developed that describes the shape of the resulting fluorescence recovery curve for these typical experimental conditions. Approximate expressions amenable to non-linear curve fitting are also given. The new theoretical formalism has been demonstrated on data for the translational mobility of a fluorescent lipid probe in phospholipid bilayers deposited on planar-fused silica substrates.     

Effects of lateral diffusion on the fluorescence anisotropy in hexagonal lipid phases. I. Theory.

It is shown that fluorescence anisotropy from lipidlike probes in the hexagonal HII phase gives information of (a) orientational order parameters, (b) the wobbling diffusion constant, and (c) the hopping diffusion constant of the probe, DH, equals DL/R2, the lateral diffusion constant over the square of the radius of the hexagonal tubes. Here we consider only lipidlike probes having the absorption transition movement and/or the emission transition moment along the long axis of the molecule. Three models are introduced for analysis of time-resolved data: the "WOBHOP," the "reduced WOBHOP," and the "P2P4HOP" model. The fluorescence anisotropy in response to a very short excitation pulse in each of the three models is a constant plus a number of exponentials. The WOBHOP and reduced WOBHOP models have 3 and 2 exponentials, respectively, and both contain four fitting parameters: r0 (the fundamental anisotropy), (P2) (the second rank orientational order parameter), DW (the wobbling diffusion constant), and DH (the hopping diffusion constant). The P2P4HOP model has eight exponentials and five fitting parameters: the four parameters listed above and (P4) (the fourth rank orientational order parameter). Analysis of fluorescence anisotropy data in the hexagonal HII phase using one of these models allows for obtaining the hopping diffusion constant, and, if the lateral diffusion constant is known, the radius of the hexagonal tubes. Substitution of DH = 0 in each of the three models yields an expression for the fluorescence anisotropy that is used in the literature for lamellar (L alpha or L beta) phases. The fluorescence anisotropy in coexisting L alpha/HII phases is discussed.     

Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery

Multiphoton fluorescence photobleaching recovery (MP-FPR) is a technique for measuring the three-dimensional (3D) mobility of fluorescent molecules with 3D spatial resolution of a few microns. A brief, intense flash of mode-locked laser light pulses excites fluorescent molecules via multiphoton excitation in an ellipsoidal focal volume and photobleaches a fraction. Because multiphoton excitation of fluorophores is intrinsically confined to the high-intensity focal volume of the illuminating beam, the bleached region is restricted to a known, three-dimensionally defined volume. Fluorescence in this focal volume is measured with multiphoton excitation, using the attenuated laser beam to measure fluorescence recovery as fresh unbleached dye diffuses in. The time course of the fluorescence recovery signal after photobleaching can be analyzed to determine the diffusion coefficient of the fluorescent species. The mathematical formulas used to fit MP-FPR recovery curves and the techniques needed to properly utilize them to acquire the diffusion coefficients of fluorescently labeled molecules within cells are presented here. MP-FPR is demonstrated on calcein in RBL-2H3 cells, using an anomalous subdiffusion model, as well as in aqueous solutions of wild-type green fluorescent protein, yielding a diffusion coefficient of 8.7 x 10(-7) cm(2)s(-1) in excellent agreement with the results of other techniques.     

Modeling genome evolution with a diffusion approximation of a birth-and-death process

MOTIVATION: In our previous studies, we developed discrete-space birth, death and innovation models (BDIMs) of genome evolution. These models explain the origin of the characteristic Pareto distribution of paralogous gene family sizes in genomes, and model parameters that provide for the evolution of these distributions within a realistic time frame have been identified. However, extracting the temporal dynamics of genome evolution from discrete-space BDIM was not technically feasible. We were interested in obtaining dynamic portraits of the genome evolution process by developing a diffusion approximation of BDIM. RESULTS: The diffusion version of BDIM belongs to a class of continuous-state models whose dynamics is described by the Fokker-Plank equation and the stationary solution could be any specified Pareto function. The diffusion models have time-dependent solutions of a special kind, namely, generalized self-similar solutions, which describe the transition from one stationary distribution of the system to another; this provides for the possibility of examining the temporal dynamics of genome evolution. Analysis of the generalized self-similar solutions of the diffusion BDIM reveals a biphasic curve of genome growth in which the initial, relatively short, self-accelerating phase is followed by a prolonged phase of slow deceleration. This evolutionary dynamics was observed both when genome growth started from zero and proceeded via innovation (a potential model of primordial evolution), and when evolution proceeded from one stationary state to another. In biological terms, this regime of evolution can be tentatively interpreted as a punctuated-equilibrium-like phenomenon whereby evolutionary transitions are accompanied by rapid gene amplification and innovation, followed by slow relaxation to a new stationary state.