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.     

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.     

Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions

We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.      

Local motion analysis reveals impact of the dynamic cytoskeleton on intracellular subdiffusion

Intracellular transport is a complex interplay of ballistic transport along filaments and of diffusive motion, reliably delivering material and allowing for cell differentiation, migration, and proliferation. The diffusive regime, including subdiffusive, Brownian, and superdiffusive motion, is of particular interest for inferring information about the dynamics of the cytoskeleton morphology during intracellular transport. The influence of dynamic cytoskeletal states on intracellular transport are investigated in Dictyostelium discoideum cells by single particle tracking of fluorescent nanoparticles, to relate quantitative motion parameters and intracellular processes before and after cytoskeletal disruption. A local mean-square displacement (MSD) analysis separates ballistic motion phases, which we exclude here, from diffusive nanoparticle motion. In this study, we focus on intracellular subdiffusion and elucidate lag-time dependence, with particular focus on the impact of cytoskeleton compartments like microtubules and actin filaments. This method proves useful for binary motion state distributions. Experimental results are compared to simulations of a data-driven Langevin model with finite velocity correlations that captures essential statistical features of the local MSD algorithm. Specifically, the values of the mean MSD exponent and effective diffusion coefficients can be traced back to negative correlations of the motion's increments. We clearly identify both microtubules and actin filaments as the cause for intracellular subdiffusion and show that actin-microtubule cross talk exerts viscosifying effects at timescales larger than 0.2 s. Our findings might give insights into material transport and information exchange in living cells, which might facilitate gaining control over cell functions.     

Role of mucus in mucosal protection through ethanol and pepsin damage models.

Gastrointestinal mucus is considered an important part of the mucosal defence mechanism against endogenous aggressors such as acid and pepsin. The mucus gel layer, adherent to the mucosal surface creates a diffusion barrier to luminal pepsin, thus protecting the underlying epithelium from the digestion by pepsin. The mucolytic pepsin will, however, digest the mucus at its luminal surface, but that lost is normally balanced by secretion of new mucus. This dynamic balance is disrupted when the mucus is exposed to excess pepsin, which causes focal haemorrhagic damage by progressively hydrolyzing the adherent mucus. The adherent mucus gel layer cannot contribute to the protection against exogen damaging agents such as ethanol and nonsteroidal anti-inflammatory drugs, as these compounds easily penetrate the mucus barrier causing, at high concentration, epithelial exfoliation. This study describes the basic properties and characteristics of gastric mucus and compares the pepsin-induced damage with the ethanol damage model.     

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.     

Transport Theory for HIV Diffusion through In Vivo Distributions of Topical Microbicide Gels

Topical microbicide products are being developed for the prevention of sexually transmitted infections. These include vaginally-applied gels that deliver anti-HIV molecules. Gels may also provide partial barriers that slow virion diffusion from semen to vulnerable epithelium, increasing the time during which anti-HIV molecules can act. To explore the barrier function of microbicide gels, we developed a deterministic mathematical model for HIV diffusion through realistic gel distributions. We applied the model to experimental data for in vivo coating distributions of two vaginal gels in women. Time required for a threshold number of virions to reach the tissue surface was used as a metric for comparing different scenarios. Results delineated how time to threshold increased with increasing gel layer thickness and with decreasing diffusion coefficient. We note that for gel layers with average thickness >∼100 μm, the fractional area coated, rather than the gel layer thickness, was the primary determinant of time to threshold. For gel layers <∼100 μm, time to threshold was brief, regardless of fractional area coated. Application of the model to vaginal coating data showed little difference in time to threshold between the two gels tested. However, the protocol after gel application (i.e., with or without simulated coitus) had a much more significant effect. This study suggests that gel distribution in layers of thickness >100 μm and fractional area coated >0.8 is critical in determining the ability of the gel to serve as a barrier to HIV diffusion.     

A Beneficiary Role for Neuraminidase in Influenza Virus Penetration through the Respiratory Mucus

Swine influenza virus (SIV) has a strong tropism for pig respiratory mucosa, which consists of a mucus layer, epithelium, basement membrane and lamina propria. Sialic acids present on the epithelial surface have long been considered to be determinants of influenza virus tropism. However, mucus which is also rich in sialic acids may serve as the first barrier of selection. It was investigated how influenza virus interacts with the mucus to infect epithelial cells. Two techniques were applied to track SIV H1N1 in porcine mucus. The microscopic diffusion of SIV particles in the mucus was analyzed by single particle tracking (SPT), and the macroscopic penetration of SIV through mucus was studied by a virus in-capsule-mucus penetration system, followed by visualizing the translocation of the virions with time by immunofluorescence staining. Furthermore, the effects of neuraminidase on SIV getting through or binding to the mucus were studied by using zanamivir, a neuraminidase inhibitor (NAI), and Arthrobacter ureafaciens neuraminidase. The distribution of the diffusion coefficient shows that 70% of SIV particles were entrapped, while the rest diffused freely in the mucus. Additionally, SIV penetrated the porcine mucus with time, reaching a depth of 65 µm at 30 min post virus addition, 2 fold of that at 2 min. Both the microscopic diffusion and macroscopic penetration were largely diminished by NAI, while were clearly increased by the effect of exogenous neuraminidase. Moreover, the exogenous neuraminidase sufficiently prevented the binding of SIV to mucus which was reversely enhanced by effect of NAI. These findings clearly show that the neuraminidase helps SIV move through the mucus, which is important for the virus to reach and infect epithelial cells and eventually become shed into the lumen of the respiratory tract.     

Infection-induced chromatin modifications facilitate translocation of herpes simplex virus capsids to the inner nuclear membrane

Herpes simplex virus capsids are assembled and packaged in the nucleus and move by diffusion through the nucleoplasm to the nuclear envelope for egress. Analyzing their motion provides conclusions not only on capsid transport but also on the properties of the nuclear environment during infection. We utilized live-cell imaging and single-particle tracking to characterize capsid motion relative to the host chromatin. The data indicate that as the chromatin was marginalized toward the nuclear envelope it presented a restrictive barrier to the capsids. However, later in infection this barrier became more permissive and the probability of capsids to enter the chromatin increased. Thus, although chromatin marginalization initially restricted capsid transport to the nuclear envelope, a structural reorganization of the chromatin counteracted that to promote capsid transport later. Analyses of capsid motion revealed that it was subdiffusive, and that the diffusion coefficients were lower in the chromatin than in regions lacking chromatin. In addition, the diffusion coefficient in both regions increased during infection. Throughout the infection, the capsids were never enriched at the nuclear envelope, which suggests that instead of nuclear export the transport through the chromatin is the rate-limiting step for the nuclear egress of capsids. This provides motivation for further studies by validating the importance of intranuclear transport to the life cycle of HSV-1.     

A Model for the Transient Subdiffusive Behavior of Particles in Mucus

In this study we have applied a model to explain the reported subdiffusion of particles in mucus, based on the measured mean squared displacements ($MSD$). The model considers Brownian diffusion of particles in a confined geometry, made from permeable membranes. The applied model predicts a normal diffusive behavior at very short and long time lags, as observed in several experiments. In between these timescales, we find that the “subdiffusive” regime is only a transient effect, $MSD\propto {\tau }^{\alpha },\alpha <1$. The only parameters in the model are the diffusion-coefficients at the limits of very short and long times, and the distance between the permeable membranes L. Our numerical results are in agreement with published experimental data for realistic assumptions of these parameters. Finally, we show that only particles with a diameter less than 40 nm are able to pass through a mucus layer by passive Brownian motion.     

Anomalous diffusion of proteins due to molecular crowding

We have studied the diffusion of tracer proteins in highly concentrated random-coil polymer and globular protein solutions imitating the crowded conditions encountered in cellular environments. Using fluorescence correlation spectroscopy, we measured the anomalous diffusion exponent alpha characterizing the dependence of the mean-square displacement of the tracer proteins on time, r(2)(t) approximately t(alpha). We observed that the diffusion of proteins in dextran solutions with concentrations up to 400 g/l is subdiffusive (alpha < 1) even at low obstacle concentration. The anomalous diffusion exponent alpha decreases continuously with increasing obstacle concentration and molecular weight, but does not depend on buffer ionic strength, and neither does it depend strongly on solution temperature. At very high random-coil polymer concentrations, alpha reaches a limit value of alpha(l) approximately 3/4, which we take to be the signature of a coupling between the motions of the tracer proteins and the segments of the dextran chains. A similar, although less pronounced, subdiffusive behavior is observed for the diffusion of streptavidin in concentrated globular protein solutions. These observations indicate that protein diffusion in the cell cytoplasm and nucleus should be anomalous as well, with consequences for measurements of solute diffusion coefficients in cells and for the modeling of cellular processes relying on diffusion.     

Subdiffusive-Brownian crossover in membrane proteins: a generalized Langevin equation-based approach

In this work, we propose a generalized Langevin equation-based model to describe the lateral diffusion of a protein in a lipid bilayer. The memory kernel is represented in terms of a viscous (instantaneous) and an elastic (noninstantaneous) component modeled through a Dirac δ function and a three-parameter Mittag-Leffler type function, respectively. By imposing a specific relationship between the parameters of the three-parameter Mittag-Leffler function, the different dynamical regimes—namely ballistic, subdiffusive, and Brownian, as well as the crossover from one regime to another—are retrieved. Within this approach, the transition time from the ballistic to the subdiffusive regime and the spectrum of relaxation times underlying the transition from the subdiffusive to the Brownian regime are given. The reliability of the model is tested by comparing the mean-square displacement derived in the framework of this model and the mean-square displacement of a protein diffusing in a membrane calculated through molecular dynamics simulations.     

Intracellular transport dynamics of endosomes containing DNA polyplexes along the microtubule network

We have explored the transport of DNA polyplexes enclosed in endosomes within the cellular environment by multiple particle tracking (MPT). The polyplex-loaded endosomes demonstrate enhanced diffusion at short timescales (t<7 s) with their mean-square displacement (MSD) Deltax(t)2 scaling as t1.25. For longer time intervals they exhibit subdiffusive transport and have an MSD scaling as t0.7. This crossover from an enhanced diffusion to a subdiffusive regime can be explained by considering the action of motor proteins that actively transport these endosomes along the cellular microtubule network and the thermal bending modes of the microtubule network itself.     

A glycosyl hydrolase activity of mammalian reovirus sigma1 protein can contribute to viral infection through a mucus layer

The mammalian reovirus sigma1 protein is responsible for viral attachment to host cells and hemagglutination properties of the virus. In the present study, sequence similarity between sigma1 and chicken-type lysozymes prompted us to investigate additional functions of the sigma1 protein. Expression in Pichia pastoris yeast cells showed that sigma1 can actually cleave lysozyme substrates, including complex sugars found in bacterial cell walls. Replacement by site-directed mutagenesis of acidic amino acid residues in sigma1 by their respective isosteric, uncharged, amino acid residues has allowed us to identify Glu36 and Asp54 as the catalytic pair involved in sigma1-mediated glycosidase activity. The enzyme appears inactive in virions but its activity is unmasked upon generation of infectious subviral particles (ISVPs) by partial proteolytic removal of the outer capsid proteins. Purified sigma1 protein and ISVPs can also hydrolyze mucins, heavily glycosylated glycoproteins that are a major component of the mucus layer overlaying the intestinal epithelium. Furthermore, reovirus infection of epithelial Madin Darby canine kidney cells was inhibited tenfold in cells expressing mucin at their apical surface, while this inhibition was overcome by ISVPs. Unmasking of sigma1 mucinolytic activity in the intestine, consecutive to proteolytic cleavage of virions to ISVPs, thus likely contributes to the known increase in infectivity of reovirus ISVPs compared to complete virions. This work presents the first evidence that some mammalian viruses have evolved mechanisms to facilitate their penetration through the protective barrier of the mucus layer in the intestinal tract.     

A Bond-Fluctuation Model of Translational Dynamics of Chain-like Particles through Mucosal Scaffolds

Mucus scaffolds represent one of the most common barriers in targeted drug delivery and can remarkably reduce the outcome of pharmacological therapies. An efficient transport of drug particles through a mucus barrier is a precondition for an efficient drug delivery. Understanding the transport mechanism is particularly important for treatment of disorders such as cystic fibrosis. These are characterized by an onset of high-density mucus scaffolds imposing an increased steric filtering. In this study, we employed the bond-fluctuation model to analyze the effect of steric interactions on slowing the translational dynamics of compound chain-like particles traversing through scaffolds of different configurations (regular isotropic and anisotropic versus irregular random). The model, which accounts for both the geometry-imposed steric interaction as well as the intrachain steric interaction between the chain subunits, yields a transient subdiffusive motional pattern persists between the short-time and long-time Gaussian diffusion limits. The motion is analyzed in terms of a mean-squared displacement, diffusion coefficient, and radius of gyration. With higher levels of restriction or larger particles, the subdiffusive motional regime persists longer. The study also demonstrates that an important feature of the motion is also geometry-induced chain accommodation. The presented model is generic and could also be applied to studying the translational dynamics of other particles with more complex architecture such as dendrites or chain-decorated nanoparticles.     

Molecular Dynamics Studies of Light Hydrocarbons Diffusion in Zeolites

Molecular dynamics simulation techniques have been used to study the diffusion of methane, ethane, propane and i-butane into the zeolite ZSM-5. From the trajectories, the mean-square displacements were obtained and the diffusion coefficients determined using Einstein's diffusion equation. The results, when compared to the available experimental data, indicate that the simulations can provide a realistic representation of the microscopic process of diffusion into the zeolite pores. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Mucin biopolymers as broad-spectrum antiviral agents

Mucus is a porous biopolymer matrix that coats all wet epithelia in the human body and serves as the first line of defense against many pathogenic bacteria and viruses. However, under certain conditions viruses are able to penetrate this infection barrier, which compromises the protective function of native mucus. Here, we find that isolated porcine gastric mucin polymers, key structural components of native mucus, can protect an underlying cell layer from infection by small viruses such as human papillomavirus (HPV), Merkel cell polyomavirus (MCV), or a strain of influenza A virus. Single particle analysis of virus mobility inside the mucin barrier reveals that this shielding effect is in part based on a retardation of virus diffusion inside the biopolymer matrix. Our findings suggest that purified mucins may be used as a broad-range antiviral supplement to personal hygiene products, baby formula or lubricants to support our immune system.     

Quantitative Analysis of Single Particle Trajectories: Mean Maximal Excursion Method

An increasing number of experimental studies employ single particle tracking to probe the physical environment in complex systems. We here propose and discuss what we believe are new methods to analyze the time series of the particle traces, in particular, for subdiffusion phenomena. We discuss the statistical properties of mean maximal excursions (MMEs), i.e., the maximal distance covered by a test particle up to time t. Compared to traditional methods focusing on the mean-squared displacement we show that the MME analysis performs better in the determination of the anomalous diffusion exponent. We also demonstrate that combination of regular moments with moments of the MME method provides additional criteria to determine the exact physical nature of the underlying stochastic subdiffusion processes. We put the methods to test using experimental data as well as simulated time series from different models for normal and anomalous dynamics such as diffusion on fractals, continuous time random walks, and fractional Brownian motion.