Anomalous Subdiffusion Research Articles

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.

Plasma Membrane Topology and Membrane Models

Single particle tracks (SPTs) can be followed over the plasma membrane with high spatial and temporal resolution. The interpretation of SPTs, combined with Monte Carlo simulations, form the basis of complex membrane models. Most interpretations of SPTs rest on the convenient assumption that cell membranes are flat. However simulations of simple diffusion over surfaces that include pillars, when the locations are known in X and Y but movement is possible in Z, substantially reduce the apparent rate of diffusion and produce tracks that could be interpreted as transient anchorage and confinement (Figure, left).In addition SPTs usually follow a tag attached to the molecule of interest, not the molecule itself. When the molecule is confined to the plasma membrane, with the tag in the extracellular fluid and the surface is flat, they may be coincident. However gradients produce a variable offset between the molecule and its tag, misreporting the actual movement (Figure, right).Plasma membranes are not flat and projections or depressions produce the appearance of anomalous subdiffusion. We suggest that complex explanations, transient anchorage or barriers, should only be considered once more obvious ones have been disproved.View Large Image | View Hi-Res Image | Download PowerPoint Slide

Modeling Subdiffusion Using Reaction Diffusion Systems

In many types of media, particularly within living cells or within their membranes, diffusing species do not follow Fick's laws but instead show transient subdiffusive behavior. Formulating spatiotemporal models that take this behavior into account is a delicate matter, as one is faced with the choice of resorting either to fractional calculus or to microscopic descriptions. In this article, we provide an alternative designed to be easier to handle analytically and numerically than the existing approaches. Specifically, starting from the continuous time random walk model, we construct linear reaction diffusion systems that can be used as components within such a model and which capture the defining properties of subdiffusion. We show how to impose physically relevant parameters, and prove stability and mass conservation. While applications to cellular biology are our main motivation, our approach is abstract and should thus be applicable to any situation where anomalous subdiffusion is observed.

Temporal Correlation Functions of Concentration Fluctuations: An Anomalous Case

We calculate, within the framework of the continuous time random walk (CTRW) model, multiparticle temporal correlation functions of concentration fluctuations (CCF) in systems that display anomalous subdiffusion. The subdiffusion stems from the nonstationary nature of the CTRW waiting times, which also lead to aging and ergodicity breaking. Due to aging, a system of diffusing particles tends to slow down as time progresses, and therefore, the temporal correlation functions strongly depend on the initial time of measurement. As a consequence, time averages of the CCF differ from ensemble averages, displaying therefore ergodicity breaking. We provide a simple example that demonstrates the difference between these two averages, a difference that might be amenable to experimental tests. We focus on the case of ensemble averaging and assume that the preparation time of the system coincides with the starting time of the measurement. Our analytical calculations are supported by computer simulations based on the CTRW model.

Size-selectivity and anomalous subdiffusion of nanoparticles through carbon nanofiber-basedmembranes

A simulation is presented here that serves the dual functions of generatinga nanoporous membrane replica and executing the Brownian motion ofnanoparticles through the virtual membrane. Specifically, the concentrationprofile of a dilute solution of fluorescent particles in a stochastic andSiO2-coated carbon nanofiber (oxCNF), nanoporous membrane was simulated. The quality ofthe simulated profile was determined by comparing the results with experimentalconcentration profiles. The experimental concentration profiles were collected adjacent tothe oxCNF membrane surface from time-lapse fluorescence microscopy images. Thesimulation proved ideal as an accurate predictor of particle diffusion—the simulatedconcentration profile merged with the experimental profiles at the inlet/exit surfaces of theoxCNF membrane. In particular, the oxCNF barrier was found to hinder the transport of50 and 100 nm particles and transmembrane trajectories were indicative of anomaloussubdiffusion; the diffusion coefficient was found to be a function of time and space.

Water in hydrated orthorhombic lysozyme crystal: Insight from atomistic simulations

Biologically important water in orthorhombic lysozyme crystal is investigated using atomistic simulations. A distinct hydration shell surrounding lysozyme molecules is found from the number distribution of water molecules. While the number of water molecules in the hydration shell increases, the percentage decreases as the hydration level rises. Adsorption of water in the lysozyme crystal shows type-IV behavior. At low hydration levels, water molecules primarily intercalate the minor pores and cavity in the crystal due to the strong affinity between protein and water. At high hydration levels, the major pores are filled with liquidlike water as capillary condensation occurs. A type-H4 hysteresis loop is observed in the adsorption and desorption isotherms. The locations of the water molecules identified from simulation match fairly well with the experimentally determined crystallographic hydration sites. As observed in experiment, water exhibits anomalous subdiffusion because of the geometric restrictions and interactions of protein. With increasing hydration level, this anomaly is reduced and the diffusion of water tends to progressively approach normal Brownian diffusion. The flexibility of protein framework slightly enhances water mobility, but this enhancement decreases with increasing hydration level.

Mechanistic insight into the biological nanopore in tetragonal lysozyme crystal

We characterize the major biological nanopore in tetragonal lysozyme crystal at the atomic resolution and investigate the mechanistic behavior of H 2O and Cl − ions in the pore. The pore is approximately cylindrical with a radius in the range of 0.6–0.7 nm. H 2O density along the pore axis is governed by the surface characteristics, while the electrostatic interactions with charged Lys and Asp residues largely determine the axial density of Cl −. In the vicinity of pore surface, Cl − and H 2O exhibit ordered structures, particularly, there are one and two peaks separately in the radial density profiles of Cl − and H 2O. Exchanges of H 2O and Cl − between inside and outside the pore are found, and the average residence times in the pore are 28.2 and 58.1 ps for H 2O and Cl −, respectively. Quasi-one-dimensional diffusion of H 2O and Cl − is observed preferentially along the pore axis, and the diffusivity is smaller than in bulk phase. In 0.1 nm thick layer in the pore, H 2O can stay continuously up to 15 ps. Due to the restrained motion near the pore surface, H 2O shows an anomalous subdiffusion and the anomaly reduces upon moving away from the surface.

Fluorescence Recovery after Photobleaching: The Case of Anomalous Diffusion

The method of FRAP (fluorescence recovery after photobleaching), which has been broadly used to measure lateral mobility of fluorescent-labeled molecules in cell membranes, is formulated here in terms of continuous time random walks (CTRWs), which offer both analytical expressions and a scheme for numerical simulations. We propose an approach based on the CTRW and the corresponding fractional diffusion equation (FDE) to analyze FRAP results in the presence of anomalous subdiffusion. The FDE generalizes the simple diffusive picture, which has been applied to FRAP when assuming regular diffusion, to account for subdiffusion. We use a subordination relationship between the solutions of the fractional and normal diffusion equations to fit FRAP recovery curves obtained from CTRW simulations, and compare the fits to the commonly used approach based on the simple diffusion equation with a time dependent diffusion coefficient (TDDC). The CTRW and TDDC describe two different dynamical schemes, and although the CTRW formalism appears to be more complicated, it provides a physical description that underlies anomalous lateral diffusion.

Fractional Cable Models for Spiny Neuronal Dendrites

Cable equations with fractional order temporal operators are introduced to model electrotonic properties of spiny neuronal dendrites. These equations are derived from Nernst-Planck equations with fractional order operators to model the anomalous subdiffusion that arises from trapping properties of dendritic spines. The fractional cable models predict that postsynaptic potentials propagating along dendrites with larger spine densities can arrive at the soma faster and be sustained at higher levels over longer times. Calibration and validation of the models should provide new insight into the functional implications of altered neuronal spine densities, a hallmark of normal aging and many neurodegenerative disorders.

Anomalous subdiffusion with multispecies linear reaction dynamics

We have introduced a set of coupled fractional reaction-diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics. The model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics. The effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms. One consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species. The general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail. We have carried out Monte Carlo simulations corresponding to these special cases and we find excellent agreement with theory.

Diffusion of Protein Receptors on a Cylindrical Dendritic Membrane with Partially Absorbing Traps

We present a model of protein receptor trafficking within the membrane of a cylindrical dendrite containing small protrusions called spines. Spines are the locus of most excitatory synapses in the central nervous system and act as localized traps for receptors diffusing within the dendritic membrane. We treat the transverse intersection of a spine and dendrite as a spatially extended, partially absorbing boundary and use singular perturbation theory to analyze the steady-state distribution of receptors. We compare the singular perturbation solutions with numerical solutions of the full model and with solutions of a reduced one-dimensional model and find good agreement between them all. We also derive a system of Fokker–Planck equations from our model and use it to exactly solve a mean first passage time (MFPT) problem for a single receptor traveling a fixed axial distance along the dendrite. This is then used to calculate an effective diffusion coefficient for receptors when spines are uniformly distributed along the length of the cable and to show how a nonuniform distribution of spines gives rise to anomalous subdiffusion.

New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation

A physical-mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order. In this paper, an anomalous subdiffusion equation (ASub-DE) is considered. A new implicit numerical method (INM) and two solution techniques for improving the order of convergence of the INM for solving the ASub-DE are proposed. The stability and convergence of the INM are investigated by the energy method. Some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and supporting theoretical results can also be applied to other fractional integro-differential equations and higher-dimensional problems.

Generalized Fick law for anomalous diffusion in the multidimensional comb model

The problem of multidimensional diffusion is considered within the framework of the comb model. It is shown that the diffusion current for the case of anomalous subdiffusion random walks is described by the generalized Fick law containing the diffusion tensor instead of the usual coefficient. The form of the diffusion tensor components is an unusual form of operator as fractional time derivatives. The orders of the fractional exponents are different for different directions.

An anomalous subdiffusion model for calcium spark in cardiac myocytes

The elementary events of excitation-contraction coupling in heart muscle are Ca2+ sparks, which arise from ryanodine receptors in the sarcoplasmic reticulum (SR). Here, an anomalous subdiffusion model is developed to explore Ca2+ spark formation in cardiac myocytes. Numerical simulations reproduce the brightness, the time course, and spatial size of a typical cardiac Ca2+ spark. It is suggested that the diffusion of Ca2+ spark in the cytoplasm may no longer obey Fickian second law, but the anomalous space subdiffusion. The physical reason is perhaps due to the effects of the electric field of the calcium ions and the viscoelasticity of the cytoplasm and its complex structures.

Anomalous diffusion of electromagnetic eddy currents in geological formations

Controlled‐source electromagnetic (EM) induction in some geological formations is shown here to be compactly described by an anomalous subdiffusion process. Such a process, which is not universal, is governed by a fractional diffusion equation or alternatively the convolutional form of Ohm's law. A subdiffusing eddy current vortex, or electromagnetic smoke ring, propagates in such a way that its position of median intensity overruns its position of peak intensity. This behavior is not allowed in classical diffusion but is a simple consequence of diffusion within a stationary fractal medium. A similar analysis has been applied to understand heavy‐tailed traveltime distributions that appear in certain hydrological time series. The tell‐tale signature of anomalous electromagnetic diffusion is a slope β of the magnetic zero‐crossing moveout curve that is constant with transmitter‐receiver (RX) offset and significantly different from unity. Neither lateral heterogeneity nor unixial anisotropy can generate such a constant‐slope moveout curve with an economy of model parameters. Controlled‐source EM data from two sites in Texas and one in New Mexico are used in this study to test the eddy current subdiffusion hypothesis.

A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model

Anomalous subdiffusion has been reported for two-dimensional diffusion in the plasma membrane and three-dimensional diffusion in the nucleus and cytoplasm. If a particle diffuses in a suitable infinite hierarchy of binding sites, diffusion is well known to be anomalous at all times. But if the hierarchy is finite, diffusion is anomalous at short times and normal at long times. For a prescribed set of binding sites, Monte Carlo calculations yield the anomalous diffusion exponent and the average time over which diffusion is anomalous. If even a single binding site is present, there is a very short, almost artifactual, period of anomalous subdiffusion, but a hierarchy of binding sites extends the anomalous regime considerably. As is well known, an essential requirement for anomalous subdiffusion due to binding is that the diffusing particle cannot be in thermal equilibrium with the binding sites; an equilibrated particle diffuses normally at all times. Anomalous subdiffusion due to barriers, however, still occurs at thermal equilibrium, and anomalous subdiffusion due to a combination of binding sites and barriers is reduced but not eliminated on equilibration. This physical model is translated directly into a plausible biological model testable by single-particle tracking.

Interleaflet Diffusion Coupling When Polymer Adsorbs onto One Sole Leaflet of a Supported Phospholipid Bilayer

ReceiVed October 9, 2006 ReVised Manuscript ReceiVed January 11, 2007 Introduction. Lipid mobility in fluid-phase lipid bilayers attracts much interest not just because this problem is fundamental but also because it is so closely related to biological processes such as membrane ion channel formation, ligandreceptor binding, and membrane adhesion and fusion.1-3 Here we consider lipid lateral diffusion in bilayers that are supported on a solid substrate. Literature on this general question is extensive, including various aspects such as free Brownian motion in singleand multicomponent bilayers,4,5 anomalous subdiffusion in heterogeneous bilayers,6 obstructed diffusion in phase-separated bilayers,7 and hop diffusion in compartmentalized cell membranes.8 However, the fact that any lipid bilayer is comprised of an inner leaflet and an outer leaflet raises the question of whether lipid moves synchronously in the two leafletssa problem of interleaflet coupling in phospholipid bilayers. Here we focus on discriminating between diffusion on the two sides of the bilayer. Previous studies of interleaflet coupling in phospholipid bilayers did not investigate the situation where the adsorbate was located onto one sole side. Important prior studies investigated lipid raft formation in the two leaflets due to temperature decrease or incorporated cholesterol molecules.9,10 Also, in the absence of adsorbate, lipid diffusion in the outer and inner leaflets of a planar-supported lipid bilayer was studied with the conclusion that strong coupling exists between diffusion in the inner and outer leaflets.11 We address the basic question of what happens when adsorbate is present. As adsorbate, we allow polymers to adsorb at low surface coverage to the outer leaflet, and using iodide quenching of diffusion in the outer leaflet, we discriminate between lipid diffusion in the outer and inner leaflets. Experimental Section. For study, phospholipid DLPC, 1,2dilauroyl-sn-glycero-3-phosphocholine, was purchased from Avanti Polar Lipids (Alabaster, AL), into which we doped at 10 ppm (0.001%) molar concentration DMPE, 1,2-dimyristoylsn-glycero-3-phosphoethanolamine, with polar head group labeled by rhodamine B. Supported bilayers of the mixture were formed on hydrophilic quartz using known protocols based on the fusion of small unilamellar vesicles.12 Fully quaternized poly(4-vinylpyridine), QPVP, was prepared by us from parent poly(vinylpyridine) (Polymer Source Inc., Quebec, Canada) by reaction with an excess of ethyl bromide.13 The surface coverage of QPVP on DLPC bilayers was quantified using Fourier transform infrared spectroscopy in the mode of attenuated total reflection (FTIR-ATR) using known methods.13 All measurements were made in PBS buffer (10 mM, pH 6.0). The lateral diffusion of dye-labeled lipids was measured by fluorescence correlation spectroscopy (FCS) in the two-photon excitation mode using a home-built apparatus.14 As illustrated in Figure 1, near-infrared femtosecond pulses from a Ti:sapphire laser (800 nm, 82 MHz, pulse width ∼100 fs) were directed into a water immersion objective lens (Zeiss Axiovert 135 TV, 63x, NA ) 1.2) via a dichroic mirror and focused onto the sample, giving an excitation spot whose diffraction-limited diameter was ∼0.35 μm, thus affording measurements that were spatially resolved depending on where the laser beam was focused. Fluorescence from the sample was collected by the same objective and split into two channels after passing through the dichroic mirror and the emission filter. Each channel was connected to a single photon counting module (Hamamatsu). The photon counting output was recorded by an integrated FCS data acquisition board (ISS, Champaign, IL) and was analyzed to give an autocorrelation function curve, G(τ), which one can fit to standard equations to give the translational diffusion coefficient (D).15,16 For the planar-supported phospholipid bilayer system studied here, it is appropriate to fit data of this kind to a 2-dimensional Gaussian model with two-photon excitation using the following equation17

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.

Molecular transport in a crowded volume created from vertically aligned carbon nanofibres:a fluorescence recovery after photobleaching study

Rapid and selective molecular exchange across a barrier is essential for emulating theproperties of biological membranes. Vertically-aligned carbon nanofibre (VACNF) forestshave shown great promise as membrane mimics, owing to their mechanical stability,their ease of integration with microfabrication technologies and the ability totailor their morphology and surface properties. However, quantifying transportthrough synthetic membranes having micro- and nanoscale features is challenging.Here, fluorescence recovery after photobleaching (FRAP) is coupled with finitedifference and Monte Carlo simulations to quantify diffusive transport in microfluidicstructures containing VACNF forests. Anomalous subdiffusion was observed forFITC (hydrodynamic radius of 0.54 nm) diffusion through both VACNFs andSiO2-coated VACNFS (oxVACNFs). Anomalous subdiffusion can be attributed tomultiple FITC–nanofibre interactions for the case of diffusion through theVACNF forest. Volume crowding was identified as the cause of anomaloussubdiffusion in the oxVACNF forest. In both cases the diffusion mode changes to atime-independent, Fickian mode of transport that can be defined by a crossover length(RCR). By identifying the space-and time-dependent transport characteristics of the VACNFforest, the dimensional features of membranes can be tailored to achieve predictablemolecular exchange.

The Monte Carlo and fractional kinetics approaches to the underground anomalous subdiffusion of contaminants

It is nowadays recognized that the experimental evidences of many transport phenomena must be interpreted in the framework of the so-called anomalous diffusion: this is the case e.g. of the spread of contaminant particles in porous media, whose mean squared displacement (MSD) has been experimentally reported to grow in time as t α (with 0 < α < 2). This is in deep contrast with the linear increase which characterizes the standard Fickian diffusive processes. Anomalous transport is currently tackled within the fractional diffusion equation (FDE) analytical model, which has blown new life into the concept of fractional derivatives, dormant since Leibniz’s first discovery, about 300 years ago. In this paper we focus on subdiffusion, i.e., the case α < 1, which provides a suitable explanation of the observed non-Fickian persistence of the contaminant particles near the source in various transport experiences in heterogeneous media. Unfortunately, the FDE approach requires several approximations to overcome its analytical complexities. To evaluate the relevance of these approximations, we propound the use of the Monte Carlo simulation as a suitable reference for the analytical FDE results. This comparison shows that the FDE results are less and less reliable as α becomes closer to 1. The approach herein proposed can be easily applied to various fields of science where anomalous diffusion phenomena may occur, from physics and chemistry to biology and finance.