Determining diffusion coefficients in inhomogeneous tissues using fluorescence recovery after photobleaching

Diffusion plays an important role in the transport of nutrients and signaling molecules in cartilaginous tissues. Diffusion coefficients can be measured by fluorescence recovery after photobleaching (FRAP). Available methods to analyze FRAP data, however, assume homogeneity in the environment of the bleached area and neglect geometrical restrictions to diffusion. Hence, diffusion coefficients in inhomogeneous materials, such as most biological tissues, cannot be assessed accurately. In this study, a new method for analyzing data from FRAP measurements has been developed, which is applicable to inhomogeneous tissues. It is based on a fitting procedure of the intensity recovery after photobleaching with a two-dimensional finite element analysis, which includes Fick's law for diffusion. The finite element analysis can account for distinctive diffusivity in predefined zones, which allows determining diffusion coefficients in inhomogeneous samples. The method is validated theoretically and experimentally in both homogeneous and inhomogeneous tissues and subsequently applied to the proliferation zone of the growth plate. Finally, the importance of accounting for inhomogeneities, for appropriate assessment of diffusivity in inhomogeneous tissues, is illustrated.     

Diffusional anisotropy in collagenous tissues: fluorescence imaging of continuous point photobleaching

Molecular transport in avascular collagenous tissues such as articular cartilage occurs primarily via diffusion. The presence of ordered structures in the extracellular matrix may influence the local transport of macromolecules, leading to anisotropic diffusion depending on the relative size of the molecule and that of extracellular matrix structures. Here we present what we believe is a novel photobleaching technique for measuring the anisotropic diffusivity of macromolecules in collagenous tissues. We hypothesized that macromolecular diffusion is anisotropic in collagenous tissues, depending on molecular size and the local organization of the collagen structure. A theoretical model and experimental protocol for fluorescence imaging of continuous point photobleaching was developed to measure diffusional anisotropy. Significant anisotropy was observed in highly ordered collagenous tissues such as ligament, with diffusivity ratios >2 along the fiber direction compared to the perpendicular direction. In less-ordered tissues such as articular cartilage, diffusional anisotropy was dependent on site in the tissue and size of the diffusing molecule. Anisotropic diffusion was also dependent on the size of the diffusing molecule, with greatest anisotropy observed for larger molecules. These findings suggest that diffusional transport of macromolecules is anisotropic in collagenous tissues, with higher rates of diffusion along primary orientation of collagen fibers.     

Effects of spatial variation in membrane diffusibility and solubility on the lateral transport of membrane components.

There exist many examples of membrane components (e.g. receptors) accumulating in special domains of cell membranes. We analyze how certain variations in lateral diffusibility and solubility of the membrane would increase the efficiency of transport to these regions. A theorem is derived to show that the mean-time-of capture, tc, for particles diffusing to a trap from an annular region surrounding it, is intermediate to the tc values that correspond to the minimum and maximum diffusion coefficients that obtain in this region. An analytical solution for tc as a function of the gradient of diffusivity surrounding a trap is derived for circular geometry. Since local diffusion coefficients can be increased dramatically by reducing the concentration of intra-membrane particles and/or allowing them to form aggregates, such mechanisms could greatly enhance the diffusion-limited transport of particular membrane components to a trap (e.g. coated pit). If the trap is surrounded by an annular region in which the probe particles' partition function is increased, say, by the local segregation of certain phospholipids, tc is shown to vary inversely with the logarithm of the relative partition function. We provide some conjectural examples to illustrate the magnitude of the effects which heterogeneities in diffusibility and solubility may have in biological membranes.     

A Biomechanical Triphasic Approach to the Transport of Nondilute Solutions in Articular Cartilage

Biomechanical models for biological tissues such as articular cartilage generally contain an ideal, dilute solution assumption. In this article, a biomechanical triphasic model of cartilage is described that includes nondilute treatment of concentrated solutions such as those applied in vitrification of biological tissues. The chemical potential equations of the triphasic model are modified and the transport equations are adjusted for the volume fraction and frictional coefficients of the solutes that are not negligible in such solutions. Four transport parameters, i.e., water permeability, solute permeability, diffusion coefficient of solute in solvent within the cartilage, and the cartilage stiffness modulus, are defined as four degrees of freedom for the model. Water and solute transport in cartilage were simulated using the model and predictions of average concentration increase and cartilage weight were fit to experimental data to obtain the values of the four transport parameters. As far as we know, this is the first study to formulate the solvent and solute transport equations of nondilute solutions in the cartilage matrix. It is shown that the values obtained for the transport parameters are within the ranges reported in the available literature, which confirms the proposed model approach.     

Parameter estimation techniques for transport equations with application to population dispersal and tissue bulk flow models

We developed techniques for estimating the coefficients, boundary data, and initial data associated with transport equations (or more generally, parabolic distributed models). Our estimation schemes are based on cubic spline approximations, for which convergence results are given. We discuss the performance of these techniques in two investigations of biological interest: (1) transport of labeled sucrose in brain tissue white matter, (2) insect dispersal that cannot be modeled by a random diffusion mechanism alone.     

What is slow axonal transport?

While the phenomenon of slow axonal transport is widely agreed upon, its underlying mechanism has been controversial for decades. There is now persuasive evidence that several different mechanisms could contribute to slow axonal transport. Yet proponents of different theories have been hesitant to explicitly integrate what were, at least initially, opposing models. We suggest that slow transport is a multivariate phenomenon that arises through mechanisms that minimally include: molecular motor-based transport of polymers and soluble proteins as multi-protein complexes; diffusion; and en bloc transport of the axonal framework by low velocity transport and towed growth (due to increases in body size). In addition to integrating previously described mechanisms of transport, we further suggest that only a subset of transport modes operate in a given neuron depending on the region, length, species, cell type, and developmental stage. We believe that this multivariate approach to slow axonal transport better explains its complex phenomenology: including its bi-directionality; the differing velocities of transport depending on cargo, as well differing velocities due to anatomy, cell type and developmental stage.     

Local macromolecule diffusion coefficients in structurally non-uniform bacterial biofilms using fluorescence recovery after photobleaching (FRAP)

Pure culture Pseudomonas putida biofilms were cultivated under controlled conditions to a desired overall biofilm thickness, then employed within classical half-cell diffusion chambers to estimate, from transient solute concentrations, the effective diffusion coefficient for several macromolecules of increasing molecular weight and molecular complexity. Results of traditional half-cell studies were found to be erroneous due to the existence of microscopic water channels or crevasses that perforate the polysaccharidic gel matrix of the biofilm, sometimes completely to the supporting substratum. Thus, half-cell devices measure a composite transfer coefficient that may overestimate the true, local flux of solutes in the biofilm polysaccharide gel matrix. An alternative analytical technique was refined to determine the local diffusion coefficients on a micro-scale to avoid the errors created by the biofilm architectural irregularities. This technique is based upon the Fluorescence Return After Photobleaching (FRAP), which allows image analysis observation of the transport of fluorescently labeled macromolecules as they migrate into a micro-scale photobleached zone. The technique can be computerized and allows one to map the local diffusion coefficients of various solute molecules at different horizontal planes and depths in a biofilm. These mappings also indirectly indicate the distribution of water channels in the biofilm, which was corroborated independently by direct microscopic observation of the settling of fluorescently-labeled latex spheres within the biofilm. Fluorescence return after photobleaching results indicate a significant reduction in the solute transport coefficients in biofilm polymer gel vs. the same value in water, with the reduction being dependent on solute molecule size and shape.     

Site-specific effects of compression on macromolecular diffusion in articular cartilage

Articular cartilage is the connective tissue that lines joints and provides a smooth surface for joint motion. Because cartilage is avascular, molecular transport occurs primarily via diffusion or convection, and cartilage matrix structure and composition may affect diffusive transport. Because of the inhomogeneous compressive properties of articular cartilage, we hypothesized that compression would decrease macromolecular diffusivity and increase diffusional anisotropy in a site-specific manner that depends on local tissue strain. We used two fluorescence photobleaching methods, scanning microphotolysis and fluorescence imaging of continuous point photobleaching, to measure diffusion coefficients and diffusional anisotropy of 70 kDa dextran in cartilage during compression, and measured local tissue strain using texture correlation. For every 10% increase in normal strain, the fractional change in diffusivity decreased by 0.16 in all zones, and diffusional anisotropy increased 1.1-fold in the surface zone and 1.04-fold in the middle zone, and did not change in the deep zone. These results indicate that inhomogeneity in matrix structure and composition may significantly affect local diffusive transport in cartilage, particularly in response to mechanical loading. Our findings suggest that high strains in the surface zone significantly decrease diffusivity and increase anisotropy, which may decrease transport between cartilage and synovial fluid during compression.     

Transport rates of proteins in porous materials with known microgeometry.

Many biological and biotechnological systems involve the diffusion of macromolecules through complicated macroporous (pore size on the order of 10-100 microns) environments. In this report, we present and evaluate an experimental system for measuring the rate of protein transport in an inert, macroporous membrane. For this particular membrane system, the microgeometry was characterized in terms of distribution of pore size, position, and orientation. Although the rate of protein desorption was much less than expected based on continuum diffusion models, we demonstrate that the measured transport rates are consistent with diffusion of protein in a complex, interconnected network of water-filled pores. The porous systems exhibit transitional behavior in quantitative agreement with the behavior of percolation lattices (mean square error 7%, n = 29). Predictive mathematical models of the diffusion process were developed: these models used percolation concepts to describe pore topology, continuum models of diffusion/dissolution to describe protein movement at each single pore, and measured pore size distributions. Effective diffusion coefficients for protein transport in aqueous, constricted macropores were predicted by this technique. Predicted diffusion coefficients, based on measured and derived microstructural parameters, agree with experimentally measured diffusion coefficients within a factor of 2. This approach may be useful in the design of porous polymer systems for biological applications and for evaluating other biological systems where conduction of mass, heat, momentum, or charge occurs in a heterogeneous environment.     

Water permeability of plant cuticles: permeance,diffusion and partition coefficients

Summary Using isolated cuticular membranes from ten woody and herbaceous plant species, permeance and diffusion coefficients for water were measured, and partition coefficients were calculated. The cuticular membranes of fruit had much higher permeance and diffusion coefficients than leaf cuticular membranes from either trees or herbs. Both diffusion and partition coefficients increased with increasing membrane thickness. Thin cuticles, therefore, tend to be better and more efficient water barriers than thick cuticles. We compared the diffusion coefficients and the water content of cuticles as calculated from transport measurements with those obtained from water vapor sorption. There is good to fair agreement for cuticular membranes with a low water content, but large discrepancies appear for polymer matrix membranes with high permeance. This is probably due to the fact that diffusion coefficients obtained from transport measurements on membranes with high permeance and water content are underestimated. Water permeabilities of polyethylene and polypropylene membranes are similar to those of leaf cuticular membranes. However, leaf cuticles have much lower diffusion coefficients and a much greater water content than these synthetic polymers. This suggests that cuticles are primarily mobility barriers as far as water transport is concerned.     

Estimation of Solvent Diffusion Coefficients Using Molecular Dynamics Simulations

Abstract

Molecular dynamics simulations have been used to investigate diffusion in two commonly used industrial solvents, toluene and tetrahydrofuran. Several different models for the solvents are compared (flexible vs. rigid, all-atom vs. united atom), and it is found that united atom and all-atom models of the solvents produce very different diffusion coefficients at the experimental density. This disagreement can be explained by the pressure dependence of the diffusion coefficient, which is found to vary in accord with the Chapman-Enskog result for hard spheres. It is recommended that force fields be parametrized carefully to produce reasonable pressures at the experimental densities, or that simulations be carried out at constant pressure, if they are to be used for the purposes of calculating transport coefficients.     

A multiscale theoretical model for diffusive mass transfer in cellular biological media

An integrated methodology is developed for the theoretical analysis of solute transport and reaction in cellular biological media, such as tissues, microbial flocs, and biofilms. First, the method of local spatial averaging with a weight function is used to establish the equation which describes solute conservation at the cellular biological medium scale, starting with a continuum-based formulation of solute transport at finer spatial scales. Second, an effective-medium model is developed for the self-consistent calculation of the local diffusion coefficient in the cellular biological medium, including the effects of the structural heterogeneity of the extra-cellular space and the reversible adsorption to extra-cellular polymers. The final expression for the local effective diffusion coefficient is: D(Abeta)=lambda(beta)D(Aupsilon), where D(Aupsilon) is the diffusion coefficient in water, and lambda(beta) is a function of the composition and fundamental geometric and physicochemical system properties, including the size of solute molecules, the size of extra-cellular polymer fibers, and the mass permeability of the cell membrane. Furthermore, the analysis sheds some light on the function of the extra-cellular hydrogel as a diffusive barrier to solute molecules approaching the cell membrane, and its implications on the transport of chemotherapeutic agents within a cellular biological medium. Finally, the model predicts the qualitative trend as well as the quantitative variability of a large number of published experimental data on the diffusion coefficient of oxygen in cell-entrapping gels, microbial flocs, biofilms, and mammalian tissues.     

Imaging Barriers to Diffusion by Pair Correlation Functions

Molecular diffusion and transport are fundamental processes in physical, chemical, biochemical, and biological systems. However, current approaches to measure molecular transport in cells and tissues based on perturbation methods such as fluorescence recovery after photobleaching are invasive, fluctuation correlation methods are local, and single-particle tracking requires the observation of isolated particles for relatively long periods of time. We propose to detect molecular transport by measuring the time cross-correlation of fluctuations at a pair of locations in the sample. When the points are farther apart than two times the size of the point spread function, the maximum of the correlation is proportional to the average time a molecule takes to move from a specific location to another. We demonstrate the method by simulations, using beads in solution, and by measuring the diffusion of molecules in cellular membranes. The spatial pair cross-correlation method detects barriers to diffusion and heterogeneity of diffusion because the time of the correlation maximum is delayed in the presence of diffusion barriers. This noninvasive, sensitive technique follows the same molecule over a large area, thereby producing a map of molecular flow. It does not require isolated molecules, and thus many molecules can be labeled at the same time and within the point spread function.     

Size-Dependent Diffusion of Dextrans in Excised Porcine Corneal Stroma

Delivery of therapeutic agents to the eye requires efficient transport through cellular and extracellular barriers. We evaluated the rate of diffusive transport in excised porcine corneal stroma using fluorescently labeled dextran molecules with hydrodynamic radii ranging from 1.3 to 34 nm. Fluorescence correlation spectroscopy (FCS) was used to measure diffusion coefficients of dextran molecules in the excised porcine corneal stroma. The preferential sensitivity of FCS to diffusion along two dimensions was used to differentially probe diffusion along the directions parallel to and perpendicular to the collagen lamellae of the corneal stroma. In order to develop an understanding of how size affects diffusion in cornea, diffusion coefficients in cornea were compared to diffusion coefficients measured in a simple buffer solution. Dextran molecules diffuse more slowly in cornea as compared to buffer solution. The reduction in diffusion coefficient is modest however (67% smaller), and is uniform over the range of sizes that we measured. This indicates that, for dextrans in the 1.3 to 34 nm range, the diffusion landscape of corneal stroma can be represented as a simple liquid with a viscosity approximately 1.5 times that of water. Diffusion coefficients measured parallel vs. perpendicular to the collagen lamellae were indistinguishable. This indicates that diffusion in the corneal stroma is not highly anisotropic. Our results support the notion that the corneal stroma is highly permeable and isotropic to transport of hydrophilic molecules and particles with hydrodynamic radii up to at least 34 nm.     

M5 mesoscopic and macroscopic models for mesenchymal motion

In this paper mesoscopic (individual based) and macroscopic (population based) models for mesenchymal motion of cells in fibre networks are developed. Mesenchymal motion is a form of cellular movement that occurs in three-dimensions through tissues formed from fibre networks, for example the invasion of tumor metastases through collagen networks. The movement of cells is guided by the directionality of the network and in addition, the network is degraded by proteases. The main results of this paper are derivations of mesoscopic and macroscopic models for mesenchymal motion in a timely varying network tissue. The mesoscopic model is based on a transport equation for correlated random walk and the macroscopic model has the form of a drift-diffusion equation where the mean drift velocity is given by the mean orientation of the tissue and the diffusion tensor is given by the variance-covariance matrix of the tissue orientations. The transport equation as well as the drift-diffusion limit are coupled to a differential equation that describes the tissue changes explicitly, where we distinguish the cases of directed and undirected tissues. As a result the drift velocity and the diffusion tensor are timely varying. We discuss relations to existing models and possible applications.Dedicated to K.P. Hadeler, a great scientist, teacher, and friend.     

Spatial-temporal dissipative structures arising in open reactive systems with a negative feedback loop

A systematic way of finding the possible spatial-temporal structures that may emerge in open reactive systems coupled with diffusive transport and containing one inhibitory (negative feedback) loop is presented. The method is illustrated on two kinetic models, one used by L. Glass, another by Higgins et al. which we coupled with diffusion. The ranges of cooperativity index and Fick transport coefficients we find for the occurrence of a spatially non-uniform time periodicity are consistent with in vitro experiments.     

Apparent Diffusivity and Taylor Dispersion of Water and Solutes in Capillary Beds

A physical theory explaining the anisotropic dispersion of water and solutes in biological tissues is introduced based on the phenomena of Taylor dispersion, in which highly diffusive solutes cycle between flowing and stagnant regions in the tissue, enhancing dispersion in the direction of microvascular flow. An effective diffusion equation is derived, for which the coefficient of dispersion in the axial direction (direction of capillary orientation) depends on the molecular diffusion coefficient, tissue perfusion, and vessel density. This analysis provides a homogenization that represents three-dimensional transport in capillary beds as an effectively one-dimensional phenomenon. The derived dispersion equation may be used to simulate the transport of solutes in tissues, such as in pharmacokinetic modeling. In addition, the analysis provides a physically based hypothesis for explaining dispersion anisotropy observed in diffusion-weighted imaging (DWI) and diffusion-tensor magnetic resonance imaging (DTMRI) and suggests the means of obtaining quantitative functional information on capillary vessel density from measurements of dispersion coefficients. It is shown that a failure to account for flow-mediated dispersion in vascular tissues may lead to misinterpretations of imaging data and significant overestimates of directional bias in molecular diffusivity in biological tissues. Measurement of the ratio of axial to transverse diffusivity may be combined with an independent measurement of perfusion to provide an estimate of capillary vessel density in the tissue.     

Intra-membrane ligand diffusion and cell shape modulate juxtacrine patterning

A key problem in developmental biology is how pattern and planar polarity are transmitted in epithelial structures. Examples include Drosophila neuronal differentiation, ommatidia formation in the compound eye, and wing hair polarization. A key component for the generation of such patterns is direct cell-cell signalling by transmembrane ligands, called juxtacrine signalling. Previous models for this mode of communication have considered homogeneous distributions in the cell membrane, and the role of polarity has been largely ignored. In this paper we determine the role of inhomogeneous protein and receptor distributions in juxtacrine signalling. We explicitly include individual membrane segments, diffusive transport of proteins and receptors between these segments, and production terms with a combination of local and global responses to ligand binding. Our analysis shows that intra-membrane ligand transport is vital for the generation of long wavelength patterns. Moreover, with no ligand transport, there is no pattern formation for lateral induction, a process in which receptor activation up-regulates ligand production. Biased production of ligand also modulates patterning bifurcations and predicted wavelengths. In addition, biased ligand and receptor trafficking can lead to regular polarity across a lattice, in which each cell has the same orientation-directly analogous to patterns of hairs in the Drosophila wing. We confirm the trends in pattern wavelengths previously observed for patterns with cellular homogeneity-lateral inhibition tends to give short-range patterns, while lateral induction can give patterns with much longer wavelengths. Moreover, the original model can be recovered if intra-membrane bound receptor diffusion is included and rapid equilibriation between the sides is considered. Finally, we consider the role of irregular cell shapes and waves in such networks, including wave propagation past clones of non-signalling cells.     

Modeling of Proton Transfer in Polymer Electrolyte Membranes on Different Time and Length Scales

Polymer electrolyte membranes (PEMs) are key component materials in fuel cell technology. Understanding the relationship between the elementary acts of proton transport and the operation of the entire cell on different time and length scales is therefore particularly rewarding. We discuss the results of recent atomistic computer simulations of proton transport in porous PEMs. Different models cover the range from individual local proton hops to diffusion processes with polymer mobility included.     

Modeling environment for numerical simulation of applied electric fields on biological cells

The application of electric pulses in cells increases membrane permeability. This phenomenon is called electroporation. Current electroporation models do not explain all experimental findings: part of this problem is due to the limitations of numerical methods. The Equivalent Circuit Method (ECM) was developed in an attempt to solve electromagnetic problems in inhomogeneous and anisotropic media. ECM is based on modeling of the electrical transport properties of the medium by lumped circuit elements as capacitance, conductance, and current sources, representing the displacement, drift, and diffusion current, respectively. The purpose of the present study was to implement a 2-D cell Model Development Environment (MDE) of ionic transport process, local anisotropy around cell membranes, biological interfaces, and the dispersive behaviour of tissues. We present simulations of a single cell, skeletal muscle, and polygonal cell arrangement. Simulation of polygonal form indicates that the potential distribution depends on the geometrical form of cell. The results demonstrate the importance of the potential distributions in biological cells to provide strong evidences for the understanding of electroporation.