Determination of diffusion coefficients in biofilms by confocal laser microscopy

Microbial exopolymer may hinder the diffusion of nutrients, antibiotics, and other materials to the cell surface. Studies of diffusion in biofilms have been limited to indirect measurements. This study demonstrated the use of fluorescein and size-fractionated fluor-conjugated dextrans in conjunction with scanning confocal laser microscopy to directly monitor and determine diffusion coefficients within biofilms. The monitoring approaches were simple and, when combined with computerized image collection, allowed assembly of a data set suitable for calculation of one-dimensional diffusion coefficients for biofilm regions. With these techniques, it was shown that regional variability in the mobility of the dextrans occurred within mixed-species biofilms. Some regions exhibited rapid diffusion of all test molecules, while adjacent regions were only penetrated by the lower-molecular-weight compounds. The effective diffusion coefficients (D(e)) determined in a mixed-species biofilm were a function of the molecular radius of the probe (i.e., fluorescein, D(e) = 7.7 x 10 cm s; 4,000 molecular weight, D(e) = 3.1 x 10 cm s; and 2,000,000 molecular weight, D(e) = 0.7 x 10 cm s). These results demonstrated that diffusion in the biofilm was hindered relative to diffusion in the bulk solution. The study indicated that in situ monitoring by scanning laser microscopy is a useful approach for determining the mobility of fluorescently labeled molecules in biofilms, allowing image acquisition, appropriate scales of study, both xy and xz monitoring, and calculation of D(e) values.     

Determination of glucose and ethanol effective diffusion coefficients in Ca-alginate gel.

Glucose and ethanol diffusion coefficients in 2% Ca-alginate gel were measured using the experimental technique based on solute diffusion into or out of gel beads in a well-stirred solution. The aim of the study was to make the measurements under typical conditions found in alcoholic fermentations, such as the concentrations of glucose (100 g l-1) and ethanol (50 g l-1), the simultaneous counter-diffusion of glucose and ethanol, and the presence of cells in the gel beads at a level of 10(9) cells g-1 of beads. Previously, an evaluation of the error associated with the methodology used indicated how the experimental procedure would minimize the error. The individual measurement of glucose and ethanol coefficients in 2% Ca-alginate with no cells gave values of 5.1 and 9.6 x 10(-6) cm2 s-1, respectively, which are lower than those in water. When the effect of counter-diffusion was investigated, both coefficients decreased: glucose by 14% and ethanol by 28%. When cells were incorporated into the beads, only the ethanol coefficient decreased significantly, while the glucose coefficient apparently increased its value to 6.9 10(-6) cm2 s-1.     

Determination of pollutant diffusion coefficients in naturally formed biofilms using a single tube extractive membrane bioreactor

A novel technique has been used to determine the effective diffusion coefficients for 1,1,2-trichloroethane (TCE), a nonreacting tracer, in biofilms growing on the external surface of a silicone rubber membrane tube during degradation of 1,2-dichloroethane (DCE) by Xanthobacter autotrophicus GJ10 and monochlorobenzene (MCB) by Pseudomonas JS150. Experiments were carried out in a single tube extractive membrane bioreactor (STEMB), whose configuration makes it possible to measure the transmembrane flux of substrates. A video imaging technique (VIT) was employed for in situ biofilm thickness measurement and recording. Diffusion coefficients of TCE in the biofilms and TCE mass transfer coefficients in the liquid films adjacent to the biofilms were determined simultaneously using a resistances-in-series diffusion model. It was found that the flux and overall mass transfer coefficient of TCE decrease with increasing biofilm thickness, showing the importance of biofilm diffusion on the mass transfer process. Similar fluxes were observed for the nonreacting tracer (TCE) and the reactive substrates (MCB or DCE), suggesting that membrane-attached biofilm systems can be rate controlled primarily by substrate diffusion. The TCE diffusion coefficient in the JS150 biofilm appeared to be dependent on biofilm thickness, decreasing markedly for biofilm thicknesses of >1 mm. The values of the TCE diffusion coefficients in the JS150 biofilms <1-mm thick are approximately twice those in water and fall to around 30% of the water value for biofilms >1-mm thick. The TCE diffusion coefficients in the GJ10 biofilms were apparently constant at about the water value. The change in the diffusion coefficient for the JS150 biofilms is attributed to the influence of eddy diffusion and convective flow on transport in the thinner (<1-mm thick) biofilms.     

Measurement of local diffusion coefficients in biofilms by microinjection and confocal microscopy

A new technique for the determination of local diffusion coefficients in biofilms is described. It is based on the microinjection of fluorescent dyes and quantitative analysis of the subsequent plume formation using confocal laser microscopy. The diffusion coefficients of fluorescein (MW 332), TRITC-IgG (MW 150000) and phycoerythrin (MW 240000) were measured in the cell clusters and interstitial voids of a heterogeneous biofilm. The diffusivities measured in the voids were close to the theoretical values in water. Fluorescein had the same diffusivity in cell clusters, voids, and sterile medium. TRITC-IgG did not diffuse in cell clusters, presumably due to binding to the cell cluster matrix. After treatment of the biofilm with bovine serum albumin, binding capacity decreased and the diffusion coefficient could be measured. The diffusivity of phycoerythrin in cell clusters was impeded by 41%, compared to interstitial voids. From the diffusion data of phycoerythrin it was further calculated that the cell cluster matrix had the characteristics of a gel with 0.6 nm thick fibers and pore diameters of 80 nm. (c) 1997 John Wiley & Sons, Inc.     

Determination of diffusion and permeability coefficients in muscle

A theory is presented for the study of diffusion in heterogeneous tissue-like structures. It is applicable to a common type of measurement in which the change of the amount of substance remaining in the tissue is determined as the substance diffuses from the tissue into an adjacent medium, for instance, Ringer's solution. The main objective of this paper is to obtain a method for the calculation of the diffusion coefficient in the intercellular space and of the permeability coefficients between this space and the cells, based on the type of measurement mentioned above. Although the fundamental ideas upon which the theory is based are applicable to any type of tissue, the formulae derived are limited to the case in which the cells form a flat bundle of parallel fibers. The theory is applied to the experimental results of E. J. Harris and G. P. Burn on diffusion of sodium in the sartorius muscle of the frog. We find that if we know the ratio of the cellular and intercellular volumes of the muscle the ratio of the equilibrium concentrations of sodium outside and inside the cells can be determined. A very simple mathematical analysis of the experimental relation between the amount of substance diffusing out of the muscle and the time of diffusion gives us this ratio. The ratio of the equilibrium sodium concentrations in the case of the sartorius frog muscle is between about 10 and 30, depending on the muscle used. The same mathematical analysis makes it possible to obtain the permeability coefficients of muscle fibers through simple calculations, if their sizes are known. The permeability coefficients for the experimental work mentioned above using sodium are 1.25 to 11.5×10⁻⁸ cm/sec for the flow into the fibers and 3.2 to 16×10⁻⁷ cm/sec for the flow in the opposite direction. The determination of the diffusion coefficient in the intercellular space is more laborious and yields only an order of magnitude: 10⁻⁶ cm²/sec.     

Determination of diffusion and permeability coefficients in nerve trunks

A mathematical theory is developed which permits the determination of certain parameters of an inhomogenous tissue, such as a nerve trunk without its epineurium. The parameters are the permeability coefficients for entrance into an exit of a substance from the nerve fibers, and the diffusion coefficient of the interstitial material. The experimental data required are the dimensions of the cross-section, the average diameter of the fibers, and the ratio of the cross-sectional are of the fibers to the total cross-section, as well as the time course of the decrease of the fraction of the substance left in the nerve trunk, when the trunk is immersed in a bathing solution containing none of it.     

A review of experimental measurements of effective diffusive permeabilities and effective diffusion coefficients in biofilms

Experimental measurements of effective diffusive permeabilities and effective diffusion coefficients in biofilms are reviewed. Effective diffusive permeabilities, the parameter appropriate to the analysis of reaction-diffusion interactions, depend on solute type and biofilm density. Three categories of solute physical chemistry with distinct diffusive properties were distinguished by the present analysis. In order of descending mean relative effective diffusive permeability (De/Daq) these were inorganic anions or cations (0.56), nonpolar solutes with molecular weights of 44 or less (0.43), and organic solutes of molecular weight greater than 44 (0.29). Effective diffusive permeabilities decrease sharply with increasing biomass volume fraction suggesting a serial resistance model of diffusion in biofilms as proposed by Hinson and Kocher (1996). A conceptual model of biofilm structure is proposed in which each cell is surrounded by a restricted permeability envelope. Effective diffusion coefficients, which are appropriate to the analysis of transient penetration of nonreactive solutes, are generally similar to effective diffusive permeabilities in biofilms of similar composition. In three studies that examine diffusion of very large molecular weight solutes (>5000) in biofilms, the average ratio of the relative effective diffusion coefficient of the large solute to the relative effective diffusion coefficient of either sucrose or fluorescein was 0.64, 0.61, and 0.36. It is proposed that large solutes are effectively excluded from microbial cells, that small solutes partition into and diffuse within cells, and that ionic solutes are excluded from cells but exhibit increased diffusive permeability (but decreased effective diffusion coefficients) due to sorption to the biofilm matrix.     

Determination of diffusion coefficients by spreading from thin layers

Two new methods for the measurement of diffusion coefficients were introduced. In the first method the spreading is measured from a. thin layer in a cylindrical tube containing a suitable gel. To avoid convection in the gel the tube is rotated horizontally round its long axis. In the second method the diffusion spreading is measured from an initial thin layer under a column of solvent in a Tiselius electrophoresis cell or a micro cuvet of a spectrophotometer. The great advantage is the economy on the material investigated.     

Diffusion coefficients of metabolites in active biofilms

Diffusion coefficients of actual metabolites in completely active biofilms can be determined by applying a new concept that is based on a constant local activity in the entire biofilm. In that case, a concentration step will be transmitted unattenuated. Subsequently, the diffusion coefficient can be calculated from the response monitored with a microelectrode positioned in the biofilm without quantitative knowledge of the local microbial kinetics. The conditions required for such a constant microbial biofilm activity were formulated in terms of the Thiele modulus and the substrate concentration in the bulk liquid. This proposed method was successfully applied to determine diffusion coefficients of oxygen and glucose in agar gels containing various fractions of active immobilized microorganisms. The values obtained were compared to experimental results from well-defined inert systems. The transient response of oxygen was far more affected by the presence of the immobilized cells than glucose. This can be explained by partition of the diffusing solute between the microbial cells and the aqueous phase.     

Determination of effective diffusion coefficients and distribution constants in polysaccharide gels with non-steady-state measurements

A non-steady-state method has been used for determining the effective diffusion coefficient, D(e), and a distribution constant, K(i), of small molecules in alginate gel beads. A mathematical model based on Pick's law and includingexternal film diffusion resistance describe the diffusion process. Criticalexperimental parameters for the estimation of D(e) and K(i), for both one- and two-parameter methods were the initial solute concentration in the bulk liquid, the void fraction inthe reactor, and the experimental starting point. In our analysis, the two-parameter method is preferable. Incorporation of an estimate of the film resistance into the overall model increased the estimated values of D(e) significantly and improved the stability of the term over a range of reactor agitation rates. (c) 1995 John Wiley & Sons Inc.     

Determination of diffusion coefficients and diffusion characteristics for chlorferon and diethylthiophosphate in Ca-alginate gel beads

Diffusion characteristics of chlorferon and diethylthiophosphate (DETP) in Ca-alginate gel beads were studied to assist in designing and operating bioreactor systems. Diffusion coefficients for chlorferon and DETP in Ca-alginate gel beads determined at conditions suitable for biodegradation studies were 2.70 x 10(-11) m(2)/s and 4.28 x 10(-11) m(2)/s, respectively. Diffusivities of chlorferon and DETP were influenced by several factors, including viscosity of the bulk solution, agitation speed, and the concentrations of diffusing substrate and immobilized cells. Diffusion coefficients increased with increasing agitation speed, probably due to poor mixing at low speed and some attrition of beads at high speeds. Diffusion coefficients also increased with decreasing substrate concentration. Increased cell concentration in the gel beads caused lower diffusivity. Theoretical models to predict diffusivities as a function of cell weight fraction overestimated the effective diffusivities for both chlorferon and DETP, but linear relations between effective diffusivity and cell weight fraction were derived from experimental data. Calcium-alginate gel beads with radii of 1.65-1.70 mm used in this study were not subject to diffusional limitations: external mass transfer resistances were negligible based on Biot number calculations and effectiveness factors indicated that internal mass transfer resistance was negligible. Therefore, the degradation rates of chlorferon and DETP inside Ca-alginate gel beads were reaction-limited.     

Determination of dendrigraft poly-L-lysine diffusion coefficients by taylor dispersion analysis

This work focuses on the physicochemical characterization of dendrigraft poly-L-lysines (DGLs) obtained by polymerization of N-carboxyanhydride in buffered water (pH 6.5). Diffusion coefficients (D) and hydrodynamic radii (Rh) of five successive DGL generations were determined by Taylor dispersion analysis (TDA). To our knowledge, this is the first experimental work using TDA for the characterization of dendrimer-like structures. Experimental Rh values obtained by TDA were compared to those derived from dynamic light scattering and size exclusion chromatography coupled to a triple detection (refractive index, viscosimetry, and static light scattering). Significant differences were obtained, especially for the highest generations, as a result of the inherent contribution of aggregates to the light scattering intensity. For that reason, TDA was found to be the most appropriate technique for determining the D values of these hyperbranched macromolecules. Regarding their physicochemical behavior, the experimental results confirm that DGLs are very similar to trifunctional dendrimers (exponential growth of the molar mass, almost linear variation of the hydrodynamic radius, high branching density, and maximum of the intrinsic viscosity or of the free volume fraction for generation 4).     

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.     

Determination of diffusion coefficients of proteins in stationary phases by frontal chromatography

A strategy to determine effective diffusion coefficients of proteins in chromatographic gels is presented in this article. An experimental methodology based on frontal liquid chromatography was combined with a numerical methodology based on a mathematical model describing the chromatographic process including the extra-column dispersion, the dispersion due to the packed bed, the external mass transfer from the bulk phase to the stationary phase, and the diffusive transport within the stationary phase. The methodology has several advantages compared to previously reported methods to determine diffusion coefficients in that no other equipment than an HPLC is required, any class of stationary phases can be investigated as long as the experiments are performed under non-binding conditions, and no modification, e.g., moulding of slabs or membranes, to the stationary phase is required. To show the applicability of the methodology, the effective diffusion coefficients of lysozyme, bovine serum albumin, and immunoglobulin gamma in Sepharose CL-4B were determined and shown to be comparable with those determined with other methods.     

Determination of catecholamine permeability coefficients for passive diffusion across phospholipid vesicle membranes

Summary A convenient catecholamine transport assay has been developed which permits continuous, instantaneous monitoring of transmembrane flux. Epinephrine transport has been examined by spectrophotometrically monitoring adrenochrome formation resulting from the passive diffusion of catecholamine into unilamellar phospholipid vesicles containing entrapped potassium ferricyanide. Ferricyanide oxidation of epinephrine under the conditions employed is fast compared to membrane transport, which obviates the need for intravesicular concentration or volume determinations. Epinephrine transport data over a pH 6 to 7 range have been fitted to an integrated rate equation from which a permeability coefficient for neutral epinephrine of 2.7±1.5×10^–6 cm/sec has been obtained.     

Reaction-diffusion models of development with state-dependent chemical diffusion coefficients

Reaction-diffusion models are widely used to model developmental processes. The great majority of current models invoke constant diffusion coefficients. However, the diffusion of metabolites or signals through tissues is frequently such that this assumption may reasonably be questioned. We consider several different physical mechanisms leading to effective diffusion coefficients in biological tissues which vary with the local conditions, including models in which juxtacrine signaling results in the diffusion of a signal in the absence of material transport. We develop a mathematical formalism for transforming local transport laws into diffusive terms. This procedure is appropriate when the typical length scale over which the concentrations change significantly is much greater than the dimensions of a cell. We review previous developmental models which considered the possibility of state-dependent diffusion coefficients. We also provide a few new motivating examples.     

Propagators and time-dependent diffusion coefficients for anomalous diffusion

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