Diffusion in cell‐free and cell immobilising κ‐carrageenan gel beads with and without chemical reaction

Diffusion into and from κ‐carrageenan gel beads was studied, both in the absence and presence of bacterial cells, both with and without biochemical reaction. The solutes were indole, L ‐serine, and L ‐tryptophan. The reaction was that of indole and L ‐serine to give L ‐tryptophan. Established theory concerning diffusion of a single solute in cell‐free gels was found to describe well the effect of the gel on diffusivity. Simultaneous diffusion of the three solutes resulted in lower diffusivities than those for individual solutes, suggesting the need to use multicomponent diffusion theory. The effect of cells on diffusion could only be accounted for by models assuming permeable cells. Diffusion with chemical reaction was reasonably well described by an effectiveness factor calculated using an effective diffusivity estimated from diffusion data without reaction. © 1999 John Wiley & Sons, Inc. Biotechnol Bioeng 63: 625–631, 1999.     

Diffusion of sucrose and dextran through agar gel membranes

Mass transfer limitations severely impede the performance of bioreactions involving large molecules by gel-entrapped microorganisms. This paper describes a quantitative investigation of such diffusional limitations in agar gel membranes. Sucrose and commercial dextran fractions with (weight-average) molecular weights ranging from 10,000 to 2,000,000 Da were used as standard diffusants. For all tested solutes but sucrose, the values of the agar/water partition coefficients highlighted steric hindrance at the entrance of the membrane pores. The effective diffusivity of sucrose in agar was similar to that in water. All dextran fractions, however, displayed restricted diffusion in the agar membranes. Their effective diffusivities were a decreasing function of the agar content of the gel membrane (0.5, 1.0, or 1.5% w/v). The effective diffusivity in a given membrane decreased as the molecular weight of the diffusing molecule increased. T500 (ucbar|M_w = 470,000 Da) and ucbar|M_w = 1,950,000 Da) fractions were unable to diffuse through 1.0 or 1.5% agar membranes. The diffusion data did not agree with the classical (Renkin) model for a hard sphere diffusing through a cylindrical pore. These results are discussed in terms of gel and diffusant characteristics.     

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.     

Diffusion in kappa-carrageenan gel beads

Using a well-mixed and temperature-led vessel, the diffusion characteristics of various solutes into spherical kappa-carrageenan gel beads were experimentally investigated. The diffusion coefficient of glucose was markedly affected by the glucose concentration and the operating temperature. In all cases the diffusivity obtained was noticeably smaller than that of glucose in pure water. The experimental data also indicated an inverse relationship between the diffusivity and the polymer concentration used in the gel preparation. As well, the glucose diffusivity was affected by the presence of other solutes in the glucose solution. Electrolytes such as ammonium sulfate, KCl, and CaCl(2) were observed to enhance the diffusion coefficient. On the other hand, the addition of arginine or bovine serum albumin had an adverse effect on the diffusivity. No diffusion of albumin into the gel beads was observed, and such a solute created a significant mass transfer resistance during the diffusion process.     

Characterisation of small molecules diffusion in hydrogel-membrane liquid-core capsules

The overall diffusion coefficients for several low molecular weight solutes, such as glucose, fructose, sucrose, lactose, and vitamin B(12) have been determined in Ca-alginate membrane liquid-core capsules using the unsteady-state method following the release of solutes from the capsules to a well-stirred solution of limited volume. The diffusion coefficients obtained for saccharides were 5-20% lower than the corresponding diffusivity in water while for vitamin B(12) about 50% that of water. The diffusion coefficients of the investigated capsules were not influenced by the change in alginate concentration in the capsule membrane from 0.5 to 1.0%. Lower diffusivities and higher deviations from the diffusivity in water were obtained for higher molecular weight solutes.     

Diffusion of lactose in kappa-carrageenan/locust bean gum gel beads with or without entrapped growing lactic acid bacteria

Effective diffusion coefficients (De) of lactose in kappa-carrageenan (2.75% wt/wt)/locust bean gum (0.25% wt/wt) (LBG) gel beads (1.5-2.0-mm diameter)with or without entrapped lactic acid bacteria (LAB) were determined at 40 degrees C. The effects of lactose concentration, bacteria strain (Streptococcus salivarius subsp. thermophilus and Lactobacillus casei subsp. casei) and cell content at various steps of the fermentation process (after immobilization, pre-incubation of the beads and successive fermentations) were measured on De as a first step for process modelling. Results were obtained from transiend concentration changes n well-stirred lactose solutions in which the beads were suspended. A mathematical model of unsteady-state diffusion in a sphere was used, and De was obtained from the best fit of the experimental data. Diffusivity of lactose in cell-tree beads was significantly lower than in pure water mainly because of the obstruction effect of the polymer chains and the hydration region. Furthermore, effective diffusivity and equilibrium partition factor were independent of lactose concentration in the range from 12.5 to 50 g/L. No significant difference was found for De (effective diffusivity) and Kp (partition) coefficients between beads entrapping S. thermophilus (approximately 5 x 10(9) CFU/mL) and cell-free beads. On the other hand higher cell counts obtained with L. casei (close to 1.8 x 10(11) CFU/mL) increased mass transfer resistance resulting in lower effective diffusivities and Kp. Finally, the effects of the type of bacteria and their distribution in the beads on the diffusivity were also discussed.     

Diffusion and partitioning of proteins in charged agarose gels.

The effects of electrostatic interactions on the diffusion and equilibrium partitioning of fluorescein-labeled proteins in charged gels were examined using fluorescence recovery after photobleaching and gel chromatography, respectively. Measurements were made with BSA, ovalbumin, and lactalbumin in SP-Sepharose (6% sulfated agarose), in phosphate buffers at pH 7 and ionic strengths ranging from 0.01 to 1.0 M. Diffusivities in individual gel beads (D) and in the adjacent bulk solution (D infinity) were determined from the spatial Fourier transform of the digitized two-dimensional fluorescence recovery images. Equilibrium partition coefficients (phi) were measured by recirculating protein solutions through a gel chromatography column until equilibrium was reached, and using a mass balance. Diffusion in the gel beads was hindered noticeably, with D/D infinity = 0.4-0.5 in each case. There were no effects of ionic strength on BSA diffusivities, but with the smaller proteins (ovalbumin and lactalbumin) D infinity increased slightly and D decreased at the lowest ionic strength. In contrast to the modest changes in diffusivity, there were marked effects of ionic strength on the partition coefficients of these proteins. We conclude that for diffusion of globular proteins through gel membranes of like charge, electrostatic effects on the effective diffusivity (Deff = phi D) are likely to result primarily from variations in phi with only small contributions from the intramembrane diffusivity.     

The effective diffusion coefficient and the distribution constant for small molecules in calcium-alginate gel beads

The effective diffusion coefficient, D(e), and the distribution constant, K(i), for selected mono- and disaccharides and organic acids were determined in homogeneous calcium-alginate gel with and without entrapped bacteria. Results were obtained from transient concentration changes in well-stirred solutions of limited volume, in which the gel beads were suspended. The effective diffusioncoefficients and the distribution constants were estimated by fitting mathematical model predictions to the experimental data using a nonlinear model fitting program (MODFIT). Both single solute diffusion and multiple solute diffusion were performed. A small positive effect was obtained onthe values of D(e) for the system of multiple solute diffusion; however, the values of K(i) were not significantly influenced. For the nine solutes tested, D(e) for 2% Ca-alginate gel beads was found to be approximately 85% of the diffusivity measured in water. The effects on D(e) and K(i), for lactose and lactic acid were determined for variations of alginate concentration, pH, temperature, and biomass content in the beads. D(e) decreased linearly for both lactose and lactic acid with increasing cell concentration in the Ca-alginate gel. K(i), was constant for both lactose and lactic acid with increasing cell concentration. D(e) was significantly lower at pH 4.5 than at pH 5.5 and 6.5 for both lactose and lactic acid. Furthermore, D(e) seemed to decrease with increased alginate concentration in the range of 1% to 4%. The diffusion rate increased with increasing temperature, and the activation energy for the diffusion process for both lactose and lactic acid was constant in the temperature range tested. (c) 1995 John Wiley & Sons Inc.     

A stirred bath technique for diffusivity measurements in cell matrices

A stirred bath technique was developed for determining effective diffusivities in cell matrices. The technique involves cell immobilization in a dilute gel which has negligible effect on solute diffusion. Agar and collagen were tested as immobilizing gels. Agar gel was shown to have minor interactions with the diffusion of various biological molecules, and was used for immobilization of Ehrlich Ascites Tumor (EAT) cells. Diffusivities of glucose and lactic acid were measured in EAT matrices for cell loadings between 20 and 45 vol %. Treatment with glutaraldehyde was effective in quenching the metabolic activity of the cells while preserving their physical properties and diffusive resistance. The measured data agree favorably with predictions based on Maxwell's equation for effective diffusion in a periodic composite material. The stirred bath technique is useful for diffusivity determinations in immobilized matrices or free slurries, and is applicable to both microbial and mammalian cell systems.     

Oxygen diffusivity in gel beads containing viable cells

This article proposes a simple steady-state method for measuring the effective diffusion coefficient of oxygen (D(e)) in gel beads entrapping viable cells. We applied this method to the measurement of D(e) in Ca- and Ba-alginate gel beads entrapping Saccharomyces cerevisiae and Pseudomonas ovalis. The diffusivity of oxygen through gel beads containing viable cells was measured within an accuracy of +/-7% and found not to be influenced by cell density (0-30 g/L gel), cell type, and cell viability in gel beads. The oxygen diffusivity in the Ca-alginate gel beads was superior to that of the Ba-alginate gel beads, and the D(e) in the Ca-alginate gel beads nearly equalled the molecular diffusion coefficient in the liquid containing the gel beads. The oxygen concentration profile in a single Ca-alginate gel bead was calculated and compared to the distribution of mycelia of Aspergillus awamori grown in that gel bead. This procedure indicated that the oxygen concentration profile is useful for the estimation of the thickness of the cell layer in a gel bead. Numerical investigation revealed that high effectiveness factors, greater than 0.8, could be obtained using microgel beads with a radius of 0.25 mm.     

A novel technique for measuring solute diffusivities in entrapment matrices used in immobilization

A novel technique has been developed for measuring effective solute diffusivities in entrapment matrices used for cell immobilization. In this technique radiotracers were used to measure effective diffusivities and equilibrium partition coefficients of the solute between the liquid and solid matrix. Ca-alginate was used in this study, because it is one of the most commonly employed matrices for the immobilization of microbial, plant and mammalian cells. The experimental apparatus consisted of a single spherical Ca-alginate bead which was attached to a rotating rod and immersed in water containing C(14)-glucose. The rotational speed of the spherical bead was controlled and resulted in excellent mixing, and negligible external film mass transfer resistance, which allowed the measurement of true effective solute diffusivity within the solid matrix. The rates of C(14)-glucose diffusion within the Ca-alginate sphere were measured using a scintillation spectrometer. A mathematical model of unsteady-state diffusion in a sphere was used with appropriate boundary conditions, and the effective diffusivity of glucose was found from the best fit of the experimental data using a computer regression analysis method. Using 2% (w/v) Ca-alginate beads in this new radiotracer technique the effective diffusivity and partition coefficient of glucose were found to be 6.62 x 10(-10) m(2)/s and 0.98, respectively. The accuracy, advantages, and simplicity of this new method for diffusivity measurements are also compared to other existing methods.     

Diffusion of (de)acylated antibiotic A40926 in alginate and carrageenan beads with or without cells and/or soybean meal

Effective diffusion coefficients (D(e)) of antibiotic A40926 and its deacylated derivative were determined in Ca-alginate (2% wt/wt) and kappa-carrageenan (2.6% wt/wt) gel beads with or without immobilized Actinoplanes teichomyceticus cells and/or soybean meal (SBM). The method used was based on transient concentration changes in a well-stirred antibiotic solution in which gel beads, initially free of solute, were suspended. Unsteady-state diffusion in a sphere was applied and D(e) determined from the best fit of experimental data. A40926 showed markedly different diffusion characteristics than its deacylated derivative. Diffusivity of deacyl-A40926 in alginate or carrageenan gel beads was six to seven times that of A40926. Large differences in partition coefficients (Kp) were also found. In case of beads without additions, A40926, in contrast to deacyl-A40926, strongly partitioned to the liquid phase. Introduction of SBM and/or mycelium in the gel beads decreased the effective diffusivity of deacyl-A40926, but increased its partitioning to the solid phase. Our findings indicate that a relatively moderate structural change of a lipoglycopeptide molecule could lead to a major change in its diffusion/partition characteristics.     

Counter-diffusion of lactose and lactic acid in kappa-carrageenan/locust bean gum gel beads with or without entrapped lactic acid bacteria.

The effective diffusion coefficient (De) and equilibrium partition factor (Kp) for lactose and lactic acid in k-carrageenan (2.75% w/w)/locust bean gum (0.25% w/w) (LBG) gel beads (1.5-2.0 mm diameter), with or without entrapped Lactobacillus casei subsp. casei (L. casei), were determined at 40 degrees C. Results were obtained from transient concentration changes in well-stirred solutions of finite volume in which the beads were suspended. Mathematical models of unsteady-state diffusion into and/or from a sphere and appropriate boundary conditions were used to calculate effective diffusion coefficients of lactose and lactic acid from the best fit of the experimental solute concentration changes. The effective diffusivities of lactose and lactic acid were 5.73 x 10(-10) and 9.96 x 10(-10) m2 s-1, respectively. Furthermore, lactic acid was found to modify gel structure since lactose diffusion characteristics (De and Kp) differed significantly from an earlier study and in the literature. In gel beads heavily colonized with L. casei, the effective diffusion coefficients of lactose and lactic acid were respectively 17% and 24% lower than for cell-free beads. Partition coefficients also confirmed the obstruction effect due to the cells, and decreased from 0.89 to 0.79, and from 0.98 to 0.87, for lactose and lactic acid, respectively. External mass transfer was estimated by an unsteady-state model in infinite volume using the Biot number. The effect of external mass transfer resistance on De results and the data reported in the literature are discussed.     

Studies on the respiration rate of free and immobilized cells of Cephalosporium acremonium in cephalosporin C production

Bioprocesses using filamentous fungi immobilized in inert supports present many advantages when compared to conventional free cell processes. However, assessment of the real advantages of the unconventional process demands a rigorous study of the limitations to diffusional mass transfer of the reagents, especially concerning oxygen. In this work, a comparative study was carried out on the cephalosporin C production process in defined medium containing glucose and sucrose as main carbon and energy sources, by free and immobilized cells of Cephalosporium acremonium ATCC 48272 in calcium alginate gel beads containing alumina. The effective diffusivity of oxygen through the gel beads and the effectiveness factors related to the respiration rate of the microorganism were determined experimentally. By applying Monod kinetics, the respiration kinetics parameters were experimentally determined in independent experiments in a complete production medium. The effectiveness factor experimental values presented good agreement with the theoretical values of the approximated zero‐order effectiveness factor, considering the dead core model. Furthermore, experimental results obtained with immobilized cells in a 1.7‐L tower bioreactor were compared with those obtained in 5‐L conventional fermentor with free cells. It could be concluded that it is possible to attain rather high production rates working with relatively large diameter gel beads (ca. 2.5 mm) and sucrose consumption‐based productivity was remarkably higher with immobilized cells, i.e., 0.33 gCPC/kg sucrose/h against 0.24 gCPC/kg sucrose/h in the aerated stirred tank bioreactor process. © 1999 John Wiley & Sons, Inc. Biotechnol Bioeng 63: 593–600, 1999.     

Diffusion of acetophenone and phenethyl alcohol in the calcium-alginate-bakers' yeast-hexane system

The intrabead diffusion coefficients of acetophenone and phenethyl alcohol were measured at 30 degrees C in the triphasic immobilized yeast-water-hexane system. Saccharomyces cerevisiae cells were deactivated with hydrochloric acid and entrapped in calcium-alginate beads. Measurements of dry cell loss during deactivation, shrinkage of the beads during deactivation and the final porosity of the beads were made for various cell loadings. Final concentrations of wet cells in the beads ranged from approximately 0.25 to 0.30 g/mL. Mass transfer in the hexane phase, external to the beads, was eliminated experimentally. The estimated error of 5% to 10% in the diffusion coefficients is within the experimental error associated with the bead method. The effect of significant sampling volumes on the diffusivities was estimated theoretically and accounted for experimentally. The intrabead concentration of acetophenone and phenethyl alcohol was 150 to 800 ppm. The deactivated cells were shown to be impervious to acetophenone so that the measured diffusivities are extracellular parameters. The cell volume fraction in the beads ranged from 0.70 to 0.90, significantly higher than previously reported data. The effective diffusion coefficients conform to the random pore model. No diffusional interaction between acetophenone and phenethyl alcohol was observed. The addition of 2 vol% ethanol or methanol slightly increased the diffusivities. The thermodynamic partition coefficients were measured in the bead-free water-organic system and found to be an order of magnitude lower than the values calculated from the analysis of the diffusion data for the organic-bead system, suggesting that bead-free equilibrium data cannot be used in triphasic systems. (c) 1994 John Wiley & Sons, Inc.     

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.     

Preparation of comb-type N-isopropylacrylamide hydrogel beads and their application for size-selective separation media

A series of the comb-type poly(N-isopropylacrylamide) (NIPAM) gel beads were prepared by inverse suspension polymerization techniques. The comb-type NIPAM gel beads exhibited large volume change at 30 degrees C, and their deswelling rate, defined as the time required for half-shrinking, was 10 times faster than that of the normal-type NIPAM gel beads. The gel beads were utilized to concentrate dilute aqueous solutions of albumin, gamma-globulin, and vitamin B(12). The separation efficiencies of albumin and gamma -globulin with the comb-type NIPAM gel were 80% and 85%, respectively. Whereas those with normal-type NIPAM gel were 55% and 60%, respectively. The incorporation of grafted chains into gel makes the effective mesh size smaller. Therefore it induces the additional obstruction effects between the solutes and network and excludes the high molecular weight solutes. After they have extracted water, their rapid deswelling property makes the gel regenerate effectively by warming to release the absorbed water.     

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.     

Effective diffusivity of galactose in calcium alginate gels containing immobilized Zymomonas mobilis.

The effective diffusivity of galactose was measured for calcium alginate gel membranes containing immobilized live Zymomonas mobilis cells at concentrations ranging from 0 to 150 g dry wt/L of gel. Since galactose is not taken up by living Z. mobilis organisms, the diffusion of this representative six-carbon sugar could be studied independently of sugar consumption. Various immobilized biomass loadings were achieved by two different techniques: addition of biomass at known concentrations to the sodium alginate solution before membrane formation and growth of cells in the gel to various biomass concentrations. The highest immobilized cell concentration, attained by in situ growth, corresponds to the maximum of this system, as growth beyond this maximum concentration led to disintegration of the gel membrane. The galactose effective diffusivity measurements for both methods of immobilized cell loading overlap within experimental error and follow the same general monotonic decline with entrapped biomass concentration. Most of the data fall below the upper bound predicted by Hashin and Shtrikman (1962) and show good agreement with the random pore model of Wakao and Smith (1962, 1964). Available effective diffusivity data from the literature provide evidence that the random pore model is an excellent predictor of sugar effective diffusivity in gel immobilized cell systems in general.     

Diffusion coefficients of glucose and ethanol in cell-free and cell-occupied calcium alginate membranes

The diffusivities of glucose and ethanol in cell-free and cell-occupied membranes of calcium alginate were measured in a diffusion cell. The lag time analysis was used. Diffusivities decreased with increasing alginate concentration and were comparable with those in water for a 2% alginate membrane. Glucose and ethanol concentrations had no effect on the respective diffusion coefficients. The ratio of ethanol diffusivity to glucose diffusivity in 2 and 4% alginate agreed closely with the inverse ratio of the hydrodynamic raii for the two molecules in water, indicating that the hydrodynamic theory of diffusion in liquids may be applicable to diffusion in dilute alginate gels. Also, the presence of 20% dead yeast cells had no effect on the diffusivities. The data reported can be used to study reaction and diffusion in immobilized cell reactors and cell physiology under immobilized conditions.