Polymorphic Protein Crystal Growth: Influence of Hydration and Ions in Glucose Isomerase

Crystal polymorphs of glucose isomerase were examined to characterize the properties and to quantify the energetics of protein crystal growth. Transitions of polymorph stability were measured in poly(ethylene glycol)/NaCl solutions, and one transition point was singled out for more detailed quantitative analysis. Single crystal x-ray diffraction was used to confirm space groups and identify complementary crystal structures. Crystal polymorph stability was found to depend on the NaCl concentration, with stability transitions requiring > 1 M NaCl combined with a low concentration of PEG. Both salting-in and salting-out behavior was observed and was found to differ for the two polymorphs. For NaCl concentrations above the observed polymorph transition, the increase in solubility of the less stable polymorph together with an increase in the osmotic second virial coefficient suggests that changes in protein hydration upon addition of salt may explain the experimental trends. A combination of atomistic and continuum models was employed to dissect this behavior. Molecular dynamics simulations of the solvent environment were interpreted using quasi-chemical theory to understand changes in protein hydration as a function of NaCl concentration. The results suggest that protein surface hydration and Na+ binding may introduce steric barriers to contact formation, resulting in polymorph selection.

Multiscale modeling of protein uptake patterns in chromatographic particles

A parallel diffusion model is presented to explain an apparent transition in uptake mechanism seen in experimental observations of protein uptake into porous adsorbents. While such models have been invoked previously, this mesoscopic description is augmented here by microscopic models for representing surface diffusion by "hopping" and adsorption within a Gibbs surface excess formulation. These contributions lead to a relation for the apparent protein diffusivity as a function of adsorption conditions, which can be used predictively with knowledge of a few readily measured physical quantities. The approach can be useful in seeking optimal conditions for preparative protein chromatography separations.

Calculation of short-range interactions between proteins

Macromolecular association is an integral component of numerous cellular and technologically relevant processes. Most molecular theories of such association neglect the explicit solvent structure and rely on continuum concepts such as surface energies for calculating short-range interactions. We present a new such method for calculating the non-electrostatic component of the interaction-free energy, based on formalisms for calculating dispersion interactions between macromolecules. The interactions are separated into a short-ranged component that is treated atomistically, and a longer range component that is treated within the continuum Lifshitz-Hamaker approach. This description avoids the singularities inherent in the continuum dispersion formulation, and its effectiveness in characterizing the shape complementarity between interacting surfaces is shown to be comparable to that of surface area-based methods of similar parametric complexity. An advantage of the new method is that it allows facile calculation of the interaction free energy as a function of intermolecular separation, including steric effects; this makes it suitable for use in simulations of protein solutions.

Sorption processes in ion-exchange chromatography of viruses

Purified viruses are used in gene therapy and vaccine production. Ion-exchange chromatography (IEC) is the most common method for large-scale downstream purification of viruses and proteins. Published IEC protocols provide details for specific separations but not general methods for selecting operating parameters. To make the selection more systematic, we study adenovirus type 5 (Ad5) as a model virus and develop batch uptake and light scattering methods for optimizing the ionic strength and pH of adsorption, as well as providing heuristics for resin geometry. The static capacity for Ad5 was found to go through a maximum with increasing ionic strength. Comparison to a protein-resin system shows that resin capacity for the virus is at least an order of magnitude lower, even on a wide-pore resin. Virus penetration into the wide-pore resin is only partial and the uptake rate is an order of magnitude slower than the uptake onto a narrow-pore resin.

Effects of pore structure and molecular size on diffusion in chromatographic adsorbents

Two computational approaches, namely Brownian dynamics and network modeling, are presented for predicting effective diffusion coefficients of probes of different sizes in three chromatographic adsorbents, the structural properties of which were determined previously using electron tomography. Three-dimensional reconstructions of the adsorbents provide detailed, explicit characteristics of the pore network, so that no assumptions have to be made regarding pore properties such as connectivity, pore radius and pore length. The diffusivity predictions obtained from the two modeling approaches were compared to experimental diffusivities measured for dextran and protein probes. Both computational methods captured the same qualitative results, while their predictive capabilities varied among adsorbents.

Light-scattering studies of protein solutions: role of hydration in weak protein-protein interactions

We model the hydration contribution to short-range electrostatic/dispersion protein interactions embodied in the osmotic second virial coefficient, B(2), by adopting a quasi-chemical description in which water molecules associated with the protein are identified through explicit molecular dynamics simulations. These water molecules reduce the surface complementarity of highly favorable short-range interactions, and therefore can play an important role in mediating protein-protein interactions. Here we examine this quasi-chemical view of hydration by predicting the interaction part of B(2) and comparing our results with those derived from light-scattering measurements of B(2) for staphylococcal nuclease, lysozyme, and chymotrypsinogen at 25 degrees C as a function of solution pH and ionic strength. We find that short-range protein interactions are influenced by water molecules strongly associated with a relatively small fraction of the protein surface. However, the effect of these strongly associated water molecules on the surface complementarity of short-range protein interactions is significant, and must be taken into account for an accurate description of B(2). We also observe remarkably similar hydration behavior for these proteins despite substantial differences in their three-dimensional structures and spatial charge distributions, suggesting a general characterization of protein hydration.

A consistent experimental and modeling approach to light-scattering studies of protein-protein interactions in solution

The osmotic second virial coefficient, B(2), obtained by light scattering from protein solutions has two principal components: the Donnan contribution and a contribution due to protein-protein interactions in the limit of infinite dilution. The Donnan contribution accounts for electroneutrality in a multicomponent solution of (poly)electrolytes. The importance of distinguishing this ideal contribution to B(2) is emphasized, thereby allowing us to model the interaction part of B(2) by molecular computations. The model for protein-protein interactions that we use here extends earlier work (Neal et al., 1998) by accounting for long-range electrostatic interactions and the specific hydration of the protein by strongly associated water molecules. Our model predictions are compared with measurements of B(2) for lysozyme at 25 degrees C over pH from 5.0 to 9.0, and 7-60 mM ionic strength. We find that B(2) is positive at all solution conditions and decreases with increasing ionic strength, as expected, whereas the interaction part of B(2) is negative at all conditions and becomes progressively less negative with increasing ionic strength. Although long-range electrostatic interactions dominate this contribution, particularly at low ionic strength, short-range electrostatic/dispersion interactions with specific hydration are essential for an accurate description of B(2) derived from experiment.

Effects of ionic strength on lysozyme uptake rates in cation exchangers. I: Uptake in SP Sepharose FF

Fluorescence scanning confocal microscopy was used in parallel with batch uptake and breakthrough measurements of transport rates to study the effect of ionic strength on the uptake of lysozyme into SP Sepharose FF. In all cases the adsorption isotherms were near-rectangular. As described previously, the intraparticle profiles changed from slow-moving self-sharpening fronts at low salt concentration, to fast-moving diffuse profiles at high salt concentration, and batch uptake rates correspondingly increased with increasing salt concentration. Shrinking core and homogeneous diffusion frameworks were used successfully to obtain effective diffusivities for the low salt and high salt conditions, respectively. The prediction of column breakthrough was generally good using these frameworks, except for low-salt uptake results. In those cases, the compressibility of the stationary phase coupled with the shrinking core behavior appears to reduce the mass transfer rates at particle-particle contacts, leading to shallower breakthrough curves. In contrast, the fast uptake rates at high ionic strength appear to reduce the importance of mass transfer limitations at the particle contacts, but the confocal results do show a flow rate dependence on the uptake profiles, suggesting that external mass transfer becomes more limiting at high ionic strength. These results show that the complexity of behavior observable at the microscopic scale is directly manifested at the column scale and provides a phenomenological basis to interpret and predict column breakthrough. In addition, the results provide heuristics for the optimization of chromatographic conditions.

Nondiffusive mechanisms enhance protein uptake rates in ion exchange particles

Scanning confocal fluorescence microscopy and multiphoton fluorescence microscopy were used to image the uptake of the protein lysozyme into individual ion exchange chromatography particles in a packed bed in real time. Self-sharpening concentration fronts penetrating into the particles were observed at low salt concentrations in all of the adsorbents studied, but persisted to 100 mM ionic strength only in some materials. In other adsorbents, diffuse profiles were seen at these higher salt concentrations, with the transition region exhibiting a pronounced fluorescence peak at the front at intermediate salt concentrations. These patterns in the uptake profiles are accompanied by significant increases in protein uptake rates that are also seen macroscopically in batch uptake experiments. The fluorescence peak appears to be a concentration overshoot that may develop, in part, from an electrokinetic contribution to transport that also enhances the uptake rate. Further evidence for an electrokinetic origin is that the effect is correlated with high adsorbent surface charge densities. Predictions of a mathematical model incorporating the electrokinetic effect are in qualitative agreement with the observations. These findings indicate that mechanisms other than diffusion contribute to protein transport in oppositely charged porous materials and may be exploited to achieve rapid uptake in process chromatography.

Determinants of protein retention characteristics on cation-exchange adsorbents

There are currently a large number of commercially available strong and weak cation-exchange adsorbents for preparative protein purification, typically prepared by coupling charged ligands to a mechanically rigid porous bead. Because of the diverse chemical nature of the base matrix (carbohydrate, synthetic polymer, inorganic) and the coupling and ligand chemistry, cation-exchange adsorbents from different suppliers can differ substantially in chemical surface properties and physical structure. The differences in chemical properties can be in ionic capacity, hydrophobicity, the presence of hydrogen bond donors/acceptors, and the nature of the charged functional groups. In order to probe the effects of these factors on protein affinity, the isocratic retention of a set of model proteins was examined on a set of cation-exchange adsorbents to obtain a quantitative assessment of retention differences between adsorbents. Two adsorbent factors were found to be the dominant determinants of overall protein retention: the anion type and the adsorbent pore size distribution. Protein retention on strong cation-exchangers was found to be greater than that on corresponding weak cation-exchangers. Protein retention was increased on adsorbents with pore size distributions that include significant amounts of pore space with dimensions similar to those of the protein solute.

Correlation between the osmotic second virial coefficient and the solubility of proteins

A correlation between the osmotic second virial coefficient and the solubility of proteins is derived from classical thermodynamics to support an empirical relation previously found by Wilson and co-workers (1). The model is based on the equality of fugacities of the protein in the equilibrium phases, with the details of the model depending on the standard state used. The parameters in this model have been fitted to data for several systems, mainly with lysozyme as the protein. The model is found to describe experimental data, with variations in protein concentration, salt type and concentration, temperature, and pH, both qualitatively and quantitatively. Agreement between the model and the experimental data is very good for protein solubilities up to 30 mg/mL. Above this value the model underpredicts the experimental data, probably as a result of multibody interactions that are not included in the model here. Variations of the model parameters with protein type, temperature, pH, and salt type are discussed.

Phase equilibria in the lysozyme-ammonium sulfate-water system

Ternary phase diagrams were measured for lysozyme in ammonium sulfate solutions at pH values of 4 and 8. Lysozyme, ammonium sulfate, and water mass fractions were assayed independently by UV spectroscopy, barium chloride titration, and lyophilization respectively, with mass balances satisfied to within 1%. Protein crystals, flocs, and gels were obtained in different regions of the phase diagrams, and in some cases growth of crystals from the gel phase or from the supernatant after floc removal was observed. These observations, as well as a discontinuity in protein solubility between amorphous floc precipitate and crystal phases, indicate that the crystal phase is the true equilibrium state. The ammonium sulfate was generally found to partition unequally between the supernatant and the dense phase, in disagreement with an assumption often made in protein phase equilibrium studies. The results demonstrate the potential richness of protein phase diagrams as well as the uncertainties resulting from slow equilibration.

Measured and calculated effects of mutations in bacteriophage T4 lysozyme on interactions in solution

Understanding the molecular determinants of protein interactions in solution has fundamental implications for understanding protein solution thermodynamics and, hence, processes as diverse as separations performance and cellular self-organization. Our earlier theoretical calculations indicate that the protein-protein interactions are dominated by a small number of configurations in which highly complementary surface regions are apposed, rather than by the overall colloidal interactions. To examine this paradigm more explicitly, we investigated the effects of protein structural modifications on protein-protein interactions. Experimental measurements are presented of B(22)(') values of a set of mutants of Ser44 in bacteriophage T4 lysozyme. Effects are seen with both charged and uncharged substitutions. The results with the charged substitutions follow the expected trends, whereas those with the uncharged substitutions may be explained by the impact of the mutations on the local protein geometry, which directly affects the complementarity of protein interactions. These effects are also captured well by molecular calculations that account for the mutations. The interaction energetics between protein pairs could provide information on the propensity for adventitious interactions, which can have important implications for separations and for normal and pathological self-assembly. Thus, protein structural data implicit in genomic information, coupled with appropriate calculational and experimental tools, can ultimately provide insights into protein interactions in vivo and in bioprocessing.

Protein crystallization by design: chymotrypsinogen without precipitants

Protein crystals are usually obtained by an empirical approach based on extensive screening to identify suitable crystallization conditions. In contrast, we have used a systematic predictive procedure to produce data-quality crystals of bovine chymotrypsinogen A and used them to obtain a refined X-ray structure to 3 A resolution. Measurements of the osmotic second virial coefficient of chymotrypsinogen solutions were used to identify suitable solvent conditions, following which crystals were grown for approximately 30 hours by ultracentrifugal crystallization, without the use of any precipitants. Existing structures of chymotrypsinogen were obtained in solutions including 10-30 % ethanol, whereas simple buffered NaCl solutions were used here. The protein crystallized in the tetragonal space group P4(1)2(1)2, with one molecule per asymmetric unit. The quality of the refined map was very high throughout, with the main-chain atoms of all but four residues clearly defined and with nearly all side-chains also defined. Although only minor differences are seen compared to the structures previously reported, they indicate the possibility of structural changes due to the crystallization conditions used in those studies. Our results show that more systematic crystallization of proteins is possible, and that the procedure can expand the range of conditions under which crystals can be grown successfully and can make new crystal forms available.

Pore size distributions of cation-exchange adsorbents determined by inverse size-exclusion chromatography

The pore dimensions, pore size distributions, and phase ratios were determined for a set of cation-exchange adsorbents using inverse size-exclusion chromatography (ISEC). The adsorbents examined represent a diverse set of materials from Pharmacia, TosoHaas, BioSepra, and EM Industries, which are widely used for protein purification. The ISEC was carried out using dextran standards with relative molecular masses of 180-6,105,000. This technique provided a comparative characterization of the accessible internal pore surface area, as a function of solute size, for the adsorbents tested. Adsorbent preparation strategies in which polymers are generated in situ or grafted onto base materials were found to have significant effects on pore dimensions and phase ratios.

A class of microstructured particles through colloidal crystallization

Microstructured particles were synthesized by growing colloidal crystals in aqueous droplets suspended on fluorinated oil. The droplets template highly ordered and smooth particle assemblies, which diffract light and have remarkable structural stability. The method allows control of particle size and shape from spheres through ellipsoids to toroids by varying the droplet composition. Cocrystallization in colloidal mixtures yields anisotropic particles of organic and inorganic materials that can, for example, be oriented and turned over by magnetic fields. The results open the way to controllable formation of a wide variety of microstructures.

Adsorbed Layers of Ferritin at Solid and Fluid Interfaces Studied by Atomic Force Microscopy

The adsorption of the iron storage protein ferritin was studied by liquid tapping mode atomic force microscopy in order to obtain molecular resolution in the adsorbed layer within the aqueous environment in which the adsorption was carried out. The surface coverage and the structure of the adsorbed layer were investigated as functions of ionic strength and pH on two different charged surfaces, namely chemically modified glass slides and mixed surfactant films at the air-water interface, which were transferred to graphite substrates after adsorption. Surface coverage trends with both ionic strength and pH indicate the dominance of electrostatic effects, with the balance shifting between intermolecular repulsion and protein-surface attraction. The resulting behavior is more complex than that seen for larger colloidal particles, which appear to follow a modified random sequential adsorption model monotonically. The structure of the adsorbed layers at the solid surfaces is random, but some indication of long-range order is apparent at fluid interfaces, presumably due to the higher protein mobility at the fluid interface. Copyright 2000 Academic Press.

Comparison of protein adsorption isotherms and uptake rates in preparative cation-exchange materials

Adsorption isotherms and effective diffusivities of lysozyme in a set of six preparative cation-exchange stationary phases were determined from batch uptake data in a stirred vessel. Both a pore diffusion and a homogeneous diffusion model were used in estimating diffusivities, with the isotherms fitted to a non-Langmuirian analytical isotherm equation. The capacities inferred from the isotherms are found to be correlated with the surface area accessible to lysozyme, the effective surface concentrations obtained being in agreement with values measured by different methods in various non-chromatographic systems. The pore diffusivities show systematic trends with protein and salt concentration, and effects of pore size and connectivity are also evident. Some trends in the homogeneous diffusivities are quite different to those in the pore diffusivities, but these differences largely disappear when the homogeneous diffusivities are rescaled to account for adsorption equilibrium behavior. Additional information is required to elucidate further the mechanisms of coupled diffusion and adsorption in stationary phases.

Protein interactions in solution characterized by light and neutron scattering: comparison of lysozyme and chymotrypsinogen

The effects of pH and electrolyte concentration on protein-protein interactions in lysozyme and chymotrypsinogen solutions were investigated by static light scattering (SLS) and small-angle neutron scattering (SANS). Very good agreement between the values of the virial coefficients measured by SLS and SANS was obtained without use of adjustable parameters. At low electrolyte concentration, the virial coefficients depend strongly on pH and change from positive to negative as the pH increases. All coefficients at high salt concentration are slightly negative and depend weakly on pH. For lysozyme, the coefficients always decrease with increasing electrolyte concentration. However, for chymotrypsinogen there is a cross-over point around pH 5.2, above which the virial coefficients decrease with increasing ionic strength, indicating the presence of attractive electrostatic interactions. The data are in agreement with Derjaguin-Landau-Verwey-Overbeek (DLVO)-type modeling, accounting for the repulsive and attractive electrostatic, van der Waals, and excluded volume interactions of equivalent colloid spheres. This model, however, is unable to resolve the complex short-ranged orientational interactions. The results of protein precipitation and crystallization experiments are in qualitative correlation with the patterns of the virial coefficients and demonstrate that interaction mapping could help outline new crystallization regions.

Molecular origins of osmotic second virial coefficients of proteins

The thermodynamic properties of protein solutions are determined by the molecular interactions involving both solvent and solute molecules. A quantitative understanding of the relationship would facilitate more systematic procedures for manipulating the properties in a process environment. In this work the molecular basis for the osmotic second virial coefficient, B22, is studied; osmotic effects are critical in membrane transport, and the value of B22 has also been shown to correlate with protein crystallization behavior. The calculations here account for steric, electrostatic, and short-range interactions, with the structural and functional anisotropy of the protein molecules explicitly accounted for. The orientational dependence of the protein interactions is seen to have a pronounced effect on the calculations; in particular, the relatively few protein-protein configurations in which the apposing surfaces display geometric complementarity contribute disproportionately strongly to B22. The importance of electrostatic interactions is also amplified in these high-complementarity configurations. The significance of molecular recognition in determining B22 can explain the correlation with crystallization behavior, and it suggests that alteration of local molecular geometry can help in manipulating protein solution behavior. The results also have implications for the role of protein interactions in biological self-organization.

2-D and 3-D Interactions in Random Sequential Adsorption of Charged Particles

We explore the influence of electrolyte concentration on the adsorption of charged spheres using modeling techniques based on random sequential adsorption (RSA). We present a parametric study of the effects of double layer interactions between the charged particles and between the particle and the substrate on the jamming limit using a two-dimensional RSA simulation similar to that of Z. Adamczyk et al. (1990, J. Colloid Interface Sci. 140, 123) along with a simple method of estimating jamming limit coverages. In addition, we present a more realistic RSA algorithm that includes explicit energetic interaction in three dimensions, that is, particle-particle and particle-surface interactions during the approach of a particle to the substrate. The calculation of interaction energies in the 3-D RSA model is achieved with the aid of a three-body superposition approximation. The 3-D RSA model differs from the 2-D model in that the extent of coverage is controlled by kinetic rather than energetic considerations. Results of both models capture the experimentally observed trend of increased surface coverage with increased electrolyte concentration, and both models require the value of a key model parameter to be specified for a quantitative match to experimental data. However, the 3-D model more effectively captures the governing physics, and the parameter in this case takes on more meaningful values than for the 2-D model. Copyright 1997 Academic Press. Copyright 1997Academic Press

Van der Waals interactions involving proteins

Van der Waals (dispersion) forces contribute to interactions of proteins with other molecules or with surfaces, but because of the structural complexity of protein molecules, the magnitude of these effects is usually estimated based on idealized models of the molecular geometry, e.g., spheres or spheroids. The calculations reported here seek to account for both the geometric irregularity of protein molecules and the material properties of the interacting media. Whereas the latter are found to fall in the generally accepted range, the molecular shape is shown to cause the magnitudes of the interactions to differ significantly from those calculated using idealized models, with important consequences. First, the roughness of the molecular surface leads to much lower average interaction energies for both protein-protein and protein-surface cases relative to calculations in which the protein molecule is approximated as a sphere. These results indicate that a form of steric stabilization may be an important effect in protein solutions. Underlying this behavior is appreciable orientational dependence, one reflection of which is that molecules of complementary shape are found to exhibit very strong attractive dispersion interactions. Although this has been widely discussed previously in the context of molecular recognition processes, the broader implications of these phenomena may also be important at larger molecular separations, e.g., in the dynamics of aggregation, precipitation, and crystal growth.

Computation of the electrophoretic mobility of proteins

A scheme is presented for computing the electrophoretic mobility of proteins in free solution, accounting for the details of the protein shape and charge distribution. The method of Teubner is implemented using a boundary integral formulation within which the velocity distribution, the equilibrium electrical potential around the molecule, and the potential distribution due to the applied field are solved for numerically using the boundary element method. Good agreement of the numerical result is obtained for spheres with the corresponding semi-analytical specialization of Henry's analysis. For protein systems, the method is applied to lysozyme and ribonuclease A. In both cases, the predicted mobility tensors are fairly isotropic, with the resulting scalar mobilities being significantly smaller than for spheres of equal volume and net charge. Comparisons with previously published experimental results for ribonuclease show agreement to be excellent in the presence of a net charge, but poorer at the point of zero charge. The approach may be useful for evaluating approximate methods for estimating protein electrophoretic mobilities and for using electrophoretic measurements to obtain insight into charge distributions on proteins.

Analysis of ordered arrays of adsorbed lysozyme by scanning tunneling microscopy

Scanning tunneling microscopy (STM) has been used to observe lysozyme at a graphite surface directly in order to gain mechanistic information about the molecular events involved in protein adsorption. The experiments were performed using an insulated tip in an aqueous protein solution, allowing the time course of the adsorption process to be followed, including the evolution of ordered arrays. Ordered arrays of protein molecules were observed, with lattice spacings that varied with bulk protein concentration and salt strength. Fourier analysis was used to determine the average cell dimensions of an array. From the observed lattice spacings, it was possible to estimate the surface coverage of the protein, and thus, by varying the conditions, adsorption isotherms could be obtained. These isotherms compare well with adsorption isotherms measured using total internal reflectance fluorescence (TIRF) spectroscopy on a hydrophobic surface. Since the protein is charged and the electrolyte has an effect on the isotherms, electrostatics are a likely controlling factor. Molecular electrostatics computations were thus used to investigate the possible origins of the lattice structure, and they suggest that favorable intermolecular interactions among adsorbed molecules are consistent with hydrophobically dominated protein-surface interactions.

STM and AFM in biotechnology

Scanning tunneling microscopy (STM) and atomic force microscopy (AFM) are capable of providing atomic-scale images, as well as insights into functional characteristics, of surfaces. Biological materials can be studied by depositing them on appropriate substrates, and samples have been imaged with varying degrees of resolution. In this review, the principles of STM and AFM are summarized, and practical aspects of implementing them for studies relevant to biotechnology are discussed, with the emphasis on investigations of biological macromolecules such as DNA and proteins. Items covered include samples, surfaces, medium, tips, and typical operating conditions. Finally, prospects are discussed for potential future applications to biotechnology.

Hydrogen washout in bone cortex and periosteum

Residence time distributions of hydrogen in bone of anesthetized dogs and rabbits were used to estimate local blood perfusion rates and to characterize the important transport processes taking place. The hydrogen was administered by inhalation, and the concentrations in the bone were measured by embedded platinum microelectrodes. Mean residence times varied significantly both with position and time, and it was found preferable to calculate residence time from moments of the residence time distribution rather than the downslope method. Moreover, the downslope on a semilogarithmic scale continued to decrease with the increase in observation time. For the tissue investigated, simple compartmental models are inadequate even for the small regions characterized by the electrodes. This means that a large number of Haldanian compartments are needed even to characterize local washout behavior. The significance of this finding for the selection of decompression schedules is briefly discussed.

The effects of axial diffusion and permeability barriers on the transient response of tissue cylinders. II. Solution in time domain

A mathematical description of transient mass transfer in a Krogh tissue cylinder, for which a solution in transform space was presented previously, is solved in the time domain. The solution is found in the form of an expansion in terms of the eigenfunctions of a non-self-adjoint differential operator, with the eigenvalues being found by way of a computational scheme which makes use of the known characteristics of the constitutive compartments of the system. The solution is compared with previous solutions of both a complete and an approximate nature, and two modifications of the single-phase axial dispersion model are found to be especially useful: the previously-used flow-limited approximation is satisfactory for highly permeable solutes, while the apparently novel barrier-limited approximation is accurate for poorly permeable solutes at the early times of most experimental interest. Although the neglect of axial diffusion does not affect the qualitative nature of the solution, e.g. in predicting a bimodal response curve, significant discrepancies shed doubt on this practice when truly impulsive inputs are used. The results obtained raise several questions regarding existing approaches to interpretation of indicator dilution experiments. These include the use of extraction ratios and of exponential extrapolation of the tails of response curves.

Microcirculatory mass transfer

The dynamics of Krogh tissue cylinders and related structures are critically reviewed to determine the roles of underlying transport and reaction processes, and the interactions between them. Emphasis is put on the gaining of insight through efficient scaling procedures, and discussion is organized about the time constants characteristic of these structures and the processes occurring in them. The basis of discussion is a new analytic solution technique which provides a formal description of indefinitely large arrays of parallel interacting elements and which includes both axial diffusion and uniform convection as well as first or zero order reaction within each element. The solution has the form of an expansion in the eigenfunctions of a non-self-adjoint differential operator. Comparison of model predictions with previously available results identifies the useful parameter ranges of analytic approximations and determines the accuracy of existing numerical procedures.