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.     

Theoretical study of flow,pressure and solute concentration in double membrane systems

A model system consisting of two rigidly held membranes in series was investigated through the application of the Kedem and Katchalsky thermodynamic single membrane flow equations. This analysis results in predictions of the steady state flow properties as well as values for the solute concentration and pressure of the internal compartment when the system is under the influence of a constant solute concentration or hydrostatic pressure gradient. It is demonstrated that although the flow properties and internal compartment pressure are complicated functions of the membrane permeability coefficients and driving gradient across the system, the relationships are greatly simplified by the explicit appearance of the internal compartment steady state solute concentration in the equations. It is shown that the steady state volume flow rate depends on the absolute value of the solute concentration in the external compartments, as well as the solute concentration gradient across the system. The properties of non-linear dependence of volume flow on concentration gradient, and rectification of volume flow are discussed and shown to be independent properties of the system. For the system under the influence of a solute concentration gradient, the internal compartment pressure can be greater or less than the ambient pressure, and depends mainly on the order in which the membranes are encountered by the volume flow. These properties are qualitatively correlated with certain available experimental observations in biological systems.     

Selective extraction using preferential transport through adsorptive membranes

Selective extraction of a protein from a mixture can be accomplished using an adsorptive membrane and low displacement recuperative parametric pumping. Low displacement recuperative parametric pumping can lead to the preferential transport of an adsorbing solute and the rejection of nonadsorbing solutes by the adsorptive membrane. Using a protein mixture consisting of lysozyme and myoglobin, we have found the conditions under which lysozyme is preferentially transported through an ion-exchange membrane cartridge while myoglobin is rejected by the membrane. Trends observed when parameters such as the desorbent concentration, feed concentration, and flow rate are varied agree with the predictions of a mathematical model. Comparison with facilitated diffusion shows that preferential transport can lead to higher solute fluxes, albeit at lower selectivity. Additionally, preferential transport can be used to transport a solute up a concentration gradient and to selectively extract a solute from a feed that contains suspended solids. (c) 1996 John Wiley & Sons, Inc.     

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.     

The osmotic migration of cells in a solute gradient.

The effect of a nonuniform solute concentration on the osmotic transport of water through the boundaries of a simple model cell is investigated. A system of two ordinary differential equations is derived for the motion of a single cell in the limit of a fast solute diffusion, and an analytic solution is obtained for one special case. A two-dimensional finite element model has been developed to simulate the more general case (finite diffusion rates, solute gradient induced by a solidification front). It is shown that the cell moves to regions of lower solute concentration due to the uneven flux of water through the cell boundaries. This mechanism has apparently not been discussed previously. The magnitude of this effect is small for red blood cells, the case in which all of the relevant parameters are known. We show, however, that it increases with cell size and membrane permeability, so this effect could be important for larger cells. The finite element model presented should also have other applications in the study of the response of cells to an osmotic stress and for the interaction of cells and solidification fronts. Such investigations are of major relevance for the optimization of cryopreservation processes.     

Determination of microvessel permeability and tissue diffusion coefficient of solutes by laser scanning confocal microscopy

Interstitium contains a matrix of fibrous molecules that creates considerable resistance to water and solutes in series with the microvessel wall. On the basis of our preliminary studies, by using laser-scanning confocal microscopy and a theoretical model for interstitial transport, we determined both microvessel solute permeability (P) and solute tissue diffusion coefficient (D) of alpha-lactalbumin (Stokes radius 2.01 nm) from the rate of tissue solute accumulation and the radial concentration gradient around individually perfused microvessel in frog mesentery. P(alpha-lactalbumin) is 1.7 +/- 0.7(SD) x 10(-6) cm/s (n = 6). D(t)/D(free) for alpha-lactalbumin is 27% +/- 5% (SD) (n = 6). This value of D(t)/D(free) is comparable to that for small solute sodium fluorescein (Stokes radius 0.45 nm), while p(alpha-lactalbumin) is only 3.4% of p(sodium fluorescein). Our results suggest that frog mesenteric tissue is much less selective to solutes than the microvessel wall.     

Molecular modeling and simulation of Mycobacterium tuberculosis cell wall permeability

The low permeability of the mycobacterial cell wall is thought to contribute to the intrinsic drug resistance of mycobacteria. In this study, the permeability of the Mycobacterium tuberculosis cell wall is studied by computer simulation. Thirteen known drugs with diverse chemical structures were modeled as solutes undergoing transport across a model for the M. tuberculosis cell wall. The properties of the solute-membrane complexes were investigated by means of molecular dynamics simulation, especially the diffusion coefficients of the solute molecules inside the cell wall. The molecular shape of the solute was found to be an important factor for permeation through the M. tuberculosis cell wall. Predominant lateral diffusion within, as opposed to transverse diffusion across, the membrane/cell wall system was observed for some solutes. The extent of lateral diffusion relative to transverse diffusion of a solute within a biological cell membrane may be an important finding with respect to absorption distribution, metabolism, elimination, and toxicity properties of drug candidates. Molecular similarity measures among the solutes were computed, and the results suggest that compounds having high molecular similarity will display similar transport behavior in a common membrane/cell wall environment. In addition, the diffusion coefficients of the solute molecules across the M. tuberculosis cell wall model were compared to those across the monolayers of dipalmitoylphosphatidylethanolamine and dimyristoylphosphatidylcholine, are two common phospholipids in bacterial and animal membranes. The differences among these three groups of diffusion coefficients were observed and analyzed.     

Coupled water transport in standing gradient models of the lateral intercellular space.

A standing gradient model of the lateral intercellular space is presented which includes a basement membrane of finite solute permeability. The solution to the model equations is estimated analytically using the "isotonic convection approximation" of Segel. In the case of solute pumps uniformly distributed along the length of the channel, the achievement of isotonic transport depends only on the water permeability of the cell membranes. The ability of the model to transport water against an adverse osmotic gradient is the sum of two terms: The first term is simply that for a well-stirred compartment model and reflects basement membrane solute permeability. The second term measures the added strength due to diffusion limitation within the interspace. It is observed, however, that the ability for uphill water transport due to diffusion limitation is diminished by high cell membrane water permeability. For physiologically relevant parameters, it appears that the high water permeability required for isotonic transport renders the contribution of the standing gradient relatively ineffective in transport against an osmotic gradient. Finally, when the model transports both isotonically and against a gradient, it is shown that substantial intraepithelial solute polarization effects are unavoidable. Thus, the measured epithelial water permeability will grossly underestimate the water permeability of the cell membranes. The accuracy of the analytic approximation is demonstrated by numerical solution of the complete model equations.     

Dynamic loading of deformable porous media can induce active solute transport

Active solute transport mediated by molecular motors across porous membranes is a well-recognized mechanism for transport across the cell membrane. In contrast, active transport mediated by mechanical loading of porous media is a non-intuitive mechanism that has only been predicted recently from theory, but not yet observed experimentally. This study uses agarose hydrogel and dextran molecules as a model experimental system to explore this mechanism. Results show that dynamic loading can enhance the uptake of dextran by a factor greater than 15 over passive diffusion, for certain combinations of gel concentration and dextran molecular weight. Upon cessation of loading, the concentration reverts back to that achieved under passive diffusion. Thus, active solute transport in porous media can indeed be mediated by cyclical mechanical loading.     

Concentrating Engines and the Kidney: II. Multisolute Central Core Systems

The analysis of the central core model of the renal medulla is extended to multisolute systems. It is shown that total solute concentration obeys the same differential equations for core and ascending limb as in a single solute system. Equations are derived for the concentration of individual solutes. Application of these equations to a two solute system shows that a central core system can concentrate with all transport being down a concentration gradient. This analysis applied to the renal medulla shows that mixing of urea from the collecting duct (CD) and salt from the loop of Henle in the central core of the inner medulla contributes to the concentration of urine during antidiuresis. It also sets criteria for completely passive function of the loop in the inner medulla, but whether these are satisfied cannot be determined from present experimental data.     

On the volume-flow mechanism of phloem transport

Summary A steady-state model of solution flow in a tubular semipermeable membrane is developed for an arbitrary distribution of solute sources and sinks along the translocation path. It is demonstrated that the volume-flow mechanism of phloem transport depends only on the two assumptions: 1. that the plasmalemma of the sieve tube is a differentially permeable membrane, and 2. that sugars are actively secreted into and absorbed from the lumen of the sieve tube. It is shown that in the absence of a pressure gradient, there is a negligible concentration gradient over most of the translocation path. However, in the presence of a pressure gradient a small concentration gradient develops as a result of the continually changing chemical potential of water along the direction of solution flow. For Poiseuille flow the concentration gradient is approximately proportional to the mean stream velocity.     

Diffusion in colloidal media

When the molecules of a solute diffuse through a medium containing large colloidal particles, which absorb the diffusing molecules, the latter are transported in the diffusion flow not as free molecules, but as absorbtion compounds: solute+colloid. When the colloidal particle is much larger than the molecule of the solute, and has therefore a much smaller mobility, this results in a reduction of the apparent diffusion coefficient for the solute. The biological implications of this are discussed.     

The steady state kinetics of some biological systems: IV. Thermodynamic aspects

The investigation initiated in III is continued. The irreversible production of entropy and the rates of dissipation in diffusion fields and due to chemical reactions are discussed. The rates of change of free energy, entropy, etc. are indicated for type-systems of biological interest in which composition and reaction rates are a function of position. Some consequences of Onsager's generalization of Fick's Law are discussed in terms of maintenance of stationary, non-equilibrium concentration distributions, transport of a solute against a concentration gradient, and the dependence of these phenomena upon metabolism.     

Fluid flow and convective transport of solutes within the intervertebral disc

Previous experimental and analytical studies of solute transport in the intervertebral disc have demonstrated that for small molecules diffusive transport alone fulfils the nutritional needs of disc cells. It has been often suggested that fluid flow into and within the disc may enhance the transport of larger molecules. The goal of the study was to predict the influence of load-induced interstitial fluid flow on mass transport in the intervertebral disc.An iterative procedure was used to predict the convective transport of physiologically relevant molecules within the disc. An axisymmetric, poroelastic finite-element structural model of the disc was developed. The diurnal loading was divided into discrete time steps. At each time step, the fluid flow within the disc due to compression or swelling was calculated. A sequentially coupled diffusion/convection model was then employed to calculate solute transport, with a constant concentration of solute being provided at the vascularised endplates and outer annulus. Loading was simulated for a complete diurnal cycle, and the relative convective and diffusive transport was compared for solutes with molecular weights ranging from 400 Da to 40 kDa.Consistent with previous studies, fluid flow did not enhance the transport of low-weight solutes. During swelling, interstitial fluid flow increased the unidirectional penetration of large solutes by approximately 100%. Due to the bi-directional temporal nature of disc loading, however, the net effect of convective transport over a full diurnal cycle was more limited (30% increase). Further study is required to determine the significance of large solutes and the timing of their delivery for disc physiology.     

Systems approach to the study of drug transport across membranes using suspension cultures of mammalian cells V. Simultaneous passive transport and biosynthesis

A physical model is described for the simultaneous enzymatic bioconversion of a nonelectrolyte solute and the passive transport of both the solute and product of the enzymatic reaction out of cells in culture suspension. The plasma membrane is assumed to be the rate-determining transport barrier. This model provides the basis for the experimental design and analysis of the Michaelis-Menten kinetic parameters of simple enzymatic reactions in situ, the phenomenological transport parameters and other factors. The primary set of differential equations describing the quasisteady state rate of change in the concentration of the solute and product within the cell due to enzyme reaction and transport are given. These are nonlinear and must be solved by numerical methods. However, analytical mathematical expressions have been derived for various cases in the limit when the rate of enzymatic reaction is first or zero order.     

Scaling phloem transport: information transmission

Sieve tubes are primarily responsible for the movement of solutes over long distances, but they also conduct information about the osmotic state of the system. Using a previously developed dimensionless model of phloem transport, the mechanism behind the sieve tube's capacity to rapidly transmit pressure/concentration waves in response to local changes in either membrane solute exchange or the magnitude and axial gradient of apoplastic water potential is demonstrated. These wave fronts can move several orders of magnitude faster than the solution itself when the sieve tube's axial pressure drop is relatively small. Unlike the axial concentration drop, the axial pressure drop at steady state is independent of the apoplastic water potential gradient. As such, the regulation of whole‐sieve tube turgor could play a vital role in controlling membrane solute exchange throughout the translocation pathway, making turgor a reliable source of information for communicating change in system state.     

Cytoplasmic permeation pathway of neurotransmitter transporters

Ion-coupled solute transporters are responsible for transporting nutrients, ions, and signaling molecules across a variety of biological membranes. Recent high-resolution crystal structures of several transporters from protein families that were previously thought to be unrelated show common structural features indicating a large structural family representing transporters from all kingdoms of life. This review describes studies that led to an understanding of the conformational changes required for solute transport in this family. The first structure in this family showed the bacterial amino acid transporter LeuT, which is homologous to neurotransmitter transporters, in an extracellularly oriented conformation with a molecule of leucine occluded at the substrate site. Studies with the mammalian serotonin transporter identified positions, buried in the LeuT structure, that defined a potential pathway leading from the cytoplasm to the substrate binding site. Modeling studies utilized an inverted structural repeat within the LeuT crystal structure to predict the conformation of LeuT in which the cytoplasmic permeation pathway, consisting of positions identified in SERT, was open for diffusion of the substrate to the cytoplasm. From the difference between the model and the crystal structures, a simple "rocking bundle" mechanism was proposed, in which a four-helix bundle changed its orientation with respect to the rest of the protein to close the extracellular pathway and open the cytoplasmic one. Subsequent crystal structures from structurally related proteins provide evidence supporting this model for transport.     

Fractionation of macromolecules in an alternating transverse electric field: simulation of the method

An electric field of alternating polarity applied in a direction transverse to the direction of solute transport is used as the basis of a method for the separation of biological macromolecules. The method derives directly from the ability of an electric field to induce movement of a charged macromolecule and from the physics of laminar fluid flow; no adsorptive immobile phase component is involved.

The method is simulated by computer for the case of solute molecules in a solvent flowing through a narrow chamber of recta generates an electric field orthogonal to the direction of solvent flow. Solute molecules repetitively traverse the solvent channel at rates determined by their electrophoretic mobility. During the transit across the channel, solute molecules are transported in the direction of solvent flow; at the channel wall, solvent velocity is negligible and solute transport is limited to that provided by transient diffusion into a mobile solvent zone. Molecules of different intrinsic electrophoretic mobility are separated.

The computer model was used to illustrate the process and to demonstrate the ‘tunability’ of the method as a function of the oscillation frequency and voltage wave form. Because of this tunability, a single instrument can function as the equivalent of several different chromatographic systems. Because fractionation is effected by direct physicochemical phenomena rather than via interaction with chromatographic sites, variations in fractionation results arising from formation of polymers for gel electrophoresis, packing of chromatography columns, or deterioration of columns with use are avoided. This method may be of particular use for the purification of nucleic acid fragments and for the analysis of protei: nucleic acid interactions.     

Analysis of proximal tubule salt and water transport in standing droplets.

A mathematical model has been developed describing solute and water movement in the renal proximal tubule standing droplet experiment. The model explicitly incorporates the constraint of isosmotic reabsorption. Solute asymmetry due to the unequal distribution of protein, bicarbonate and other solutes between plasma and the standing droplet is shown to be one of the major reabsorptive forces; however, the introduction of an additional reabsorptive mechanism into the system equations is required in order to obtain a quantitative fit with experimental observations. The model demonstrates that limiting concentration gradients can be obtained in the absence of active transport and that their magnitudes will vary inversely with the permeability of the poorly permeant solute. Conversely, situations can occur where active transport will not elicit a limiting gradient. Consequently, previous interpretations of the meaning of limiting gradients and their magnitudes need to be reconsidered. The model further predicts that the technique for measuring non-electrolyte permeability using standing droplet experiments is likely to underestimate the true permeability. Finally, it is shown that a previous theoretical model of standing droplets, which does not explicitly include the constraint of isomotic reabsorption, cannot fit experimental data from proximal tubules.     

Non-polar solutes in water and in aqueous solutions of protein denaturants. Modeling of solution and transfer processes

A simple molecular model for the thermodynamic behavior of non-polar solutes in water and in aqueous solutions of protein denaturants is presented. Three contributions are considered: (i) combinatorial arising from the mixing process, (ii) interactional characterizing the molecular interactions occurring in the mixture and (iii) a contribution originating from the structural changes occurring in the first shell of water molecules around the solute. The latter is modeled assuming that water molecules in contact with the solute are involved in a chemical equilibrium between two states. The model describes well the temperature and denaturant concentration dependences of the Gibbs energies of solution and transfer for benzene, toluene and alkanes in water and aqueous solutions of urea and guanidine hydrochloride. Model parameters are physically meaningful, allowing a discussion of the molecular interactions involved. A preferential solvation of the solute by the denaturant is found. However, the non-polar solute-denaturant interaction is not specific, i.e. leading to a distinct chemical entity. Urea and guanidine hydrochloride are non-polar solubilizing agents because their interactions with the solute are less unfavorable than those between water and the solute.