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.     

Thermodynamic Model Equations for Heterogeneous Multicomponent Non-Ionic Solution Transport in a Multimembrane System

Non-equilibrium thermodynamic model equations for non-ionic and heterogeneous n-component solution transport in a m-membrane system are presented. This model is based on two equations. The first one describes the volume transport of the solution and the second the transport of the solute. Definitions of the hydraulic permeability, reflection and diffusive permeability coefficients of the m-membrane system and relations between the coefficients of the m-membrane system and the respective membranes of the system are also given. The validity of this model for binary and ternary solutions was verified, using a double-membrane cell with a horizontally mounted membrane. In the cell, volume and solute fluxes were measured as a function of concentration and gravitational configuration.     

Physiological significance of the structure and components of the apoplast canal system for water absorption in plants

The non-linear differential equation that describes the coupling between water transport and solute transport in the apoplast canal system in plants was proposed by Katou and Furumoto in 1986. In the present paper, we analytically solved the equation in order to find the law describing the canal system. In the canal system, water transport is regulated linearly by solute transport under physiological conditions. The approximate solution of the differential equations defines the conditions of the structure and components of the apoplast canal for optimal water absorption. Water absorption during cell elongation in plants requires that the apoplast canal be composed of a cell wall with an appropriate diffusion coefficient for solute.     

Osmotic Transport across Cell Membranes in Nondilute Solutions: A New Nondilute Solute Transport Equation

The fundamental physical mechanisms of water and solute transport across cell membranes have long been studied in the field of cell membrane biophysics. Cryobiology is a discipline that requires an understanding of osmotic transport across cell membranes under nondilute solution conditions, yet many of the currently-used transport formalisms make limiting dilute solution assumptions. While dilute solution assumptions are often appropriate under physiological conditions, they are rarely appropriate in cryobiology. The first objective of this article is to review commonly-used transport equations, and the explicit and implicit assumptions made when using the two-parameter and the Kedem-Katchalsky formalisms. The second objective of this article is to describe a set of transport equations that do not make the previous dilute solution or near-equilibrium assumptions. Specifically, a new nondilute solute transport equation is presented. Such nondilute equations are applicable to many fields including cryobiology where dilute solution conditions are not often met. An illustrative example is provided. Utilizing suitable transport equations that fit for two permeability coefficients, fits were as good as with the previous three-parameter model (which includes the reflection coefficient, σ). There is less unexpected concentration dependence with the nondilute transport equations, suggesting that some of the unexpected concentration dependence of permeability is due to the use of inappropriate transport equations.     

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.     

A closed‐form solution for steady‐state coupled phloem/xylem flow using the Lambert‐W function

A closed‐form solution for steady‐state coupled phloem/xylem flow is presented. This incorporates the basic Münch flow model of phloem transport, the cohesion model of xylem flow, and local variation in the xylem water potential and lateral water flow along the transport pathway. Use of the Lambert‐W function allows this solution to be obtained under much more general and realistic conditions than has previously been possible. Variation in phloem resistance (i.e. viscosity) with solute concentration, and deviations from the Van't Hoff expression for osmotic potential are included. It is shown that the model predictions match those of the equilibrium solution of a numerical time‐dependent model based upon the same mechanistic assumptions. The effect of xylem flow upon phloem flow can readily be calculated, which has not been possible in any previous analytical model. It is also shown how this new analytical solution can handle multiple sources and sinks within a complex architecture, and can describe competition between sinks. The model provides new insights into Münch flow by explicitly including interactions with xylem flow and water potential in the closed‐form solution, and is expected to be useful as a component part of larger numerical models of entire plants.     

Osmotic Biosensors. How to Use a Characean Internode for Measuring the Alcohol Content of Beer

Isolated internodes of Chara corallina and Nitella flexilis have been used to determine the concentration of one passively permeating solute in the presence of non-permeating solutes. The technique was based on the fact that the shape of the peaks of the biphasic responses of cell turgor (as measured in a conventional way using the cell pressure probe) depended on the concentration and composition of the solution and on the permeability and reflection coefficients of the solutes. Peak sizes were proportional to the concentration of the permeating solute applied to the cell. Thus, using the selective properties of the cell membrane as the sensing element and changes of turgor pressure as the physical signal, plant cells have been used as a new type of biosensor based on osmotic principles. Upon applying osmotic solutions, the responses of cell turgor (P) exactly followed the P(t) curves predicted from the theory based on the linear force/flow relations of irreversible thermodynamics. The complete agreement between theory and experiment was demonstrated by comparing measured curves with those obtained by either numerically solving the differential equations for volume (water) and solute flow or by using an explicit solution of the equations. The explicit solution neglected the solvent drag which was shown to be negligible to a very good approximation. Different kinds of local beers (regular and de-alcoholized) were used as test solutions to apply the system for measuring concentrations of ethanol. The results showed a very good agreement between alcohol concentrations measured by the sensor technique and those obtained from conventional techniques (enzymatic determination using alcohol dehydrogenase or from measurement of the density and refraction index of beer). However, with beer as the test solution, the characean internodes did show irreversible changes of the transport properties of the membranes leading to a shift in the responses when cells were treated for longer than 1 h with diluted beer. The accuracy and sensitivity of the osmotic biosensor technique as well as its possible applications are discussed.     

Epithelial cell volume modulation and regulation

Summary Epithelial cell volume is a sensitive indicator of the balance between solute entry into the cell and solute exit. Solute accumulation in the cell leads to cell swelling because the water permeability of the cell membranes is high. Similarly, solute depletion leads to cell shrinkage. The rate of volume change under a variety of experimental conditions may be utilized to study the rate and direction of solute transport by an epithelial cell. The pathways of water movement across an epithelium may also be deduced from the changes in cellular volume. A technique for the measurement of the volume of living epithelial cells is described, and a number of experiments are discussed in which cell volume determination provided significant new information about the dynamic behavior of epithelia. The mechanism of volume regulation of epithelial cells exposed to anisotonic bathing solution is discussed and shown to involve the transient stimulation of normally dormant ion exchangers in the cell membrane.     

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.     

The point of maximum cell water volume excursion in case of presence of an impermeable solute

A relativistic permeability model of cell osmotic response (Cryobiology 40:64-83; 41:366-367) is applied to a two-solute system with one impermeable solute. The use of the normalized water volume (w), and the amount of intracellular permeable solute (x), which is the product of the water volume and intracellular osmolality (y), as the main variables allowed us to obtain a homogeneous differential equation dx(Delta)/dw(Delta)=f(x(Delta)/w(Delta)), where w(Delta)=w-w(f), x(Delta)=x-x(f), and f refers to the final (equilibrium) values. The solution of this equation is an explicit function, w(Delta)=g(x(Delta)), which is given in the text. This approach allows us to obtain an analytical (exact) expression of the water volume at the moment of the maximum excursion (water extremum w(m)). Results are compared with numeration of basic osmotic equations and with approximation given in (Cryobiology 40:64-83). Assumption that, dw/dt approximately 0 gives good approximations of the kinetics of water and permeable CPA after the point of maximum volume excursion (the slow phase of osmotic response). Practical aspects of the relativistic permeability approach are also discussed.     

NMR spin exchange kinetics at equilibrium in membrane transport and enzyme systems

A set of differential equations is formulated to describe the rapid exchange (time scale, approximately 0.01 to approximately 10 s) of a labelled solute across the membranes of cells in suspension. The labelling is achieved with nuclear magnetic resonance by exposure of the system to a high intensity radio-frequency pulse, and the excited nuclei relax to the equilibrium state with a short half life. An analytical expression for the decay of the magnetic resonance signal is presented; the solution involves the determination of eigenvalues, of an array of Laplace-Carson transformed differential equations, by use of the general solution of a quartic polynomial. Simulations of the behaviour of the exchange system using various conditions of cell number, rate constants and nuclear magnetic relaxation times are presented. The marked concentration dependence of the extent of reaction at a given time has not previously been reported for nuclear magnetic resonance exchange systems and is a feature anticipated from the known saturability of several membrane transport systems including glucose transport into human erythrocytes. The theory is readily generalized to other model systems by appropriate reinterpretation of the physical meaning of various parameters; the general form of the solution holds in many biological contexts other than membrane transport and includes equilibrium enzyme kinetics.     

Membrane permeability equations and their solutions for red cells

Summary The mathematical equations for the transport of nonelectrolytes across cell membranes are critically examined and cast in forms suitable for solution which involve fewer approximations than has heretofore been commonly done. For the case of red cells, the equations are developed to include the effect of the variation in apparent nonosmotic water owing to the variation in hemoglobin concentration as the cell swells or shrinks. Two methods of solution of the equations are developed and studied and sample calculations are provided. It is shown that the solutions to the linearized equations commonly found in the literature are insufficiently accurate for some purposes and this inaccuracy is avoided by the methods given here. The importance of retaining the effects of variations in apparent nonosmotic water and in solute volume in the cell is demonstrated.     

Membrane transport of electrolyte ions and time-dependent membrane potential

Equations are derived for the transport of a symmetrical electrolyte, consisting of cations and anions of equal valency, through a neutral membrane that separates two solutions of finite volume under quasi-steady-state conditions. The time-dependent membrane potential produced by the flow of ions is taken into account. Deviation of the time course of the solute concentrations from that of neutral solutes is found to be determined by the permeability ratio of cations and anions (when this ratio equals unity, the derived membrane transport equations reduce to those for neutral substances). Simple approximate expressions for the solute concentrations and of the membrane potential as functions of time are proposed, which are in excellent agreement with the exact numerical results.     

A model for radial water transport across plant roots

Summary A model based on the canal theory (Katou andFurumoto 1986 a, b) is proposed for the absorption of solute and water at the root periphery. The present canal model in the periphery and the model which was previously proposed for the exudation in the stele (Katou et al. 1987), are organized into a model for radial transport across excised plant roots, in the light of anatomical and physiological knowledge of maize roots. The canal equations for both canals are numerically solved to give quite a good explanation for the observed exudation of maize roots. It is found that the regulation of solute transport has a primary importance in the regulation of water transport across excised roots. The internal cell pressure of the symplast adjusts the water absorption at the root periphery to the water secretion into the vessels. There seems no need for this explanation of the radial water transport across roots to assume cell membranes with low reflection coefficient or variable water permeability. It would seem that the apoplast wall layers play a crucial role in metabolic control of water transport in roots as well as in hypocotyls.Abbreviations J _s ^ex* the theoretically estimated rate of solute exudation per unit surface area of model maize roots - J that of volume exudation per unit surface area of model maize roots - the reflection coefficient of the cell membrane against solutes     

Modeling epithelial cell homeostasis: Assessing recovery and control mechanisms

Critical to epithelial cell viability is prompt and direct recovery, following a perturbation of cellular conditions. Although a number of transporters are known to be activated by changes in cell volume, cell pH, or cell membrane potential, their importance to cellular homeostasis has been difficult to establish. Moreover, the coordination among such regulated transporters to enhance recovery has received no attention in mathematical models of cellular function. In this paper, a previously developed model of proximal tubule (Weinstein, 1992, Am. J. Physiol. 263, F784–F798), has been approximated by its linearization about a reference condition. This yields a system of differential equations and auxiliary linear equations, which estimate cell volume and composition and transcellular fluxes in response to changes in bath conditions or membrane transport coefficients. Using the singular value decomposition, this system is reduced to a linear dynamical system, which is stable and reproduces the full model behavior in a useful neighborhood of the reference. Cost functions on trajectories formulated in the model variables (e.g., time for cell volume recovery) are translated into cost functions for the dynamical system. When the model is extended by the inclusion of linear dependence of membrane transport coefficients on model variables, the impact of each such controller on the recovery cost can be estimated with the solution of a Lyapunov matrix equation. Alternatively, solution of an algebraic Riccati equation provides the ensemble of controllers that constitute optimal state feedback for the dynamical system. When translated back into the physiological variables, the optimal controller contains some expected components, as well as unanticipated controllers of uncertain significance. This approach provides a means of relating cellular homeostasis to optimization of a dynamical system.     

Determinants of epithelial cell volume

Epithelial cell volume is determined by the concentration of intracellular, osmotically active solutes. The high water permeability of the cell membrane of most epithelia prevents the establishment of large osmotic gradients between the cell and the bathing solutions. Steady-state cell volume is determined by the relative rates of solute entry and exit across the cell membranes. Inhibition of solute exit leads to cell swelling because solute entry continues; inhibition of solute entry leads to cell shrinkage because solute exit continues. Cell volume is then a measure of the rate and direction of net solute movements. Epithelial cells are also capable of regulation of the rate of solute entry and exit to maintain intracellular composition. Feedback control of NaCl entry into Necturus gallbladder epithelial cells is demonstrable after inhibition of the Na,K-ATPase or reduction in the NaCl concentration of the serosal bath. Necturus gallbladder cells respond to a change in the osmolality of the perfusion solution by rapidly regulating their volume to control values. This regulatory behavior depends on the transient activation of quiescent transport systems. These transport systems are responsible for the rapid readjustments of cell volume that follow osmotic perturbation. These powerful transporters may also play a role in steady-state volume regulation as well as in the control of cell pH.     

Computer Simulation of the Rapid Adaptation of Elongation Growth to Osmotic Stress in Segments of Cowpea Stem by Application of the Apoplast Canal Model

The transport of water during the adaptive rapid recovery ofelongation growth upon additional osmotic stress was examinedin the model stem segments of cowpea (Vigna unguiculata L.)by numerical solution of the extended canal equation, whichconsists of a set of time-dependent non-linear partial differentialequations. The calculated dynamic behaviour of the depletionof solute within the apoplast canal effectively explained thereported transient changes in water flow and, therefore, ingrowth during the adaptation to stress if the stress-inducedenhancement of net uptake of solute from the apoplast canalwas assumed. The extended canal model was also adequate forsimulation of the elastic shrinkage of hypocotyl segments uponexposure to osmotic stress which took place when the supplyof energy was limited. It appeared that the function of thecanal system in absorbing water is intrinsically stable againstperturbations of the water environment. (Received September 5, 1990; Accepted January 11, 1991)     

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.     

An efficient numerical method for distributed-loop models of the urine concentrating mechanism

In this study we describe an efficient numerical method, based on the semi-Lagrangian (SL) semi-implicit (SI) method and Newton's method, for obtaining steady-state (SS) solutions of equations arising in distributed-loop models of the urine concentrating mechanism. Dynamic formulations of these models contain large systems of coupled hyperbolic partial differential equations (PDEs). The SL method advances the solutions of these PDEs in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The SI approach controls stiffness arising from transtubular transport terms by averaging these terms in time along flow trajectories. An approximate SS solution of a dynamic formulation obtained via the SLSI method can be used as an initial guess for a Newton-type solver, which rapidly converges to a highly accurate numerical approximation to the solution of the ordinary differential equations that arise in the corresponding SS model formulation. In general, it is difficult to specify a priori for a Newton-type solver an initial guess that falls within the radius of convergence; however, the initial guess generated by solving the dynamic formulation via the SLSI method can be made sufficiently close to the SS solution to avoid numerical instability. The combination of the SLSI method and the Newton-type solver generates stable and accurate solutions with substantially reduced computation times, when compared to previously applied dynamic methods.     

The physico-chemical mechanism of mediated transport. I. Facilitated diffusion

On the basis of the currently accepted model for the cell membrane structure, a physico-chemical model for mediated transport is developed and solved for the case of polar non-electrolyte migration through the cell membrane. The model considers the interstitial space defined by the transport protein subunits to be the migration pathway for polar solutes. A Langmuir-type adsorption equilibrium is assumed at the interfaces and a multicomponent diffusion mechanism of solute and water is postulated within the migration pathway, where the polar residues of the transport protein represent another component of the system. Membrane selectivity is governed by the adsorption constants, which are shown to affect strongly the kinetics of transport. Isosmotic transport and the volume change of the cell are important features incorporated in the model, which is shown to fulfill the peculiar properties of facilitated diffusion systems. It is concluded that the same type of pathway can be used for the transport of other polar solutes through existing or induced hydrophilic channels, for which a similar approach is suggested.