A model relating root permeability to flux and potentials: application to existing data from soybean and other plants

A model that relates hydraulic permeability to water flux and to gradients in pressure potential and solute potential was tested using soybean (Glycine max) plants. Water flux was varied by additions of polyethylene glycol 6,000 around one portion of a divided root system and by changing the light intensity and CO₂ concentration around the plants. The data are compatible with the model only if the hydraulic permeability varies with flux; however, the data were insufficient for rigorous testing. Three sets of published data fit the model only if hydraulic permeability varies. Evidence originally presented as involving constant hydraulic permeability is shown, rather, to require variable hydraulic permeability.     

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.     

Transport of gases into leaves

Abstract. Transport of gases between the intercellular spaces of plant leaves and the surrounding air is analysed in terms of multicomponent collision processes through an isothermal, porous septum. Interaction of diffusing species with each other and with the pore walls is described using a modified Stefan–Maxwell equation and an equation relating the pressure gradient to the sum of the diffusive fluxes, weighted by their appropriate Knudsen diffusivities. Viscous How arising from an excess pressure within the leaf is also considered.
Equations are derived which describe the flux densities of water vapour and CO₂ through the stomata. The analysis is general and is applicable to trace gases other than CO₂. A simple conductance is defined for water vapour to relate the flux and mol fraction difference across the stomata, viz. N_w =− g_w , δ x_w / x_a . A simple conductance cannot be defined for CO₂ because the flux of water vapour has a significant influence on the CO₂ gradient. The equation derived for the intercellular mol fraction of CO₂ is in terms of the fluxes of CO₂ and water vapour and represents a 'large-pore' ( d > μm) approximation which requires no information about stomalal geometry. Analogous equations are developed for transfer of gases through the leaf boundary layer. Sample calculations are presented to illustrate the effect of neglecting the interaction of water vapour and CO₂ on the calculated intercellular and surface concentrations of CO₂. Equations for computing water vapour and CO₂ flux densities from leaf chamber measurements are also presented.     

A simple kinetic model of contraction of striated muscle: Full activation at full filament overlap in sarcomeres

A simple kinetic model of muscle contraction is suggested. The rates of cross-bridge transitions from one kinetic state to another one are supposed to depend on the strain averaged over an ensemble of actin-bound bridges. With a proper set of strain-dependent rate constants, the model fits well a broad range of experimental data. Owing to that the strain-stress relation is described by a set of ordinary differential equations, the model can be used for simulating complex 3D muscle contractions.     

Oscillations in stomatal conductance and plant functioning associated with stomatal conductance: Observations and a model

Summary Measurements of transpiration, leaf water content, and flux of water in a cotton plant exhibiting sustained oscillations, in stomatal conductance are presented, and a model of the mechanism causing this behaviour is developed. The dynamic elements, of the model are capacitors—representing the change of water content with water potential in mesophyll, subsidiary and guard cells—interconnected by resistances representing flow paths in the plant. Increase of water potential in guard cells causes an increase in stomatal conductance. Increase of water potential in the subsidiary cells has the opposite effect and provides the positive feed-back which can cause stomatal conductance to oscillate. The oscillations are shown to have many of the characteristics of free-running oscillations in real plants. The behaviour of the model has been examined, using an analogue computer, with constraints and perturbations representing some of those which could be applied to real plants in physiological experiments. Aspects of behaviour which have been simulated are (a) opening and closing of stomata under the influence of changes in illumination, (b) transient responses due to step changes in potential transpiration, root permeability and potential of water surrounding the roots, (c) the influence of these factors on the occurrence and shape of spontaneous oscillations, and (d) modulation of sustained oscillations due to a circadian rhythm in the permeability of roots.     

Responding phospholipid membranes--interplay between hydration and permeability.

Osmotic forces are important in regulating a number of physiological membrane processes. The effect of osmotic pressure on lipid phase behavior is of utmost importance for the extracellular lipids in stratum corneum (the outer part of human skin), due to the large gradient in water chemical potential between the water-rich tissue on the inside, and the relative dry environment on the outside of the body. We present a theoretical model for molecular diffusional transport over an oriented stack of two-component lipid bilayers in the presence of a gradient in osmotic pressure. This gradient serves as the driving force for diffusional motion of water. It also causes a gradient in swelling and phase transformations, which profoundly affect the molecular environment and thus the local diffusion properties. This feedback mechanism generates a nonlinear transport behavior, which we illustrate by calculations of the flux of water and solute (nicotine) through the bilayer stack. The calculated water flux shows qualitative agreement with experimental findings for water flux through stratum corneum. We also present a physical basis for the occlusion effect. Phase behavior of binary phospholipid mixtures at varying osmotic pressures is modeled from the known interlamellar forces and the regular solution theory. A first-order phase transformation from a gel to a liquid--crystalline phase can be induced by an increase in the osmotic pressure. In the bilayer stack, a transition can be induced along the gradient. The boundary conditions in water chemical potential can thus act as a switch for the membrane permeability.     

Pressure-induced water transport in membrane channels studied by molecular dynamics

A method is proposed to measure the water permeability of membrane channels by means of molecular dynamics simulations. By applying a constant force to the bulk water molecules and a counter force on the complementary system, a hydrostatic pressure difference across the membrane can be established, producing a net directional water flow. The hydraulic or osmotic permeability can then be determined by the ratio of the water flux and the pressure difference. The method is applied and tested on an aquaglyceroporin channel through a series of simulations totaling 5 ns in duration.     

Ein mathematisches Modell der Massenentwicklung planktischer Diatomeen

A Mathematical Model of Blooms of Plankton Diatoms Blooms of plankton diatoms are of great importance for the management of drinking water reservoirs in the southern regions of the German Democratic Republic. A mathematical model has been developed by the author in order to find out optimal management alternatives. The model consists of coupled differential equations. The first step is a very simple model which is compared with a mass development of Asterionella formosa in the Saidenbach reservoir in the Erzgebirge. The dynamic properties of the improved model are examined with respect to the influence of various ecological parameters and that of initial values. It could be shown that loss rate and initial biomass of phytoplankton exert the greatest influence of bloom dynamics. In the last step, the results of the model are compared with a bloom ef Asterionella in laboratory cultures and with recent data obtained from the reservoir. A qualitative comparison between model behaviour and observations in the lake gives information about the validity range of the model parameters and provides ideas for further development. The applicability to water quality management of the model is considered.     

Interaction between Osmotic- and Pressure-induced Water Flow in Plant Roots

When the pressure gradient across a root alters, there is often an apparent change in the permeability of the root to water. Fiscus (Plant Physiol. 1975. 55: 917-922) has suggested that this can be explained by a simple two-compartment model which takes into account rates of solute uptake into the xylem. A method of testing actual data against the Fiscus model is proposed; this shows that in some cases the apparent changes in permeability cannot be explained by the model. The model is not adequate to predict the amounts of solute reaching the xylem by passive drag: a three-compartment model would be more realistic.     

A mathematical model of solute coupled water transport in toad intestine incorporating recirculation of the actively transported solute

A mathematical model of an absorbing leaky epithelium is developed for analysis of solute coupled water transport. The non-charged driving solute diffuses into cells and is pumped from cells into the lateral intercellular space (lis). All membranes contain water channels with the solute passing those of tight junction and interspace basement membrane by convection-diffusion. With solute permeability of paracellular pathway large relative to paracellular water flow, the paracellular flux ratio of the solute (influx/outflux) is small (2-4) in agreement with experiments. The virtual solute concentration of fluid emerging from lis is then significantly larger than the concentration in lis. Thus, in absence of external driving forces the model generates isotonic transport provided a component of the solute flux emerging downstream lis is taken up by cells through the serosal membrane and pumped back into lis, i.e., the solute would have to be recirculated. With input variables from toad intestine (Nedergaard, S., E.H. Larsen, and H.H. Ussing, J. Membr. Biol. 168:241-251), computations predict that 60-80% of the pumped flux stems from serosal bath in agreement with the experimental estimate of the recirculation flux. Robust solutions are obtained with realistic concentrations and pressures of lis, and with the following features. Rate of fluid absorption is governed by the solute permeability of mucosal membrane. Maximum fluid flow is governed by density of pumps on lis-membranes. Energetic efficiency increases with hydraulic conductance of the pathway carrying water from mucosal solution into lis. Uphill water transport is accomplished, but with high hydraulic conductance of cell membranes strength of transport is obscured by water flow through cells. Anomalous solvent drag occurs when back flux of water through cells exceeds inward water flux between cells. Molecules moving along the paracellular pathway are driven by a translateral flow of water, i.e., the model generates pseudo-solvent drag. The associated flux-ratio equation is derived.     

Analysis of ultrafiltration and mass transfer in a bioartificial pancreas

A bioartificial pancreas is an implantable device which contains insulin secreting cells (Langerhans islets), separated from the circulating blood by a semi-permeable membrane to avoid rejection. This paper describes the operation of such a device and evaluates the respective contributions of diffusion and ultrafiltration to the glucose and insulin mass transfer. It is shown that the pressure drop along the blood channel produces across the first half of the channel an ultrafiltration flux toward the islet compartment followed in the second half by an equal flux in reverse direction from islets to blood. The mass transfer analysis is carried out for an optimal geometry in which a U-shaped blood channel surrounds closely a very thin islet compartment formed by a folded flat membrane. A complete model of insulin release by this device is developed and is compared with in vitro data obtained with rats islets. Satisfactory kinetics is achieved with a polyacrylonitrile membrane used in hemodialysis. But the model shows that the membrane hydraulic permeability should be increased by a factor of 10 to significantly improve the performance.     

Singular perturbation analysis of the nonlinear, flow-dependent compressive stress relaxation behavior of articular cartilage

The dominant mechanism giving rise to the viscoelastic response of articular cartilage during compression is the nonlinear diffusive interaction of the fluid and solid phases of the tissue as they flow relative to one another. The present study is concerned with the role of this interaction under uniaxial stress relaxation in compression. The model is a biphasic mixture of fluid and solid which incorporates the strain-dependent permeability found earlier from permeation experiments. When a ramp-displacement is imposed on the articular surface, simple, but accurate, asymptotic approximations are derived for the deformation and stress fields in the tissue for slow and moderately fast rates of compression. They are shown to agree very well with experiment and they provide a simple means for determining the material parameters. Moreover, they lead to important insights into the role of the flow-dependent viscoelastic nature of articular cartilage and other hydrated biological tissues.     

Convection and Diffusion in Charged Hydrated Soft Tissues: A Mixture Theory Approach

The extracellular matrix of cartilage is a charged porous fibrous material. Transport phenomena in such a medium are very complex. In this study, solute diffusive flux and convective flux in porous fibrous media were investigated using a continuum mixture theory approach. The intrinsic diffusion coefficient of solute in the mixture was defined and its relation to drag coefficients was presented. The effect of mechanical loading on solute diffusion in cartilage under unconfined compression with a frictionless boundary condition was analyzed numerically using the model developed. Both strain-dependent hydraulic permeability and diffusivity were considered. Analyses and results show that (1) In porous media, the convective velocity for each solute phase is different. (2) The solute convection in tissue is governed by the relative convective velocity (i.e., relative to solid velocity). (3) Under the assumption that all the frictional interactions among solutes are negligible, the relative convective velocity for α-solute phase is equal to the relative solvent velocity multiplied by its convective coefficient (H ^α) which is also known as the hindrance factor in the literature. The relationship between the convective coefficient and the relative diffusivity of solute is presented. (4) Solute concentration profile within the cartilage sample depends on the phase of dynamic compression.     

Exact solutions of a two parameter flux model and cryobiological applications

Solute-solvent transmembrane flux models are used throughout biological sciences with applications in plant biology, cryobiology (transplantation and transfusion medicine), as well as circulatory and kidney physiology. Using a standard two parameter differential equation model of solute and solvent transmembrane flux described by Jacobs [The simultaneous measurement of cell permeability to water and to dissolved substances, J. Cell. Comp. Physiol. 2 (1932) 427-444], we determine the functions that describe the intracellular water volume and moles of intracellular solute for every time t and every set of initial conditions. Here, we provide several novel biophysical applications of this theory to important biological problems. These include using this result to calculate the value of cell volume excursion maxima and minima along with the time at which they occur, a novel result that is of significant relevance to the addition and removal of permeating solutes during cryopreservation. We also present a methodology that produces extremely accurate sum of squares estimates when fitting data for cellular permeability parameter values. Finally, we show that this theory allows a significant increase in both accuracy and speed of finite element methods for multicellular volume simulations, which has critical clinical biophysical applications in cryosurgical approaches to cancer treatment.     

Leaf water stress in engelmann spruce: influence of the root and shoot environments

The response of xylem pressure potential of Engelmann spruce (Picea engelmannii Engelm.) to environmental factors was studied in the natural subalpine environment. Data were analyzed in the context of a leaf water potential model based upon the van den Honert model for water transport through the soil-plant-atmosphere continuum. At soil temperatures of 10 to 15 C, xylem pressure potential decreased to about −10 bars as the ratio of leaf to air absolute humidity difference to leaf diffusion resistance (an estimate of transpiration) increased. The potentials were slightly lower at all flux rates above zero when the soil temperature was 5 to 10 C, and at temperatures of 0 to 5 C the potentials decreased sharply to as low as −20.4 bars, even though the soil water supply was adequate. The relative viscosity of water and soil to leaf resistances for flow were compared for Engelmann spruce and citrus at low soil temperatures. These comparisons indicated that decreased root permeability was probably not an important factor causing higher stresses in spruce at 5 to 10 C, but for citrus, root permeability became limiting at soil temperatures as high as 13.5 C. Xylem pressure potential was correlated with net radiation during the daytime when soil temperature was above 7 C. Under other conditions, however, xylem potential and net radiation apparently had a different relationship. The relationship between flux density and potential was the same on unshaded and shaded portions of the crown, with differences in potential related to differences in flux density.     

A nonlinear numerical model of percutaneous drug absorption.

A nonlinear mathematical model developed by Chandrasekaran et al. is examined to monitor pharmacokinetic profiles in percutaneous drug absorption and is addressed to several associated problems that could occur in the data analysis of in vitro experiments. The formulation of the model gives rise to a nonlinear partial differential equation (PDE) of parabolic type, and a family of finite-difference methods is developed for the numerical solution of the associated initial/boundary-value problem. The value given to a parameter in this family determines the stability properties of the resulting method and whether the solution is obtained explicitly or implicitly. In the case of implicit members of the family it is seen that the solution of the nonlinear PDE is obtained by solving a linear algebraic system, the coefficient matrix of which is tridiagonal. The behaviors of two methods of the family are examined in a series of numerical experiments. Numerical differentiation and integration procedures are combined to monitor the cumulative amount of drug eliminated into the receptor cell per unit area as time increases. It is found that the use of the equation for the simple membrane model to estimate the permeability coefficient and lag time is warranted even if the system should be described by the dual-sorption model, provided cumulative amount versus time data collected for a sufficiently long time are used. However, being different from the behavior in the simple membrane model, the lag time, which can be estimated in this way, is dose-dependent and decreases with increasing donor cell concentration. On the other hand, the permeability coefficient in the dual-sorption model remains constant irrespective of the donor cell concentrations as in the simple membrane model.     

The influence of calcium on potassium fluxes across the root of Ricinus communis

Summary The effect of calcium on the flux of potassium to the exudate of detached root systems of Ricinus communis has been investigated. Previous analyses have indicated the presence of a water dependent and a water independent flux of potassium which vary with the concentration of potassium in the bathing medium, in the presence of 0.1 mM CaCl₂. In the present study it has been observed that at a higher concentration of calcium in the bathing medium (2.5 mM CaCl₂) the water dependent flux of potassium is greatly reduced while the water independent flux is not affected. It is proposed that the differential effect of calcium on these two fluxes is a reflection of the degree of dependence of these fluxes on the permeability of the plasmamembranes within the root.     

A probabilistic approach for estimating water permeability in pressure-driven membranes

A probabilistic approach is proposed to estimate water permeability in a cellulose triacetate (CTA) membrane. Water transport across the membrane is simulated in reverse osmosis mode by means of non-equilibrium molecular dynamics (MD) simulations. Different membrane configurations obtained by an annealing MD simulation are considered and simulation results are analyzed by using a hierarchical Bayesian model to obtain the permeability of the different membranes. The estimated membrane permeability is used to predict full-scale water flux by means of a process-level Monte Carlo simulation. Based on the results, the parameters of the model are observed to converge within 5-ns total simulation time. The results also indicate that the use of unique structural configurations in MD simulations is essential to capture realistic membrane properties at the molecular scale. Furthermore, the predicted full-scale water flux based on the estimated permeability is within the same order of magnitude of bench-scale experimental measurement of 1.72×10^?5 m/s.     

Cellular Inhomogeneity in Dog Red Cells As Revealed by Sodium Flux

Unidirectional ²⁴Na fluxes across the dog red blood cell membrane were measured. The kinetics were incompatible with a single time constant but could be accounted for in terms of a two-series compartment cell model, with approximately 1% of cell Na in the outer compartment. Dog red blood cells are known to be inhomogeneous in their Na and K permeabilities. Theoretical analysis showed that such cellular inhomogeneity in the Na permeability coefficient might in principle account for the flux data. In order to evaluate the inhomogeneity effect, a technique based on the differential response of cells suspended in isosmolar high K buffers was devised to measure the variations in Na permeability in the cell population. A variation in the Na permeability coefficient of approximately 30% was found. This inhomogeneity is insufficient to account for the flux data.     

A two-compartment model of osmotic lysis in Plasmodium falciparum-infected erythrocytes

We recently identified a voltage-dependent anion channel on the surface of human red blood cells (RBCs) infected with the malaria parasite, Plasmodium falciparum. This channel, the plasmodial erythrocyte surface anion channel (PESAC), likely accounts for the increased permeability of infected RBCs to various small solutes, as assessed quantitatively with radioisotope flux and patch-clamp studies. Whereas this increased permeability has also been studied by following osmotic lysis of infected cells in permeant solutes, these experiments have been limited to qualitative comparisons of lysis rates. To permit more quantitative examination of lysis rates, we have developed a mathematical model for osmotic fragility of infected cells based on diffusional uptake via PESAC and the two-compartment geometry of infected RBCs. This model, combined with a simple light scattering assay designed to track osmotic lysis precisely, produced permeability coefficients that match both previous isotope flux and patch-clamp estimates. Our model and light scattering assay also revealed Michaelian kinetics for inhibition of PESAC by furosemide, suggesting a 1:1 stoichiometry for their interaction.