Cellular response of adipose derived passage-4 adult stem cells to freezing stress

A differential scanning calorimeter technique was used to generate experimental data for volumetric shrinkage during cooling at 20 degrees C/min in adipose derived adult stem cells (ASCs) in the presence and absence of cryoprotective agents (CPAs). By fitting a model of water transport to the experimentally determined volumetric shrinkage data, the membrane permeability parameters of ASCs were obtained. For passage-4 (P4) ASCs, the reference hydraulic conductivity Lpg and the value of the apparent activation energy ELP were determined to be 1.2 X 10(-13) m3/Ns and 177.8 kJ/mole, respectively. We found that the addition of either glycerol or dimethylsulfoxide (DMSO) significantly decreased the value of the reference hydraulic conductivity Lpg(cpa) and the value of the apparent activation energy ELp(cpa) in P4 ASCs. The values of Lpg(cpa) in the presence of glycerol and DMSO were determined as 0.39 x 10(-13) and 0.50 X 109-13) m3/Ns, respectively, while the corresponding values of ELp(cpa) were 51.0 and 61.5 kJ/mole. Numerical simulations of water transport were then performed under a variety of cooling rates (5-100 degreesC/min) using the experimentally determined membrane permeability parameters. And finally, the simulation results were analyzed to predict the optimal rates of freezing P4 adipose derived cells in the presence and absence of CPAs.     

Structural mechanisms of acute VEGF effect on microvessel permeability

To investigate the ultrastructural mechanisms of acute microvessel hyperpermeability by vascular endothelial growth factor (VEGF), we combined a mathematical model (J Biomech Eng 116: 502-513, 1994) with experimental data of the effect of VEGF on microvessel hydraulic conductivity (L(p)) and permeability of various-sized solutes. We examined the effect of VEGF on microvessel permeability to a small solute (sodium fluorescein, Stokes radius 0.45 nm), an intermediate solute (alpha-lactalbumin, Stokes radius 2.01 nm), and a large solute [albumin (BSA), Stokes radius 3.5 nm]. Exposure to 1 nM VEGF transiently increased apparent permeability to 2.3, 3.3, and 6.2 times their baseline values for sodium fluorescein, alpha-lactalbumin, and BSA, respectively, within 30 s, and all returned to control within 2 min. On the basis of L(p) (DO Bates and FE Curry. Am J Physiol Heart Circ Physiol 271: H2520-H2528, 1996) and permeability data, the prediction from the model suggested that the most likely structural changes in the interendothelial cleft induced by VEGF would be a approximately 2.5-fold increase in its opening width and partial degradation of the surface glycocalyx.     

Permeability of barnacle muscle fibers to water and nonelectrolytes

Summary The permeability of isolated muscle fibers of the giant barnacleMegabalanus psittacus to water and nonelectrolytes was studied by determining their reflection and permeability coefficients. Reflection coefficients were obtained by comparing the osmotic fluxes produced by a test molecule to that produced by the impermeant sucrose molecule. Permeability coefficients were determined for measurements of tracer uptake.The results indicate that, in these fibers, nonelectrolyte permeability is closely related to lipid solubility and molecular size.A value of 2.16×10^–12 cm³/sec dyne for the hydraulic conductivity and a value of 10.45×10^–4 cm/sec for³HHO permeability coefficient were obtained.The effect of membrane surface invaginations and clefts on the determination of permeability coefficients is discussed.     

Predicted permeability parameters of human ovarian tissue cells to various cryoprotectants and water

This study presents a generic numerical model to simulate the coupled solute and solvent transport in human ovarian tissue sections during addition and removal of chemical additives or cryoprotective agents (CPA). The model accounts for the axial and radial diffusion of the solute (CPA) as well as axial convection of the CPA, and a variable vascular surface area (A) during the transport process. In addition, the model also accounts for the radial movement of the solvent (water) into and out of the vascular spaces. Osmotic responses of various cells within an human ovarian tissue section are predicted by the numerical model with three model parameters: permeability of the tissue cell membrane to water (L(p)), permeability of the tissue cell membrane to the solute or CPA (omega) and the diffusion coefficient of the solute or CPA in the vascular space (D). By fitting the model results with published experimental data on solute/water concentrations within an human ovarian tissue section, I was able to determine the permeability parameters of ovarian tissue cells in the presence of 1.5M solutions of each of the following: dimethyl sulphoxide (DMSO), propylene glycol (PROH), ethylene glycol (EG), and glycerol (GLY), at two temperatures (4 degrees C and 27 degrees C). Modeling Approach 1: Assuming a constant value of solute diffusivity (D = 1.0 x 10(-9) m(2)/sec), the best fit values of L(p) ranged from 0.35 x 10(-14) to 1.43 x 10(-14) m(3)/N-sec while omega ranged from 2.57 x 10(-14) to 70.5 x 10(-14) mol/N-sec. Based on these values of L(p) and omega, the solute reflection coefficient, sigma defined as sigma = 1-omega v(CPA)/L(P) ranged from 0.9961 to 0.9996. Modeling Approach 2: The relative values of omega and sigma from our initial modeling suggest that the embedded ovarian tissue cells are relatively impermeable to all the CPAs investigated (or omega approximately 0 and sigma approximately 1.0). Consequently the model was modified and used to predict the values of L(p) and D assuming omega = 0 and sigma = 1.0. The best fit values of L(p) ranged from 0.44 x 10(-14) to 1.2 x 10(-14) m(3)/N-sec while D ranged from 0.85 x 10(-9) to 2.08 x 10(-9) m(2)/sec. Modeling Approach 3: Finally, the best fit values of D from modeling approach 2 were incorporated into model 1 to re-predict the values of L(p) and omega. It is hoped that the ovarian tissue cell parameters reported here will help to optimize chemical loading and unloading procedures for whole ovarian tissue sections and consequently, tissue cryopreservation procedures.     

Permeability properties of the subepithelial tissues of Necturus gallbladder

The permeability properties of the subepithelial connective tissue of Necturus gallbladder were evaluated by measurement of electrical resistance, dilution potentials and hydraulic water permeability. The gallbladder epithelial cells were removed by scraping and the underlying connective tissue placed in an Ussing chamber. The electrical resistance was 2.2 +/- 0.8 omega X cm2; the tissue was slightly cation selective relative to free solution. The subepithelial tissues restricted the rate of diffusion of small solutes to 50% of the free solution value. The hydraulic water permeability averaged 2.1 X 10(-2) cm/s per atm. We conclude that limitations of the area of subepithelium available for fluid movement are the most important factors in determining the restrictions to solute and water flow offered by the subepithelial tissues.     

Partition and permeation of dextran in polyacrylamide gel.

Partition of sized FITC-dextrans in polyacrylamide gel showed a relationship between Kav and solute radius as predicted by the theory of Ogston, which is based solely on geometry of the spaces. Permeability data for the same dextrans were fit to several theories, including those based on geometry and those based on hydrodynamic interactions, and the gel structure predicted by the partition and permeability data were compared. The Brinkman effective-medium model (based on hydrodynamic interactions and requiring a measure of the hydraulic conductivity of the matrix) gave the best fit of permeability data with the values for fiber radius (rf) and void volume of the gel (epsilon) that were obtained from the partition data. The models based on geometry and the hydrodynamic screening model of Cukier, using the rf and epsilon from partition data, all predicted higher rates of permeation than observed experimentally, while the effective-medium model with added term for steric interaction predicted lower permeation than that observed. The size of cylindrical pores appropriate for the partition data predicted higher rates of permeation than observed. These relative results were unaffected by the method of estimating void volume of the gel. In sum, it appears that one can use data on partition of solute, combined with measurement of hydraulic conductivity, to predict solute permeation in polyacrylamide gel.     

Effect of sucrose on the dimerization of alpha-chymotrypsin. Allowance for thermodynamic nonideality arising from the presence of a small inert solute

The space-filling effects of sucrose on the dimerization of alpha-chymotrypsin have been investigated by sedimentation equilibrium studies on the enzyme in acetate-chloride buffer, pH 3.9, I 0.2. From the extent of enhancement of the apparent dimerization constant in the presence of 0.05-0.16 M sucrose, it is concluded that this effect of thermodynamic nonideality finds quantitative explanation in terms of excluded volume. However, the suggested approximation that the radius of an inert small solute would be sufficiently small to be neglected in the calculation of covolumes (D.J. Winzor and P.R. Wills, Biophys. Chem. 25 (1986) 243) has not withstood the more stringent test afforded by the present study of alpha-chymotrypsin dimerization. A value of 0.34 nm for the effective thermodynamic radius of sucrose was inferred from the covolume for self-interaction obtained by frontal gel chromatography on Sephadex G-10 under the conditions of the ultracentrifugal studies. Finally, results of sedimentation equilibrium experiments on alpha-chymotrypsin in the presence of 0.1 M glycerol were also shown to be consistent with interpretation in terms of the model of space-filling effects entailing complete exclusion of small solute from the hydrated protein domain.     

Comparative permeability of canine visceral and parietal pleura

To determine the permeability of canine pleural mesothelium, visceral and intercostal parietal pleura from mongrel dogs was carefully stripped from the underlying tissue and mounted as a planar sheet in a Ussing-type chamber. The hydraulic conductivity (Lp) was determined from the rate of volume flux in response to hydrostatic pressure gradients applied to either the mucosal or serosal surface of the pleural membrane. The diffusional permeability (Pd) of radiolabeled water, sucrose, inulin, and albumin was determined under equilibrium conditions from the unidirectional tracer flux. The Lp of the visceral pleura was 0.39 +/- 0.032 (SE) X 10(-4) ml.s-1.cmH2O-1.cm-2 and that Lp of parietal pleura was 1.93 +/- 0.93 X 10(-4) ml.s-1.cmH2O-1.cm-2 (P less than 0.001). The Pd of the visceral pleura ranged from 12.21 +/- 0.45 X 10(-4) cm/s for 3H2O to 0.34 +/- 0.03 X 10(-4) cm/s for [3H]albumin. The Pd of the parietal pleura for water and sucrose was similar to that of the visceral membrane, whereas its Pd for the larger inulin and albumin molecules was greater than that of visceral pleura (P less than 0.01). A spontaneous potential difference could not be detected across either membrane. The relatively higher parietal pleural Lp and Pd for larger solutes is probably due to the presence of stomata in this membrane. These results indicate that both the parietal and the visceral pleura are extremely permeable tissues which offer little resistance to water and solute flux.     

Expression of aquaporin-3 improves the permeability to water and cryoprotectants of immature oocytes in the medaka (Oryzias latipes)

The permeability of the plasma membrane plays a crucial role in the successful cryopreservation of oocytes and embryos. Several efforts have been made to facilitate the movement of water and cryoprotectants across the plasma membrane of fish oocytes/embryos because of their large size. Aquaporin-3 is a water/solute channel that can also transport various cryoprotectants. In this study, we tried to improve the permeability of immature medaka (Oryzias latipes) oocytes to water and cryoprotectants by artificially expressing aquaporin-3. The oocytes were injected with aquaporin-3 cRNA and cultured for 6-7 h. Then, hydraulic conductivity (L(P)) and cryoprotectant permeability (P(S)) were determined from volume changes in a hypertonic sucrose solution and various cryoprotectant solutions, respectively, at 25 degrees C. The L(P) value of the cRNA-injected oocytes was 0.22+/-0.04 microm/min/atm, nearly twice larger than that of intact or water-injected oocytes (0.14+/-0.02 and 0.14+/-0.03 microm/min/atm, respectively). P(S) values of intact oocytes for ethylene glycol, propylene glycol, and DMSO were 1.36+/-0.34, 1.97+/-0.20, and 1.17+/-0.52 x 10(-3) cm/min, respectively. The permeability to glycerol could not be calculated because oocytes remained shrunken in the glycerol solution. On the other hand, cRNA-injected oocytes had significantly higher P(S) values (glycerol, 2.20+/-1.29; ethylene glycol, 2.98+/-0.36; propylene glycol, 3.93+/-1.70; DMSO, 3.11+/-0.74 x 10(-3) cm/min) than intact oocytes. When cRNA-injected oocytes were cultured for 12-14 h, 51% matured to the metaphase II stage, and 43% of the matured oocytes were fertilized and hatched following in vitro fertilization and 14 days of culture. Thus, the permeability of medaka oocytes to water and cryoprotectants was improved by the artificial expression of aquaporin-3, and the oocytes retained the ability to develop to term.     

Transient transcapillary exchange of water driven by osmotic forces in the heart

Osmotic transient responses in organ weight after changes in perfusate osmolarity have implied steric hindrance to small-molecule transcapillary exchange, but tracer methods do not. We obtained osmotic weight transient data in isolated, Ringer-perfused rabbit hearts with NaCl, urea, glucose, sucrose, raffinose, inulin, and albumin and analyzed the data with a new anatomically and physicochemically based model accounting for 1) transendothelial water flux, 2) two sizes of porous passages across the capillary wall, 3) axial intracapillary concentration gradients, and 4) water fluxes between myocytes and interstitium. During steady-state conditions approximately 28% of the transcapillary water flux going to form lymph was through the endothelial cell membranes [capillary hydraulic conductivity (Lp) = 1.8 +/- 0.6 x 10-8 cm. s-1. mmHg-1], presumably mainly through aquaporin channels. The interendothelial clefts (with Lp = 4.4 +/- 1.3 x 10-8 cm. s-1. mmHg-1) account for 67% of the water flux; clefts are so wide (equivalent pore radius was 7 +/- 0.2 nm, covering approximately 0.02% of the capillary surface area) that there is no apparent hindrance for molecules as large as raffinose. Infrequent large pores account for the remaining 5% of the flux. During osmotic transients due to 30 mM increases in concentrations of small solutes, the transendothelial water flux was in the opposite direction and almost 800 times as large and was entirely transendothelial because no solute gradient forms across the pores. During albumin transients, gradients persisted for long times because albumin does not permeate small pores; the water fluxes per milliosmolar osmolarity change were 200 times larger than steady-state water flux. The analysis completely reconciles data from osmotic transient, tracer dilution, and lymph sampling techniques.     

Determination of the permeability of human lymphocytes with a microscope diffusion chamber

A diffusion chamber similar to that proposed by J.J. McGrath (J. Microsc., in press) was constructed which allows microscopic observation of osmotically induced volume changes of individual cells in small (microliter) sample volumes. The cells are kept fixed in position in the upper compartment of the chamber by means of a highly permeable membrane and exposed to a step-like change in concentration generated in the lower compartment. An electrical conductivity probe in the upper compartment was used to monitor the temporal change of salt concentration as experienced by the cells. The rise from isotonic to hypertonic can be approximated by an exponential function. Its time constant of tau = 2.08 sec seems to be mainly determined by the change in flushing solution as tau = 1.48 sec was measured with no membrane installed. With human lymphocytes, no loss of cell volume was detected before 5 sec, i.e., when 95% of the final concentration was reached extracellularly. A step change can hence be assumed when modeling exosmosis for determining the lymphocyte membrane permeability. The equations for coupled transport of water and salt were solved numerically and fitted to the experimental data. The results were also compared to various other transport models described in the literature. Human lymphocytes are almost ideally semipermeable with a hydraulic reference permeability of Lp = 4.23 X 10(-4) cm/sec (3.13 X 10(-3) micron X atm-1 X sec-1) at T = 23 degrees C. The temperature and concentration dependence are described by an activation energy Ea = 14.3 kJ/mol and a concentration coefficient alpha 2 = 0.261 osmol/kg. An osmotically inactive volume fraction of 36.9% was determined from the final cell volumes reached asymptotically after shrinkage.     

Microvascular albumin permeability in isolated perfused lung: effects of EDTA

We examined the effects of decreases in perfusate concentrations of calcium and magnesium on the pulmonary vascular permeability in the isolated perfused rabbit lung. The albumin permeability-surface area product (PS) and the albumin reflection coefficient (sigma) were determined in the same lung using 125I- and 131I-labeled albumin tracers. Decreases in vascular Ca2+ and Mg2+ concentrations were induced by adding ethylenediaminetetraacetic acid (EDTA) to the perfusate. Decreases in the concentration of these cations resulted in an increase in the PS from a control value of 1.18 +/- 0.13 X 10(-3) to 7.69 +/- 0.75 X 10(-3) cm3 X min-1 X g wet lung wt-1 and a decrease in the sigma from 0.96 +/- 0.01 to 0.74 +/- 0.02. The decrease in sigma suggests an increase in the calculated equivalent pore radius from 44 to 63 A. The results indicate that Ca2+ and Mg2+ play a role in the maintenance of normal pulmonary vascular permeability to proteins.     

OZONE-INDUCED MEMBRANE PERMEABILITY CHANGES

The effect of 0.5 ppm ozone for 0.5-1 hr on plant cell membrane permeability was ascertained. Permeabilities to both water and solutes were estimated by measuring leaf disc weight changes and following tritiated water and ⁸⁶Rb fluxes. Measurements were made immediately after ozone exposure and 24 hr after exposure. The reflection coefficient, σ, an index of solute permeability, decreased in ozone-treated primary leaves of pinto bean (Phaseolus vulgaris). The latter indicates an increase in membrane solute permeability or internal solute leakage. Water and THO flux estimates both indicated a decrease in membrane permeability to water; both the hydraulic conductivity (L_p) and the water diffusional coefficient (L_D) apparently decreased, an anomaly which is discussed. These data indicate that ozone has a direct effect on membrane function by altering permeability characteristics. We assume from these data that cell membranes are primary target sites for ozone injury.     

Determination of bull sperm membrane permeability to water and cryoprotectants using a concentration-dependent self-quenching fluorophore

The objective of this study was to determine the membrane permeability characteristics of bovine spermatozoa. These included the hydraulic conductivity (Lp), the permeability coefficients (Ps) of four common cryoprotective agents (CPAs) and the associated reflection coefficients (sigma). Stopped-flow fluorometry was applied in order to capture rapid cell volume changes under anisosmotic conditions in the absence or presence of permeant solutes (CPAs). This technique utilizes a concentration-dependent self-quenching entrapped fluorophore. The resulting cell volume changes were used in three-parameter fitting calculations to compute Lp in the absence of permeant solutes and Ps and Lp in the presence of permeating solutes (CPAs) at 22 degrees C. The hydraulic conductivity in the absence of permeating solutes was estimated to be 0.68+/-0.05 microm/min/atm (mean+/-SEM). Hydraulic conductivity (Lp) in the presence of CPAs was 0.91+/-0.27 (mean+/-SEM), 0.29+/-0.04, 0.42+/-0.05, and 0.39+/-0.03 microm/min/atm in the presence of dimethylsulfoxide (Me(2)SO), glycerol, propylene glycol (PG), and ethylene glycol (EG), respectively. The values for Ps were estimated to be 1.72+/-0.36, 1.75+/-0.03, 2.47+/-0.24, and 1.49+/-0.33 x 10(-3)cm/min for Me(2)SO, glycerol, PG, and EG, respectively. The data were used to simulate volume excursions during addition and removal of CPA, to predict the different effects of the four CPAs.     

Survival of frozen-thawed human red cells as a function of the permeation of glycerol and sucrose

Previous studies have demonstrated that glycerol does not have to permeate bovine red cells to protect them against subsequent freezing and thawing. The present study is concerned with the relation between solute permeation and freezing injury of human red cells. Cells were held in 2 m glycerol for 30 sec to 10 min at 0 °C and then frozen to ?196 °C at 60 °C/min. Cells cooled at this rate have a very low probability of undergoing intracellular freezing. Percent survivals (≡percent unhemolyzed) increased by 21% (from 66 to 80%) over the first 3-min period. Extrapolation to zero time (and zero glycerol permeation) yields a survival of 57%. Between 30 sec and 3 min the calculated osmolal ratio of intracellular glycerol to other solutes increased 240% (from 2.5 to 5.7). The human red cell is impermeable to sucrose at 0 °C. Cells suspended in 1.40 m sucrose (equiosmolal to 2.0 m glycerol) for 0.5 to 10 min prior to freezing yielded as high survivals after thawing as did cells in glycerol.These data indicate that prior permeation of additive is not a prerequisite for the survival of red cells subjected to subsequent freezing and thawing. Although sucrose and glycerol protect equally well to this point, differences appear when attempts are made to remove the additive. Over 90% of the cells survive the removal of glycerol. Only some 30% survive the removal of sucrose. Cells frozen in an equisomolal solution of sodium chloride do not even survive the initial freezing and thawing.The findings indicate that slow freezing injury cannot be accounted for in terms of the attainment of a critical minimum volume, nor can it be considered to be equivalent to posthypertonic hemolysis.     

Determination of Solute Permeability in Chara Internodes by a Turgor Minimum Method : Effects of External pH

The analysis of Sha'afi et al. (Sha'afi, Rich, Mickulecky, Solomon 1970 J Gen Physiol 55: 427-450) for determining solute permeability in red blood cells has been modified and applied to turgid plant cells. Following the addition of permeating solute to the external medium, a biphasic response of cell turgor can be measured with the pressure probe in isolated internodes of Chara corallina. After an initial decrease in turgor due to water flow (water phase), turgor increases due to the uptake of the solute (solute phase) until the original turgor is reattained. From the pressure/time course in the neighborhood of the minimum turgor, the permeability of the osmotic solute can be determined. The data obtained by the minimum method for rapidly permeating (ethanol, methanol) and slowly permeating (formamide, dimethylformamide) solutes are similar to those calculated from the half-time of pressure changes during the solute phase and to data obtained by other workers using radioactive tracers. The methods employing the pressure probe were applied to examine the effect of high pH (up to pH 11) on the membrane permeability. There appeared to be no effect of high pH on the permeability coefficients, reflection coefficients, and hydraulic conductivity.     

Transport of water and solutes across bullfrog alveolar epithelium

Water and solute transport properties of the alveolar epithelium of isolated bullfrog lungs were studied. Lungs from Rana catesbeiana were removed and mounted in an Ussing chamber. Unstirred layers on both sides of the tissue were estimated from the time courses of dilution potential development, and the measured transport parameters were corrected for the effect of the unstirred layers. Spontaneous potential difference, short-circuit current, tissue resistance, instantaneous voltage-current relationships, diffusional permeabilities of water and hydrophilic solutes, and hydraulic conductivities were determined. The hydraulic conductivity obtained from hydrostatically driven water flow anomalously decreased with time, and was initially 100 -1,000 times higher than osmotically determined hydraulic conductivity. The equivalent pore radius of the bullfrog alveolar epithelium was estimated to be 0.8-0.9 nm. We conclude that the alveolar epithelium is extremely tight, presenting a major barrier to water and solute flow. This high resistance to water and solute flow may be helpful in maintaining the alveolar lumen relatively free of fluid under normal physiological conditions.     

Determination of reflection coefficients for various ions and neutral molecules in sarcoplasmic reticulum vesicles through osmotic volume change studied by stopped flow technique

Osmotic volume change of sarcoplasmic reticulum vesicles was studied by following the change in light-scattering intensity using a stopped flow apparatus. From the analysis of the initial rate of scattering change, reflection coefficients for various ions and neutral molecules were determined. The following are typical results: K+, 0.72; Tris+, 0.98; choline 1; NO3-, 0.32; Cl-, 0.46; methanesulfonate, 0.62; gluconate, 0.96; glycerol, 0.86; and glucose, 1. When the K+ permeability was increased in the presence of 10(-6) g valinomycin/ml, the reflection coefficient for K+ changed from 0.72 to 0.31. It was found that there was a close relationship between the reflection coefficients and the permeabilities of the solutes. Hydraulic conductivity was also determined from the initial rate of light scattering change and was not different for the different solutes. The water permeability was estimated to be 2.1 x 10(-3) cm/sec at 23 degrees C.     

The Measurement of Permeability to water in Disks of Storage Tissues4

1. Experiments have been performed to examine methods to determinehydraulic conductivity and diffusional permeability in disksof red beet (Beta vulgaris) and Jerusalem artichoke (Helianthustuberosus). 2. The half-time for the weight change caused by transferringdisks between pure water and a sucrose solution, or vice-versa,has been shown to be linearly dependent on the square of thedisk thickness. 3. The efflux kinetics of sucrose from these disks are alsocharacterized by a half-time linearly dependent on the squareof the disk thickness. 4. It is concluded that extracellular solute diffusion, andnot hydraulic conductivity, is the principal rate-controllingfactor in experiments of this type. 5. Efflux studies with tritiated water also show a characteristichalf-time which is linearly dependent on the square of the diskthickness. The cell membrane does not constitute a sufficientbarrier to diffusion fot its diffusional permeability to bedetermined.     

Revisiting the Münch pressure-flow hypothesis for long-distance transport of carbohydrates: modelling the dynamics of solute transport inside a semipermeable tube

A mathematical model of the Münch pressure-flow hypothesis for long-distance transport of carbohydrates via sieve tubes is constructed using the Navier-Stokes equation for the motion of a viscous fluid and the van't Hoff equation for osmotic pressure. Assuming spatial dimensions that are appropriate for a sieve tube and ensuring suitable initial profiles of the solute concentration and solution velocity lets the model become mathematically tractable and concise. In the steady-state case, it is shown via an analytical expression that the solute flux is diffusion-like with the apparent diffusivity coefficient being proportional to the local solute concentration and around seven orders of magnitude greater than a diffusivity coefficient for sucrose in water. It is also shown that, in the steady-state case, the hydraulic conductivity over one metre can be calculated explicitly from the tube radius and physical constants and so can be compared with experimentally determined values. In the time-dependent case, it is shown via numerical simulations that the solute (or water) can simultaneously travel in opposite directions at different locations along the tube and, similarly, change direction of travel over time at a particular location along the tube.