JXB Advance Access originally published online on November 3, 2006
Journal of Experimental Botany 2006 57(15):4133-4144; doi:10.1093/jxb/erl190
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© 2006 The Author(s).
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/2.0/uk/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.This paper is available online free of all access charges (see http://jxb.oxfordjournals.org/open_access.html for further details)
RESEARCH PAPER |
Further quantification of the role of internal unstirred layers during the measurement of transport coefficients in giant internodes of Chara by a new stop-flow technique
Department of Plant Ecology, Bayreuth University, D-95440 Bayreuth, Germany
To whom correspondence should be addressed. E-mail: ernst.steudle{at}uni-bayreuth.de
Received 8 June 2006; Accepted 6 September 2006
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Abstract |
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A new stop-flow technique was employed to quantify the impact of internal unstirred layers on the measurement of the solute permeability coefficient (Ps) across the plasma membrane of internodes of the giant-celled alga Chara corallina using a cell pressure probe. During permeation experiments with rapidly permeating solutes (acetone, 2-propanol, and dimethylformamide), the solute concentration inside the cell was estimated and the external medium was adjusted to stop solute transport across the membrane, after which responses in turgor were measured. This allowed estimation of the solute concentration right at the membrane. Stop-flow experiments were also simulated with a computer. Both the stop-flow experiments and simulations provided quantitative data about internal concentration gradients and the contribution of unstirred layers to overall measured values of
for the three solutes. The stop-flow experimental results agreed with stop-flow simulations assuming that solutes diffused into a completely stagnant cell interior. The effects of internal unstirred layers on the underestimation of membrane Ps declined with decreasing Ps. They were no bigger than 37% in the presence of the most rapidly permeating solute, acetone (=4.2x10–6 m s–1), and 14% for the less rapidly permeating dimethylformamide (=1.6x10–6 m s–1). It is concluded that, even in the case of rapidly permeating solutes such as isotopic water and, even when making pessimistic assumptions about the internal mixing of solutes, an upper limit for the underestimation of Ps due to internal unstirred layers was 37%. The data are discussed in terms of recent theoretical estimates of the effect of internal unstirred layers and in terms of some recent criticism of cell pressure probe measurements of water and solute transport coefficients. The current stop-flow data are in line with earlier estimations of the role of unstirred layers in the literature on cell water relations. Key words: Cell pressure probe, Chara corallina, internal unstirred layers, solute permeability coefficient, stop-flow technique
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Introduction |
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Whenever transport across membranes or other types of permeation barriers is measured and quantified in terms of certain coefficients, such as the permeabilities of water and solutes, unstirred layers (USLs) may affect the measurement and should be accounted for. This is so, because the permeation of substances across the membrane (or barrier) should cause a depletion of the permeant on one side of the membrane while increasing its concentration on the other. The relative contribution of USLs to the overall measured permeability depends on the rate at which solutes or water move across the membrane as compared with the rate at which substances diffuse from the bulk solution to the membrane surface or vice versa. Stirring of the media separated by the membrane can substantially reduce the thickness of USLs but can never completely eliminate them (Ye et al., 2006). Relative contributions of USLs increase with increasing permeability of solutes. Whenever the actual transport properties are required for a single membrane rather than for an entire barrier, the effects of USLs have to be quantified.
A USL is a region of slow laminar flow parallel to the membrane in which the only mechanism of transport is by diffusion (Dainty, 1963). When dealing with non-electrolytes, there are two different kinds of effects of USLs. These depend on (i) whether transport across the complex barrier consisting of USLs and membrane is just diffusional in nature, or (ii) is both diffusional and convective in nature. For the first model, when a permeating solute diffuses across a membrane, depending on the diffusional supply from the bulk solution to the membrane, the actual concentration gradient driving the solution permeation across the membrane may be smaller than that measured in the bulk solution. This type of effect of USL has been termed the ‘gradient-dissipation effect’ (Barry and Diamond, 1984). In the second model, solutes are swept by convection with the water in the perpendicular direction to the membrane where they are concentrated on one side but depleted on the other. Concentration gradients built up in the solution and adjacent to the membrane will be opposed by a back-diffusion within USLs. This type of USL effect in the presence of water flow across the membrane has been termed ‘sweep-away effect’ (‘convection versus diffusion’; Dainty, 1963). Both types of effects of USLs result in overall permeabilities for water and solutes that are smaller than those of just the membrane.
In a previous paper, Ye et al. (2006) examined how USLs contributed to the measurement of transport coefficients such as the hydraulic conductivity (Lp, water permeability), permeability coefficient (Ps, solute permeability), and reflection coefficients of solutes (
s) in giant cylindrical internodes of Chara corallina. The big, isolated cells (diameter: 2R 
1 mm; length 40–120 mm) could be measured in fairly turbulent media to reduce the thickness of external USLs, thus minimizing their effects. However, as the cell interior could not be stirred, this could have caused the build-up of substantial internal USLs. The results of Ye et al. (2006) showed that sweep-away effects are usually negligible, but gradient-dissipation effects could be significant in the presence of rapidly permeating solutes (the typical solute used was acetone with a high membrane Ps). In the latter case, the diffusional resistances of internal USLs (
i/Ds in which i is the thickness of internal USLs and Ds is the diffusion coefficient) were not negligible. For example, Tyree et al. (2005) claimed that, for the rapidly permeating test solute acetone, the real membrane permeability could be as big as five times the measured permeability, suggesting a rate-limitation by USLs. These authors assumed fairly thick internal and external USLs that dominated the overall solute transport. Ye et al. (2006) applied non-steady-state diffusion kinetics to Chara internodes that were taking up or losing a rapidly permeating solute [e.g. the test solute acetone as measured with the cell pressure probe (CPP)]. The upper limit of the equivalent i of internal USLs was 97 µm in this case, which referred to an upper limit of 40% of the contribution of internal USLs to the overall measured permeability coefficient (
). Ye et al. stress that this referred to a completely stagnant cell interior with no mixing by cytoplasmic streaming, local differences in density, or shaking and bending of cells during the experiments. Ye et al. (2006) estimated the real equivalent i to be around 50 µm. Using i values of 50 and 100 µm for acetone and a of 4.2x10–6 m s–1, the actual Ps of the membrane equalled 5.2x10–6 m s–1 or 7.0x10–6 m s–1, respectively (Ds=1.2x10–9 m2 s–1; R=0.4 mm). This means that the contribution of internal USLs could have been as large as 19% or 40%, respectively.
The problem is important in cell water relations whenever comparisons are made between the osmotic water permeability, Pf (Pf=Lp·Vw/RT where Vw is the molar volume of liquid water) and the diffusional water permeability, Pd, measured with isotopic water (Steudle and Henzler, 1995; Sehy et al., 2002; Henzler et al., 2004). It has been readily shown that the ratio of Pf/Pd is equal to the number (N) of water molecules sitting in an aquaporin pore, provided that effects of USLs (namely in Pd) can be neglected (Levitt, 1974; Finkelstein, 1987; Ye et al., 2005). This is always the case when cells or vesicles are small, but may present a problem for big cells such as Chara internodes or Xenopus oocytes (Sehy et al., 2002).
The present paper continues to quantify the role of diffusional USLs, focusing on internal diffusional USLs. Ye et al. (2006) showed that vigorous external stirring minimized the effects of external USLs, but the role of the internal USLs could not yet be verified experimentally. To do this, a new stop-flow technique (SFT) was developed to work out the solute profile in the cell, which is otherwise difficult to measure. Knowing the right concentration at the membrane surface is necessary to evaluate the ‘true’ membrane permeability of a solute, Ps. In the SFT, it was intended to stop the solute flow across the membrane by applying the same concentration in the external medium as that of the cell interior by trial and error. In this way, the right concentration at the membrane was accessed or estimated to get an idea of whether or not there was a significant USL inside the cell. Besides the stop-flow measurements, a computer simulation of stop-flow experiments was performed, assuming either a well-stirred or a completely stagnant cell interior (minimum or maximum contribution of USLs). Results from simulations were compared with those of experiments to provide deeper insights into the effects of internal USLs. In both the experiments and simulations, solutes of different permeability were used. The Ps of the most rapid solute acetone was similar to that of isotopic water so that effects were tested in the presence of an extremely rapid permeant.
The results of the present paper were in line with recent findings of Ye et al. (2006) who showed that (i) during sweep-away, effects of USLs on Lp measurements were negligible and that (ii) the effects on permeability coefficients were up to 40% in the case of the most rapid solute acetone. The latter conclusion was drawn from a theoretical consideration of the diffusion within the cylindrical internodes. The current results agreed with conclusions from other workers in the field (e.g. Steudle and Tyerman, 1983; Hertel and Steudle, 1997; Henzler and Steudle, 2000; Ye et al., 2005). They disagreed with recent estimates of Tyree et al. (2005), who claimed that USLs may dominate the measurement of transport coefficients with cell pressure probes, at least in the presence of rapidly permeating substances (such as the test solute acetone; see discussion in Ye et al., 2006).
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Theory and results from computer simulations |
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When a permeating solute is added to an isolated cell sitting in a well-stirred medium, the water (JV) and solute flow (Js) are calculated by two coupled differential equations derived from irreversible thermodynamics (‘phenomenological equations’; Kedem and Katchalsky, 1958; Dainty, 1963; Steudle and Tyerman, 1983; Steudle, 1993; Steudle and Henzler, 1995):
 | (1) |
 | (2) |
is the number of moles of permeating solute ‘s’ in the cell, P denotes the actual turgor pressure, Lp is the hydraulic conductivity, Ps is the permeability coefficient of the membrane for a given solute, s is the reflection coefficient of solute ‘s’, which denotes the passive selectivity of the membrane for a given solute, 
is the concentration of ‘s’ inside of the membrane, 
is the concentration outside of the membrane, and 
is the mean concentration in the membrane [= + /2]. The Ci and Co are concentrations of impersoluble solutes inside and outside of the membrane. On the right side of equation 1, it is indicated that there are two components of water flow, one hydrostatic and the other osmotic. The solute flow in equation 2 has three components: one is diffusional [Ps(– )], another is solvent-drag [(1–s)· ·JV)], and the third is an active flow of solute ‘s’, 
. Active transport was neglected for the test solutes used in the current work. Usually, the solvent drag does not contribute much to the overall solute flow and may be neglected as well, as shown in earlier comparisons with experimental findings and computer simulations (Rüdinger et al., 1992; see below). Using these assumptions, Steudle and Tyerman (1983) provided an analytical solution of equations 1 and 2 that represents the time-course of biphasic pressure relaxations in the presence of permeating solutes, added at t=0:
 | (3) |
is the cell elastic modulus, 
is the change in the external osmotic pressure of permeating solute added at t=0 to produce the osmotic response, 
i is osmotic pressure of the cell, and kw and ks are the rate constants for water and solute exchange, respectively, whereby kw >> ks usually holds.
Equation 3 states that, following a rapid water phase, solute flow will be following first-order kinetics, i.e. the solute phase should be exponential after sufficiently long time intervals, when the term exp(–kw·t) vanishes. In the Steudle–Tyerman theory, USLs either play no role or are quickly constant and contribute to the overall measured value of solute permeability (), which depends on the solute permeability of the membrane (Ps), the thickness of internal USLs (i), and the diffusion coefficient of the solute in the cell (Ds; see above), i.e.
 | (4) |
 | (5) |
i=R–r. In the presence of an external USL, equation 5 may be extended (Ye et al., 2006). Different from equations 4 and 5, real diffusive USLs do not have well-defined edges, and the term ‘thickness’ is used somewhat vaguely. There are two definitions: (i) USLs with a sharp physical boundary to the rest of the cell that is vigorously stirred, and (ii) USLs created during non-steady-state diffusion, when the contribution of the USL becomes virtually constant (‘equivalent unstirred layer thickness’). Definition (i) is a useful hypothetical situation because it assumes that the interior of a compartment, such as a Chara internode, can be well stirred except for a layer of i thickness. Provided that i is small compared with the radius, the concentration gradient within the USL should be linear to a good approximation, following a rather short time interval after adding the solute to the medium. In definition (ii), solutes cross the plasma membrane (PM) and then diffuse into a completely stagnant cell interior. During this non-steady process, the actual thickness will grow continuously until, at a certain thickness of the diffusive layer, it will tend to become quasi-steady. During this state, an ‘equivalent USL thickness’ may be defined (Ye et al., 2006). This case appears to be more realistic than case (i). In the present paper, the concept of an equivalent USL thickness was used. Despite the difficulties in defining equivalent thicknesses of USLs, the USL concept is useful. It provides figures to judge the contribution of boundary layers close to membranes to overall measured permeabilities, namely in the presence of rapidly permeating solutes. Other concepts, such as the use of time constants, result in similar difficulties (Sehy et al., 2002) that are due to the complex nature of diffusion kinetics in the non-steady case.
When a steady internal thickness of USLs is assumed and hence a steady 
, equation 3 should hold, as indicated by numerous earlier results from pressure probe measurements using a suite of test solutes of different permeability (Steudle and Tyerman, 1983; Tyerman and Steudle, 1984; Rüdinger et al., 1992; Steudle and Henzler, 1995; Henzler and Steudle, 1995; Schütz and Tyerman, 1997; Hertel and Steudle, 1997). However, measurements with most rapidly permeating solutes, including heavy water and acetone, may cause a problem (Ye et al., 2004, 2006; Tyree et al., 2005). For these solutes, an estimated upper limit for the contribution of internal USLs was between 25% and 40% (Ye et al., 2005, 2006).
Simulations of diffusion inside the cell
To quantify the role of internal diffusive transport in Ps measurement further, classical numerical approaches for diffusion in a cylinder were used (see computational fluid dynamics textbooks such as that by Abbot and Basco, 1990). Internal USLs were modelled assuming that diffusion within Chara internodes was between thin concentric cylindrical shells according to Fick's first law at a given time. Taking into account the cylindrical geometry (equation 5), the differential equation for the diffusion of solute ‘s’ in a cylinder (e.g. Steudle and Frensch, 1989) is:
 | (6) |
 | (7) |
 | (8) |
 | (9) |
[i] is the concentration of solute in shell i at a given time, radius r[i] is the distance from the centre of the cylinder to the centre of the shell i, and L is the cell length. In the simulations, time intervals of dt=0.01 s and radius increments of the shells, 
r=5 µm, were used (the radius of the cell, R=400 or 500 µm). Further reduction of dt to 0.001 s and of r to 1 µm did not further refine the internal profiles of concentration (data not shown). Also, it was verified at given times, 
, and different distances from the membrane, x, so that x2/ was constant for a given concentration as predicted by basic diffusion kinetics (relation of Einstein and Smoluchowski; see textbooks on diffusion such as Jost, 1960).
Simulation of effects of internal USLs during stop flow
For membrane transport in a Chara internode, equations 1 and 2 were incorporated into simulations. During the simulations, solvent drag did not occur between shells, but instead was incorporated at the membrane. In equations 2 and 7, the transfer of solutes across the membrane and between adjacent shells is denoted, respectively. A standard computer program was used to integrate equations 1, 2, and 6 numerically. This resulted in (i) concentration profiles within the cells and (ii) overall rates of uptake or losses by cells. Hence, it was tested in simulations whether or not the overall or mean concentration of solute ‘s’ in the cell (‘prospected concentration’; as assumed by Steudle and Tyerman, 1983; equation 3) was a good estimate for the concentration adjacent to the membrane.
The Steudle–Tyerman theory assumes either that there are no effects of USLs or that these effects are incorporated into the value of . So far, the evidence that the contribution of internal USLs is relatively small is derived from the comparison of measured rates of solute flow (CPP) with rates expected from diffusion kinetics (Ye et al., 2006). Hence, the evidence is indirect. The problem is 2-fold: (i) by contrast to the outer solution, the cell interior could not be stirred to test for the contribution of internal USLs; (ii) the concentration adjacent to the inner side of the PM could not be directly measured in order to determine how it would deviate from that used to calculate Ps, assuming that the cell interior was sufficiently stirred by mixing. In the following, results of simulations (Figs 1–4), which are later compared with those of experiments, are presented (Figs 5–7).
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In the SFT, the concentration in the cell was first calculated using the Steudle–Tyerman theory (equation 3) for a given time, i.e. a mean concentration was calculated and named the ‘prospected concentration’, Cprosp. The internal concentration could be calculated at each pressure (time) point of the solute phase by the difference between the extrapolated pressure to time zero and the original turgor, Po (
Pprosp), because the solute phase and extrapolated curve of it represented the solute uptake or loss in the cell (Fig. 1). In the endosmotic experiment (solute added outside and moving into the cell), once the cell's internal concentration had reached the Cprosp, the Cprosp was then applied to the outside medium, which tended to stop the solute flow across the membrane and bring the turgor pressure back to its original value, Po. If there were no internal USLs, when the Cprosp was added to external media, the Cprosp was the same as the concentration at the membrane, 
, and no solute phase was observed in the simulation (Fig. 2b, solid line). However, assuming an internal USL, the solute flow did not stop when the Cprosp was added to the external media (Fig. 2b, dashed line). This was due to the concentration gradient developed in the cell by the diffusion process. Concentrations that were higher and lower than Cprosp were selected and applied to external media to find the appropriate value at which solute flow would stop. Depending on whether the selected concentration was smaller or bigger, this resulted in either an overshoot (Fig. 2a) or undershoot (Fig. 2c) of the turgor pressure, respectively. After solution exchange, pressure responses showed another ‘water phase’ as in Fig. 1. Some water had to be transferred to adjust the pressure when the solution was changed. The true stop-flow pressure, Pstop, could be obtained by extrapolating to the time when the solution was changed, as shown in Fig. 2.
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Simulated values of
P=Pstop–Pe were plotted against the concentration applied to stop the solute flow where Pstop was the turgor pressure extrapolated to the time, and Pe, the final steady-state turgor. The internal mean concentration at the point at which the solution was changed, as suggested by stop-flow (CSF), was determined by the intercept with the abscissa (Fig. 3). Figure 3 shows that in the absence of USLs, the simulation produced the CSF value identical to the Cprosp, which was the internal mean concentration when the solution was changed (500 mM in the example shown in Fig. 3). However, in the presence of internal, diffusive USLs, the intercept occurred at a concentration of 477 mM of CSF, which is smaller than 500 mM by 5% (Fig. 3). This was observed because at the solution changing point, the concentration inside the cell at the membrane () was bigger than Cprosp, and solutes moved out of the cell when the external concentration of Cprosp was added. This first outward solute movement was buried in the water phase. After a certain amount of solutes was moving out of the cell, the direction of solute movement was reversed to go into the cell and this was observed in a solute phase. Because the solute phase showing the solutes going into the cell was used, CSF value was estimated to be smaller than the Cprosp. At the solution changing point, the was obtained as 703 mM by the computer simulation of the internal concentration profile (data not shown). The was identified only by simulations and not by experiment. By contrast, CSF was measured using both experiments and simulations. CSF is an internal mean concentration calculated from stop-flow curves and is a good indicator of whether or not there is a USL inside the cell. In the corresponding exosmotic experiment (such as in Fig. 1; solutes moving out of the cell), CSF was 5% greater than the prospected 500 mM. This means that the effects were symmetrical and independent of the direction of solute flow, as one would expect (data not shown).
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When the measured permeability coefficient,
, was varied during simulations, but the other parameters were left constant, the relative contribution of internal USLs increased as increased, as one would expect (Fig. 4). A new parameter, the relative difference of concentrations [RDC=(CSF–Cprosp)/Cprosp] between CSF and Cprosp was introduced. The RDC was a direct measure that could quantify the effects of internal USLs. In their absence, RDCs were zero for every Ps, and measured permeabilities were identical with the true membrane Ps (Fig. 4a, b). However, in the presence of USLs, RDCs increased for a bigger Ps tending to increase the underestimation of solute permeability (Fig. 4a, b). The relative contribution of USLs was bigger in the values of than in those of RDC, because the calculation of Ps accounted for the concentration profile in the cell (Fig. 4b).
In simulations based on the measured permeability coefficient for acetone (=4.2x10–6 m s–1), the RDC was ±6% in exosmotic and endosmotic experiments, respectively (data not shown). The corresponding underestimation of Ps (accounting for the concentration profile in the cell) was 37% in both cases (data not shown). For the less rapidly permeating dimethylformamide (DMF) (=1.6x10–6 m s–1) simulations resulted in an underestimation of RDC by only 1%, and the underestimation in Ps was 14% (data not shown; see comparison with experiments in Results).
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Materials and methods |
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Algal material
Chara corallina was grown in artificial pond water (APW; composition in mM: 1.0 NaCl, 0.1 KCl, 0.1 CaCl2, and 0.1 MgCl2 at a pH
5.5). For detailed information of the growing conditions, the reader is referred to Henzler et al. (2004) and Ye et al. (2005). Chara internodes were 40–120 mm long and 0.8–1.0 mm in diameter and could be treated as cylinders to a good approximation.
Determination of transport parameters (LP, Ps, and s) and cell wall elasticity ()
Using the CPP, three transport parameters were measured (Steudle, 1993): (i) hydraulic conductivity (Lp) is a measure of water permeability across the cell membrane; (ii) permeability coefficient (Ps) denotes the passive permeability of the cell membrane for a given solute; (iii) reflection coefficient (s) is a quantitative measure of the ‘passive selectivity’ of the cell membrane for a solute as compared with that of water. The elastic coefficient of the cell wall was measured as well (elastic modulus, ). This parameter is required to relate the pressure/time curves measured with the probe to volume/time curves to determine water flow and cell Lp. Hydraulic conductivity was calculated from half-times of hydrostatic pressure relaxations (
). In the presence of permeating solutes, P(t) curves were biphasic with a water and solute phase. From the latter phase, Ps values were calculated from the half times (
). Reflection coefficients were obtained from maximum changes in pressure following step changes in the concentration of permeating solutes. Equations used for calculating , Lp, Ps, and s were:
 | (10) |
 | (11) |
 | (12) |
 | (13) |
i is osmotic pressure of cell sap, ks is the rate constant of solute exchange, Po–Pmin(max) is the maximum change in cell turgor pressure (Fig. 1), RT· is the given change of osmotic pressure of the medium, and tmin(max) is the time spent to reach the minimum (maximum) change in cell turgor pressure during the first phase (water phase) of the osmotic experiments.
‘Stop-flow technique (SFT)’ in osmotic experiments with permeating solutes
Stop-flow experiments were performed with three different solutes that varied in their rates of membrane permeation. Differences between prospected and measured concentrations (Cprosp and CSF, respectively) were expected to be largest for the most rapidly permeating solute, acetone (=4.2x10–6 m s–1; Ds=1.2x10–9 m2 s–1). These differences were expected to decrease for 2-propanol (=2.1x10–6 m s–1 and Ds=1.0x10–9 m2 s–1) and DMF (=1.6x10–6 m s–1 and Ds=1.0x10–9 m2 s–1). In the experiments, 1000 mM of acetone, 600 mM of 2-propanol, and 300 mM of DMF were used, and the concentrations of these solutes in the cell changed with time as they permeated into or out of the cell. Prospected concentrations of these solutes inside of the cell at certain times (pressures) during biphasic pressure/time-courses (Fig. 1) were chosen to be around 500 mM, 300 mM, and 150 mM for acetone, 2-propanol, and DMF, respectively (Cprosp). First, a cell was exposed to either acetone (500 mM), 2-propanol (300 mM), or DMF (150 mM), and biphasic curves were observed. The solute phase was fitted exponentially, and pressure differences (Pprosp) were measured after extrapolating the solute phase back to the exact point where the solute was added, namely, at time zero to consider the solute movement during the water phase (Fig. 1). Next, a higher concentration of acetone (1000 mM), 2-propanol (600 mM), or DMF (300 mM) was added to the external solution. At that time (pressure) point calculated from Pprosp, the solute phase was interrupted when the internal cell concentration was expected to be 500 mM (acetone), 300 mM (2-propanol), or 150 mM (DMF). Depending on the internal concentration chosen, the concentration series for the exchange solution was 150, 300, 500, 700, 850 mM for acetone; 100, 200, 300, 400, 500 mM for 2-propanol; and 50, 100, 150, 200, 250 mM for DMF. For each stop-flow pressure/time curve produced, the true stop-flow pressure, Pstop, was found by extrapolation to the time when the solution was changed and the pressure differences (P=Pstop–Pe) were measured. Internal concentration at the solution changing point, suggested by stop-flow (CSF), was obtained from the intercept with the abscissa (P=0) by plotting P versus the concentration added during the solute phase.
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Results |
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Typical curves from stop-flow experiments are shown in Fig. 5. In the first example, cell A was initially treated with 1000 mM acetone, causing an endosmotic response in pressure (solute uptake). After reaching solute flow equilibrium, there was the same concentration of acetone at all points within the cell (1000 mM). When the acetone solution was replaced by APW free of acetone, the acetone permeated out of the cell due to the concentration difference between the cell interior and the medium (
was kept at zero). Consequently, an exosmotic pressure curve was produced. During the solute phase, when acetone diffused out of the cell, the internal acetone concentration decreased from 1000 mM. Solute efflux was interrupted at the point where the internal acetone concentration was expected to be Cprosp=490 mM by adding acetone solutions with concentrations of between 125 mM and 785 mM (see Materials and methods). In Fig. 5a, b, the lowest and highest concentrations in the series are given as examples. Addition of 125 mM produced an undershoot (Fig. 5a) and 785 mM an overshoot (Fig. 5b) in pressure relative to the baseline (Po). Since the pressure/time curve during the solute phase was exponential, the pressure difference (P=Pstop–Pe) was measured after extrapolation of the solute phase to the time when the solution was changed. The values of P were used to determine CSF as the intercept with the abscissa (Fig. 6), as was also done in the simulations (Fig. 3). It should be noted that during stop-flow experiments, the solute phases following the water exchange had similar as those measured in unchanged biphasic responses (Fig. 5a, b).
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) of the three solutes were given as mean ±standard deviation (n=7–12 cells).In the second example, cell B was first treated with 600 mM of the less-permeating 2-propanol. After replacing the 2-propanol solution with APW, the solute phase was interrupted by the addition of 100 mM (Fig. 5c) or 500 mM (Fig. 5d) 2-propanol, at Cprosp=278 mM. Again,
P values were worked out after extrapolation to the point of solution exchange (Fig. 5c, d).
In the last example, cell C was treated with DMF but the results were presented for an endosmotic stop-flow experiment (solute uptake; Fig. 5e, f). First, 300 mM DMF was added to the medium and, after it began to diffuse into the cell, the movement was interrupted by adding 150 mM (Fig. 5e) or 50 mM (Fig. 5f) DMF at a time when Cprosp=154 mM. Again, P values were measured after extrapolation to the point of solution exchange (Fig. 5e, f).
Data from the stop-flow experiments were used to plot P versus the external concentration, allowing for the calculation of CSF (Fig. 6). Representative examples of endosmotic stop-flow experiments show the deviation between CSF and Cprosp for the three solutes: CSF and Cprosp were (i) for acetone, 464 and 490 mM, respectively (Fig. 6a), (ii) for 2-propanol, 272 and 278 mM (Fig. 6b), and (iii) for DMF, 152 and 154 mM (Fig. 6c). Hence, deviations from the prospected values were biggest for the most rapidly permeating solute (acetone), and lowest for the slowly permeating DMF.
To avoid variability between cells, RDCs were measured from stop-flow experiments in repetition. Results are summarized in Fig. 7 (mean ±standard deviation; n=2 or 3 cells). The values of Cprosp were around 500 mM, 300 mM, and 150 mM for acetone, 2-propanol, and DMF, respectively. The mean values of RDC decreased for acetone (–13/+11%: endo/exo), 2-propanol (–3/+1%), and DMF (–2/+1%) with a decreasing permeability coefficient of solutes (Ps). This was in agreement with the results obtained during the simulation (±6% for acetone; ±2% for 2-propanol; ±1% for DMF; see results from computer simulations). This indicated that effects of internal USLs differed depending on the Ps of the solute used (see Discussion). Effects were biggest for acetone and smallest for DMF due to the relative importance of USLs (i/Ds) as compared with 1/Ps (see the Theory and results from computer simulations section).
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Discussion |
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Both the experimental data obtained from stop-flow experiments and the simulations indicated a similar pattern of effects of internal USLs. As the solute permeability of the membrane decreased, the role of internal USLs also decreased. Effects due to internal USLs were only substantial for the most rapidly permeating solute, acetone, where the underestimation of Ps was found to be up to 37% according to the results from simulations which refer to a completely stagnant cell interior. When Ps was decreased by a factor of two (2-propanol) or more (DMF), the effects were as small as 19% or 14%, respectively. Permeability coefficients of the solutes used (4.2 to 1.6x10–6 m s–1) would still be regarded as rather high when compared with those of nutrient ions, sugars, or other metabolites in the cell sap (where values range between 10–9 and 10–11 m s–1; Nobel, 1999). It is concluded that, in the case of endogenous solutes, the effects of USLs are negligible. However, absolute values of solute permeability play a role during the measurement of the permeability of heavy water (Pd). A comparison between the diffusional water permeability, Pd, and the osmotic or hydraulic water permeability (Pf, Lp) is often made to calculate the number of water molecules sitting in aquaporin pores (N=Pf/Pd; Levitt, 1974; Finkelstein, 1987; Henzler et al., 2004; Ye et al., 2005, 2006; see Introduction). When Pd is underestimated due to the existence of internal USLs, N may be overestimated. The Pd for water is bigger than that of acetone by a factor of two (Henzler et al., 2004; Ye et al., 2006), but this effect may be cancelled by the bigger diffusion coefficient of heavy water as compared with acetone (Ye et al., 2006). Hence, the Ps values derived for acetone are important for predicting the effects of internal USLs during the measurement of the permeability of isotopic water. Overall, the data presented indicate an underestimation of up to 37% for the most rapidly permeating solutes; a result that coincides with the earlier estimates of Ye et al. (2006).
It should be noted that the estimates of the role of internal USLs were based on the most pessimistic assumption, i.e. that the cell interior was completely stagnant. Of course this condition is not true because there would have been some stirring in the cytoplasm due to cyclosis. There would also be some stirring caused by local differences in density (e.g. when concentration gradients of solutes with a density lower than that of cell sap develop in the cell) and the shaking of the internodes while they sit in a turbulent medium during the experiments. Also, reference was made just to the permeability of the PM, but the tonoplast membrane may also have influenced USL formation. However, for Chara and other plant cells, the tonoplast is known to be rather permeable for water and solutes and may be incorporated into the PM or internal USLs (Kiyosawa and Tazawa, 1977; Maurel et al., 1997). Overall, the idea of a relatively small contribution by internal USLs agrees with the finding of first-order kinetics for both the water and solute permeation throughout the pressure/time course of biphasic pressure relaxations as expected from the Steudle–Tyerman theory (equation 3). Hence, the present data confirm this older view for the contribution of internal USLs as a function of solute permeability (Fig. 4).
During stop-flow simulations, the assumption of a stagnant cell interior corresponded with experimental findings, suggesting that the model was close to reality. The majority of cells behaved as predicted by the model. Results from the present work indicated that the recent estimates of Ye et al. (2006) provided an upper limit for the contribution of internal USLs. Clearly, the present data show that the claims made by Tyree et al. (2005) that USLs may dominate the permeability of rapidly permeating solutes, as measured with the pressure probe, are wrong. These authors claimed that the diffusional resistance across external and internal USLs would be larger by a factor of up to five than that across the PM. As already discussed extensively by Ye et al. (2006), Tyree et al. (2005) were misled to this conclusion by several incorrect assumptions about pressure probe experiments, as well as by some misunderstandings concerning the physics underlying USL concepts.
The SFT produced a parameter, RDC, that was used to quantify the effect of internal USLs on the solute permeability measurements. Comparisons of RDC values obtained by simulation with those obtained in experiments suggested that the inside of the cell was nearly stagnant. Both stop-flow simulations (assuming stagnant conditions) and experiments revealed similar differences in the prospected internal concentration and the right concentration at the membrane as simulated/measured at the RDC (Figs 4, 7). This means that the model used during simulations was fairly realistic, assuming stagnant conditions. This may indicate that the mixing by cytoplasmic streaming was smaller than one may assume. However, cytoplasmic streaming should affect the solute concentration close to the PM, which is important during osmotic water flow and can, in turn, determine the turgor pressure measured by the CPP. Hence, the inclusion of cytoplasmic streaming into the simulation models is, perhaps, necessary in a future refinement of simulation models. Since stagnant conditions appeared to be a realistic approach, effects of local changes in density, and the shaking of cells within the turbulent external solution may be smaller than expected (Stevenson et al., 1975).
The simple comparison of shapes of measured biphasic pressure/time curves (Fig. 1) with simulations, as done by Tyree et al. (2005), cannot be used to elucidate the effects of USLs. It has limitations because there are too many unknown factors to be considered in order to obtain a perfect fit between the measured biphasic curve and the simulated one. The water phase of the osmotic curve was the critical zone necessary for a good fit. In the experiment, the Lp should have decreased as the concentration applied to the cell increased (Ye et al., 2004). However, the exact function of Lp with regards to the concentration is not yet clear and, therefore, it is hard to incorporate it. In addition, the instantaneous solution exchange in the external media was not possible experimentally and it has to be considered if one intends to get perfect fitting curves. In the simulation, choosing the correct Lp and considering a non-instantaneous medium exchange would greatly change the shape of the water phase in the biphasic osmotic curve. This was overlooked by Tyree et al. (2005). By contrast with their curve-fitting approach, the SFT has the advantage that it is free from these problems. The resulting solute-concentration profile in the cell is virtually free from the changes in Lp.
In the past, effects of internal USLs, such as those investigated here for Chara internodes, have also been discussed for plant tissues such as roots (Steudle and Frensch, 1989; Ye and Steudle, 2006). In roots, the endodermis should be the main osmotic barrier where both diffusional USLs and concentration polarization effects may play a role (sweep away; Dainty, 1963; Steudle and Frensch, 1989). Different from the Chara internode, effects of USLs in roots may be more pronounced because the diffusive mobility of solutes such as nutrient ions in the root apoplast may be substantially smaller than in bulk solution. Furthermore, the thickness of USLs may be bigger than in Chara, and even as big as the entire thickness of the cortex or the radius of the stele (Steudle and Frensch, 1989). Experiments are underway to determine the effects of USLs in young corn roots using the root pressure probe. In parallel to this, the effects are simulated in analogy to the stop-flow experiments of this paper.
In conclusion, the new SFT used in the current study to quantitatively elucidate the contribution of internal USLs on the measurement of solute permeability in Chara revealed that the effects are lower than previously estimated. Even in the presence of a rapidly permeating test solute (acetone), and assuming a completely stagnant cell interior, the underestimation was no bigger than 37%. A similar figure may be obtained for heavy water. Both the simulation and experimental results obtained by the new SFT indicated estimates of the contribution of USLs similar to those predicted earlier by Ye et al. (2006). Recent simulations on the effects of USLs in Chara by Tyree et al. (2005) are false because of erroneous assumptions and physical mistakes. Since the ratio between the permeability of the PM of Chara and diffusion coefficient of acetone in the water is similar to that of heavy water, the data are of importance for the measurement of diffusional permeability (Pd; isotopic water), which is used to estimate the volume of aquaporins by comparison of Pd with bulk flow permeability (Pf, Lp).
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Acknowledgements |
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The authors are indebted to Chris J Meyer, Department of Biology, University of Waterloo, Ontario, Canada, for critically reading the manuscript. They gratefully acknowledge the technical assistance of Burkhard Stumpf, Department of Plant Ecology, University of Bayreuth. They thank Ralf Geyer, Department of Plant Ecology, University of Bayreuth for his great help in coding the simulation program. This work was supported by a grant of the Deutscher Akademischer Austauschdienst, DAAD, to YK.
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Footnotes |
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* Present address: Biological Laboratories, Harvard University, 16 Divinity Avenue, Cambridge, MA 02138, USA.
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Abbreviations |
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APW, artificial pond water; CPP, cell pressure probe; Cprosp, prospected concentration; CSF, internal concentration suggested by stop-flow;
i, thickness of internal USLs; DMF, dimethylformamide; Ds, diffusion coefficient of a solute in water; Js, solute flow; JV, water flow; Lp, hydraulic conductivity; Pd, diffusional water permeability; Pf, osmotic water permeability; PM, plasma membrane; Ps, true membrane permeability for a solute; , measured overall permeability including USLs for a solute; RDC, relative difference between Cprosp and CSF; 
, concentration inside the cell at the membrane; s, reflection coefficient; SFT, stop-flow technique; USL, unstirred layer. |
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