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J Exp Bot. 2010 May; 61(9): 2507–2518.
Published online 2010 April 30. doi:  10.1093/jxb/erq082
PMCID: PMC2877899

The surface charge density of plant cell membranes (σ): an attempt to resolve conflicting values for intrinsic σ

Abstract

The electrical potentials at membrane surfaces (ψ0) strongly influence the physiological responses to ions. Ion activities at membrane surfaces may be computed from ψ0, and physiological responses to ions are better interpreted with surface activities than with bulk-phase activities. ψ0 influences the gating of ion channels and the driving force for ion fluxes across membranes. ψ0 may be computed with electrostatic models incorporating the intrinsic surface charge density of the membrane (σ0), the ion composition of the bathing medium, and ion binding to the membrane. Some of the parameter values needed for the models are well established: the equilibrium constants for ion binding were confirmed for several ions using multiple approaches, and a method is proposed for the computation of other binding constants. σ0 is less well established, although it has been estimated by several methods, including computation from the near-surface electrical potentials [zeta (ζ) potentials] measured by electrophoreses. Computation from ζ potentials yields values in the range –2 mC m−2 to –8 mC m−2, but other methods yield values in the range –15 mC m−2 to –40 mC m−2. A systematic discrepancy between measured and computed ζ potentials was noted. The preponderance of evidence supports the suitability of σ0= –30 mC m−2. A proposed, fully paramatized Gouy–Chapman–Stern model appears to be suitable for the interpretation of many plant responses to the ionic environment.

Keywords: Charge density, membrane, plant, surface potential, ζ potential

Introduction

Membrane surface potentials, arising from membrane surface charges, profoundly affect plant function, especially interactions between roots and ions in the rooting medium. Knowledge of the plasma membrane (PM) surface electrical potential (ψ0) enables the computation of ion activities at the PM surface. The Nernst equation, {IZ}0={IZ}exp[–ZIFψ0/(RT)], relates the activity of ion I with charge Z at the PM surface to the activity of the ion in the bulk-phase medium (see Methods). (The subscript 0 in {IZ}0 and ψ0 denotes quantities at zero distance from the PM surface; the absence of a subscript in {IZ} indicates activity in the bulk-phase medium.) To illustrate, a value of –59.2 mV for ψ0 will enrich (relative to the bulk-phase medium) ion activities at the PM surface 10-, 100-, and 1000-fold for monovalent, divalent, and trivalent cations, respectively. Anions will be depleted in a reciprocal fashion. Root responses to ions (e.g. elongation, intoxication, alleviation of intoxication, transport across PMs, and PM enzyme activity) often correlate poorly with {IZ} and often correlate well with {IZ}0 (Gibrat et al., 1985; McLaughlin, 1989; Zhang et al., 2001; Yermiyahu and Kinraide, 2005; Kinraide, 2006; Wang et al., 2008). In addition to the enrichment or depletion of ions at membrane surfaces, ψ0 influences the surface-to-surface potential difference across the membranes. This potential difference influences ion channel gating and the driving force for ion fluxes across membranes (Hille, 2001). Intracellular functions, such as electron transport in mitochondria (Møller et al. 1984a) and chloroplasts (Conjeaud and Mathis, 1986), are also responsive to ψ0.

In this article we shall argue that measured or computed values of ψ0 are proportional to actual values but may be different from the actual values themselves. In many instances, proportional values for ψ0 are sufficient for the interpretation of results such as the alleviation of the toxicity of toxic cations (Al3+, Cu2+, and H+) or the enhancement of the toxicity of toxic anions (SeO42– and H2AsO4) by treatments that reduce PM surface negativity (e.g. increases in the ionic strength or decreases in the pH of bathing media) (Gimmler et al., 1991; Yermiyahu and Kinraide, 2005; Kinraide, 2006; Wang et al., 2008). However, results have been reported in which detailed interpretations require more precise knowledge of actual values for ψ0. An example of this is presented in Wang et al. (2008) in which the extrinsic sensitivity to {Cu2+} (sensitivity to bulk-phase {Cu2+}) is reduced by reductions in ψ0 negativity, as expected, but the intrinsic sensitivity to Cu2+ (sensitivity to {Cu2+}0) is increased. This apparent increase in intrinsic sensitivity could be genuine (our view) or it could be the consequence of erroneous values for ψ0 derived from erroneous values for surface charge density (σ) all leading to a miscalculation of {Cu2+}0.

Computation of PM surface activities (e.g. {Cu2+}0) requires knowledge of ψ0, and computation of ψ0 requires knowledge of PM σ. σ is the contingent surface charge density, that is, σ in a bathing medium where ions may be bound to the membrane surface, thereby altering σ from the condition where no ions are bound. In the absence of ion binding, σ is designated the intrinsic surface charge density, denoted σ0, and σ0 must be known in order to compute σ.

In this article we shall attempt to assess the ability to measure or compute σ0, ψ0, and membrane surface ion activities at the PM and other cell membranes. The computation of ψ0 requires an appropriate electrostatic model, and the model in most common use is the Gouy–Chapman–Stern (GCS) model (Barber, 1980; Kinraide, 1994; Tatulian, 1999; Yermiyahu and Kinraide, 2005). This model incorporates σ0, the ion composition of the bathing medium, and ion binding to the membrane. In our view, σ0 is the parameter simultaneously of greatest importance and greatest uncertainty, and in this study we marshal the evidence for the suitability of σ0= –30 mC m−2 as the best general value while noting the occurrence of variability among membranes. This variability is small relative to the difference in estimates of σ0 by electrophoreses [–4.3±0.9 mC m−2 (mean±SE)] and all other methods [–27.5±2.1 mC m−2 (mean±SE)]. These values for σ0 would lead to ψ0= –18.8 mV and ψ0= –36.4 mV, respectively, for membranes bathed in 1 mM CaCl2 at pH 6. {Ca2+}0 would be 3.4 mM and 13.3 mM, respectively. The datum tables presented later provide further illustrations of the interactions among σ0, ψ0, and ion activities.

Methods

Reported values for σ0, as assessed by several methods, are presented in Table 1. Zeta (ζ) potential measurements taken from published reports and contributed data are presented in Table 2. The ζ potential is the near-surface potential of membrane vesicles or cell protoplasts measured by electrophoresis. ψ0 is related to the ζ potential, which is the electrical potential at a distance s from the membrane surface; s is the location of the hydrodynamic plane of shear (the slipping plane). Thus ζ potential=ψs, and ψs0exp[–3.29sμ1/2] where μ is the ionic strength and 3.29 is a constant appropriate at 25 °C when s is expressed in nm and μ is expressed in M (Morel and Hering, 1993). The negative exponential ensures that |ψs| < |ψ0|.

Table 1.
Computed intrinsic surface charge densities (σ0) of plant membranes
Table 2.
ζ potentials of plant membranes

If ψ0 is known or can be computed, then σ0 may be computed by the GCS model, and vice versa (see below). For these computations the detailed ionic composition of the bathing medium must be known (detailed chemical speciation is required), and some quantitative parameters for the model, including equilibrium constants for ion binding to the membrane surface, must be known. Some parameter values for a model currently in use are presented in the second column of Table 3, and henceforth the GCS model using those parameter values will be referred to as the ‘standard model’.

Table 3.
Parameter values for a Gouy–Chapman–Stern model for plant plasma membranes

For the GCS model, the PM was modelled as though it were composed of negatively charged (R) and neutral (P0) sites to which ions (IZ) may bind. Ions may bind according to the reactions R+IZRIZ–1 and P0+IZPIZ for which the binding constants KR,I {=[RIZ–1]/([R][IZ]0)} and KP,I {=[PIZ]/([P0][IZ]0)} are needed. [R], [P0], [RIZ–1], and [PIZ] denote PM surface densities in mol m−2, and [IZ]0 {=[IZ]exp[–ZIFψ0/(RT)], a Boltzmann equation} denotes the concentration in M of an unbound ion at the PM surface. F, R, and T are the Faraday constant, the gas constant, and the temperature, respectively; F/(RT)=1/25.7 at 25 °C for ψ0 expressed in mV. The contingent surface charge density (σ) can be expressed as the sum of the products of the surface density of each species and the charge of each species all times F in order to express σ in units C m−2:

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(1)

Equation 1 expresses the Stern portion of the GCS model. Stern reactions are strong interactions represented as binding of ions to PM sites as noted above. The computation of σ by Equation 1 requires both binding constants and ψ0 (in order to obtain [IZ]0 by the Boltzmann equation above). The Gouy–Chapman portion of the model is expressed in this equation:

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(2)

rε0RT=0.00345 at 25 °C for bulk-phase concentrations of IZ expressed in M and σ2 expressed in (C m−2)2r is the dielectric constant for water, ε0 is the permittivity of a vacuum). Thus in any bathing medium there are two equations with two unknowns (ψ0 and σ). To compute ψ0, trial values were assigned to it, and σ in Equations 1 and 2 were computed until the values for σ from the two equations converged. See Yermiyahu and Kinraide (2005) for more detailed descriptions and references.

The possibly problematical necessity of the simultaneous truth of both the Nernst and Boltzmann equations (the first incorporating activities and the second concentrations, {IZ}0={IZ}exp[–ZIFψ0/(RT)] and [IZ]0=[IZ]exp[–ZIFψ0/(RT)], respectively) has been discussed, but not resolved (Kinraide, 1994; Rytwo, 2004). This matter will be discussed later in conjunction with a systematic discrepancy between measured and computed ζ potentials observed in the present study. A computer program for the GCS model may be obtained from the authors.

Results

Discrepancies in reported values for σ0

Fluorescent dye quenching methods of σ0 assessment, and several other methods, produce values ranging, usually, from –15 mC m−2 to –40 mC m−2 (Table 1), but they include one exceptional value: –5.7 mC m−2 (Gibrat et al., 1985). Measurements with 9-aminoacridine (9-AA) have undergone several improvements and reassessments (Bérczi and Møller, 1993; Brauer et al., 2000), and fair agreement between σ0 computed by the 9-AA fluorescence method and σ0 computed on the basis of liposome composition (where σ0 is known) has been claimed (Brauer et al., 2000). Estimates from ζ potential measurements yield dramatically different results. For them, the estimates of σ0 range, usually, from –2 mC m−2 to –8 mC m−2, and the averages of all values from the two sets differ by 6.4-fold [–4.3±0.9 versus –27.5±2.1 (means±SEs)].

Within-study variation among σ0 values

The 6.4-fold bimodal distribution of σ0 values was large compared with the generally <2-fold variation among σ0 values for different membranes within a study or for membranes of different type, sidedness, or experimental treatment. Membranes vary according to the plant genotype, the tissue, and the organelles from which the membranes were derived. In addition, treatments such as ionic, water, and temperature stress may affect the membranes. Except for the extreme value of –5.7 mC m−2, the values assessed by fluorescent dye quenching vary by 2.8-fold in Table 1. In the study by Møller et al. (1984b) the inside and outside surfaces of PM vesicles were compared. The σ0 for inside surfaces was 56% more negative than for outside surfaces. Yermiyahu et al. (1997a) compared Scout and Atlas cultivars of wheat (Triticum aestivum L.) and found Scout PM vesicles to be 26% more negative than Atlas. This difference was consistent with greater Scout sensitivity to cationic toxicants and greater Atlas sensitivity to anionic toxicants (Yermiyahu et al., 1997a). The difference between dicotyledons and monocotyledons in Table 1 is statistically insignificant. Finally, weak red radiation produced a 31% reduction in negativity in wheat thylakoids relative to greenhouse controls (Dahlin, 2003).

Estimation of GCS model parameters from ζ potentials

Among the σ0 estimates presented in Table 1 are several from electrophoresis studies. Table 1 also presents estimates (–1.6 mC m−2 to –10.4 mC m−2) from the present study based upon the ζ potentials in Table 2. To achieve these estimates, values for ψ0 were computed by a GCS model using parameter values selected to optimize the correspondence between the ζ potentials and ψ0. For these estimates, we assumed initially that ζ potential ≈ψ0 as explained below. Optimization entailed the incorporation of starting values for the parameters into an iterative computer program to compute ψ0. Then one of the parameters was changed incrementally until the sum of squares [SS=(ζ potential–ψ0)2] was minimized. The program fixed that value and moved on to the next parameter. This process was repeated until all parameters were evaluated, then the size of the incremental changes was reduced and new rounds of evaluations occurred until the SS was minimized. The parameters so derived are presented in Tables 3 and 4, the latter presenting the computed ψ0 for each of the 59 solutions and the computed σ0 for each of the nine studies. For the regression of ζ potentials versus computed ψ0, r2=0.917 (Fig. 1).

Fig. 1.
Measured ζ potentials plotted against ψ0. ψ0 is the plasma membrane surface potential computed with a Gouy–Chapman–Stern model with parameters optimized for best correspondence between ζ potentials and ψ ...

Equilibrium constants for ion binding

Among the parameter values derived by optimization are the equilibrium constants for ion binding to negative binding sites (R) on the PM surface (KR,K, KR,Na, KR,Ca, KR,Mg, KR,La, and KR,H). A notable feature of these results is the close correspondence between the binding constants evaluated by two entirely different techniques—electrophoresis, as just described, and adsorption (second column in Table 3). For the latter, the binding constants for ions were computed on the basis of measured ion adsorption to PM vesicles of wheat root cells (Yermiyahu et al., 1997b; Vulkan et al., 2004). Figure 2A illustrates this close correspondence in a plot of rescaled values (cube roots) for which r2=0.998. Estimation of equilibrium constants for ion binding to neutral binding sites (P0) on the PM surface (KP,K, KP,Na, KP,Ca, KP,Mg, KP,La, and KP,H) will be described in the next section.

Fig. 2.
Comparison of equilibrium constants for ion binding to negative sites on plasma membranes assessed by different methods. Plots are of the cube roots of binding constants assessed by ion adsorption, electrophoresis, and relative binding strength to hard ...

The column designated General binding strengths in Table 3 is derived from the relative strength of ion binding to hard ligands (e.g. ligands with F or O donors) as opposed to soft ligands (e.g. ligands with I or S donors). These relative binding strengths to 13 hard ligands were determined in a survey of 81 metal ions (Kinraide, 2009). The scale presented in that survey is related linearly to the logK values for binding to the hard ligands, and that scale may be converted to PM binding constants by the formula KR,I=10[3+1.75(Hard Ligand Scale)]. Figure 2B illustrates the close correspondence between these estimates and binding constants evaluated by adsorption. The Hard Ligand Scale (Kinraide, 2009) allows the attribution of binding strengths to numerous additional ions, with some uncertainty.

Surface densities of negative binding sites (R) and neutral binding sites (P0)

σ0 may be stated in several ways (–1 mC m−2=0.0104 μmol negative charges m−2=16 000 Å2 per negative charge=0.00624 e nm−2). We use RT as an expression of σ0. It is the total number of negative binding sites (R) per m2, and a value of 0.3074 μmol m−2 has been assigned previously (Yermiyahu et al., 1997b; and see Table 3). In the analysis of the ζ potential data (Table 2), values for RT were assigned individually, by optimization, for each of the nine studies. Inspection of the table reveals apparent differences in σ0 for the studies. For example, one would expect the ζ potential for solution no. 2 to be similar to or slightly less negative than the ζ potential for solution no. 22. The opposite occurs, indicating that σ0 may be more negative for the tobacco leaf protoplasts (Nagata and Melchers, 1978) than for the barley leaf protoplasts (Obi et al., 1989a). A similar argument applies in the case of solutions 5 and 20.

PT is the total number of neutral binding sites (P0) per m2, and a single value of PT=2.4 μmol m−2 was assigned on the basis of the number of phospholipids per m2 in a typical membrane (Akeson et al., 1989; Kinraide et al., 1998; and see Table 3). Separate equilibrium constants for ion binding to P0 were not determined. Instead, it was assumed, as an approximation, that the binding strengths for the neutral sites are a constant fraction of the binding strengths for negative sites. In a previous study (Kinraide et al., 1998), an optimized value of 1/180 was determined for cations; that is (binding strength to P0)/(binding strength to R)=KP,I/KR,I=1/180. In the present analysis of ζ potentials (Table 2) the optimized value for KP,I/KR,I was 1/1100. This value is greatly different from the previous 1/180, but the smaller value (1/1100) is a consequence of the less negative values for σ0 estimated from ζ potentials (analysis not shown). A sensitivity analysis of the parameters (see below) also indicated the relative insensitivity to changes in KP,I/KR,I, which in any case is a small number (<0.01).

The very close agreement between previous (adsorption) and present (electrophoresis) estimates of ion binding strength at negative sites (R) has been noted (Table 3 and Fig. 2A), but the disparities between previous and present estimates of RT in Table 3 (second and third columns) are typical of the disparities observed in Table 1. In Table 3 the value RT=0.3074 μmol m−2 (second column) was computed by the 9-AA fluorescent dye quenching method (Yermiyahu et al., 1997b), and the values 0.017–0.108 μmol m−2 (third column) were computed from electrophoresis data as just noted.

The parameter s (distance of the slipping plane from the PM surface)

In the initial evaluation of parameters, ζ potential was taken to equal ψ0; that is, s=0. This assignment should not introduce a great error if s≈0.2 nm, as is commonly assumed (McLaughlin, 1989). At the small μ of 1 mM, ψ0= –20.4 mV when ψs= –20 mV, and at the large μ of 100 mM, ψ0= –24.6 mV when ψs= –20 mV.

In order to evaluate s by the iterative optimization method, the model-computed value for ψ0 was converted to the electrical potential at s nm from the membrane surface with the equation ψs0exp[–3.29sμ1/2]. Nonsensically, s evaluated to –0.76 nm, indicating, impossibly, that the slipping plane was inside the vesicle and that the potential became more negative with increased distance from the membrane surface. The incorporation of s= –0.76 nm into the model caused minor changes in the other parameters: σ0 became even less negative; SS declined from 895 to 839; and r2 increased from 0.917 to 0.921. The apparently negative value of s may be related to a systematic error noted in the next section.

In further analyses, s was assigned positive values in order to force values for σ0 into a range comparable with those computed by other methods (Table 1). When s=2 nm, σ0 ranged from –2.2 mC m−2 to –51.8 mC m−2 (–17.0 average). However, SS {=[ζ potential–ψ(s=2)]2} rose to 1557, and r2 declined to 0.866. Thus ζ potential-derived values for σ0 are compatible with other values if s is increased 10-fold the usually accepted value, but then the correspondence between measured and computed potentials declines.

The GCS model generates a systematic error relative to ζ potentials

The values for ψ0 computed with optimized parameter values (third column in Table 3) differ systematically with measured ζ potentials: the differences (Table 4, column D) became more negative with increasing μ (Fig. 3A); that is, the model underestimates the depolarizing effectiveness of low μ media and overestimates the depolarizing effectiveness of high μ media as assessed by electrophoresis. To test the possibility that this effect of μ was the accidental consequence of random error or a consequence of very large μ, the data were trimmed. Solutions 12, 16, 29, 30, and 55 were eliminated because of relatively large residuals (>| 8 |), and solutions 4, 6, and 8 were eliminated because of large μ (>50 mM), then optimal parameters were recomputed. The error persisted with the trimmed data (Fig. 3B), and s continued to evaluate to negative values. When differences between ψ0 computed by the ‘standard model’ and ζ potentials (Table 4, column F) were plotted against the square root of μ, a statistically significant relationship qualitatively similar to that in Fig. 3A was obtained also (not shown).

Fig. 3.
Residual values plotted against the square root of ionic strength (μ in mM). (A) Residuals from the first set of difference values in Table 4 (column D). (B) Residuals from a new analysis, using newly optimized parameters, with eight solutions ...
Table 4.
Measured ζ potentials and computed ψ0 of plant membranes (refer to Table 2)

As mentioned earlier, the ζ potential=ψs, the potential at the slipping plane s nm from the membrane surface. Because of the dependence of ψs upon μ and s according to the equation ψs0exp(–3.29sμ1/2), we assessed the influence of s upon the μ-related error just noted. Adjustments of s from 0 nm to 2 nm failed to eliminate the error.

σ0 inferred from toxicological studies

In toxicological studies one may compute σ0 by making the (possibly false) assumption that the best value for σ0 is the one that leads to the best correlation between the computed surface activity of the toxicant and the physiological response. Consider a study (Kinraide, 2006) evaluating wheat root elongation in response to variable pH and variable concentrations of CuSO4, CaCl2, and NaCl. Relative root length (RRL) declined as a function of {Cu2+}, but the correlation (r2=0.565) was poor (Fig. 3A in Kinraide. 2006). The correlation was much better (r2=0.823) for RRL versus {Cu2+}0 (Fig. 3B in Kinraide, 2006) where {Cu2+}0 was computed by the ‘standard model’. If {Cu2+}0 was computed after optimization of RT, then r2=0.846 (Table 5, experiment no. 8). This optimized value for RT (1/2 the standard value) was still in the range of 9-AA-computed values. Furthermore, the elimination of the single greatest outlier increased r2 to 0.856 for RRL versus {Cu2+}0, and now the optimized value for RT was equal to the standard value.

Table 5.
Plant responses to ion activities in the bathing medium ({IZ}) or at the PM surface ({IZ}0)

Table 5 and Fig. 4 present reanalyses of eight toxicity studies, including the one just mentioned. Only in the case of experiment 3 did σ0 fall into the –2 mC m−2 to –8 mC m−2 range of ζ potential-derived values. On average, the optimized value for σ0 was –46 mC m−2. An attempt was made to avoid prejudice in the selection of studies. In fact, it was not possible to predict whether optimization would lead to greater or smaller values for σ0.

Fig. 4.
Se uptake plotted against SeO42– activity in the rooting medium or at the plasma membrane surface. {SeO42–}0 was computed using the ‘standard model’ value for σ0 (RT=0.3074 μmol m−2) or with a value ...

Discussion

The distribution of computed values for σ0 is bimodal, with almost no overlap. Electrophoretic studies yield values in the range –2 mC m−2 to –8 mC m−2, and other studies yield values in the range –15 mC m−2 to –40 mC m−2. We know of no certain criterion by which to reject either of the two ranges, but the following arguments militate in favour of the more negative values: (i) the more negative values were obtained by five methods rather than just one (Table 1); (ii) values for σ0 computed by the 9-AA fluorescence method correspond well to values computed on the basis of liposome composition (where σ0 is known) (Brauer et al., 2000); (iii) optimization of σ0 in physiological studies generally led to more negative values (–46 mC m−2 on average; Table 5); and (iv) a comparison of measured near-surface potentials (ζ potentials) and computed surface potentials (ψ0) revealed a systematic error: a GCS model, whether with optimized parameter values or with ‘standard model’ parameter values, underestimated the depolarizing effectiveness of low μ media and overestimated the depolarizing effectiveness of high μ media. Furthermore, the evaluation of s from electrophoretic measurements yielded the nonsensical value of –0.76 nm.

It is uncertain whether these errors reflect a deficiency in the measurement of ζ potentials or in the electrostatic model for computing ψ0. The errors may relate to a possible problem in the GCS model previously discussed, but not resolved (Kinraide, 1994; Rytwo, 2004). The Nernst equation is accepted as the appropriate thermodynamic expression for the distribution of ions in an electric field [or at least for the equilibrium partition of ions between the bathing medium and the usually negative cell interior (Nobel, 2005)]. In contrast, the derivation of the Gouy–Chapman theory incorporates the Boltzmann equation as the thermodynamic expression for the distribution of ions at a charged membrane surface (Barber, 1980; Tatulian, 1999; and see Methods). For both equations to be correct the activity coefficients of the ions (γi) must remain constant at all distances from the charged surface. Only then can {IZ}0/{IZ}=γi,0[IZ]0/(γi[IZ])=[IZ]0/[IZ]=exp[–ZIFψ0/(RT)].

Should these errors call into question other parameter values computed from ζ potentials? Binding constants assessed from ζ potentials agree remarkably well with those assessed from other methods (Table 3 and Fig. 2). Furthermore, ζ potentials and computed potentials are usually proportional to one another and highly correlated within a study (compare values in column B with values in column C in Table 4). For the equation ζ potential=a+bψ0, r2 ranged from 0.708 to 0.998, b ranged from 0.630 to 1.24, and, for seven of nine studies from Table 4, a was not statistically different from zero (a=0 is, of course, a feature of proportionality). The statistics just cited are based upon ψ0 computed with the optimized parameter values. Using ψ0 computed from the ‘standard model’ (column E in Table 4), these statistics are obtained: r2 ranged from 0.833 to 0.987, b ranged from 0.199 to 0.721, and a was not statistically different from zero for eight of the nine studies. Thus, the proportionality within studies was even stronger using potentials computed from the ‘standard model’.

In conclusion, –30 mC m−2 appears to be the best available value for σ0 for plant cell membranes. Within-study differences for σ0 among membranes differing by membrane type, sidedness, or experimental treatment are small (<2-fold) compared with the 6.4-fold difference between ζ potential-derived values and values derived from all other methods. Other parameter values can be taken from the ‘standard model’ in Table 3, and additional binding constants can be computed as instructed there. The much smaller negative values for σ0 computed from ζ potentials may be related to the preparation of vesicles or protoplasts or to some unexpected electrophoretic behaviour. Alternatively, these values may reflect some deficiency in theory, and perhaps the present study will inspire theoreticians to address the problem. For many applications the actual value of σ0 over a wide range is relatively unimportant because of the probable proportionality between computed ψ0 and actual ψ0. That is, physiological responses correlate better with computed surface activities of ions than with bulk-phase activities irrespective of assumed σ0 over a wide range: note the relatively small effect upon r2 of changes for RT from standard to optimal in Table 5. Use of the ‘standard model’ has contributed to several novel insights into plant–ion interactions (Yermiyahu and Kinraide, 2005) that are not negated by the use of σ0 values differing by several fold from the standard value. Nevertheless, some interpretations are dependent upon exact values, rather than proportional values, for computed ion activities at cell surfaces (Wang et al., 2008). Finally, it is important to realize that all references to and measurements of σ in this study refer to global or average values. Local values at ion channel openings, for example, may be different (Hille, 2001). Nevertheless, even global values for σ have contributed greatly to the interpretation of plant–ion interactions.

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