Computation of Surface Electrical Potentials of Plant Cell Membranes

Abstract

A Gouy-Chapman-Stern model has been developed for the computation of surface electrical potential (ψ₀) of plant cell membranes in response to ionic solutes. The present model is a modification of an earlier version developed to compute the sorption of ions by wheat (Triticum aestivum L. cv Scout 66) root plasma membranes. A single set of model parameters generates values for ψ₀ that correlate highly with published ζ potentials of protoplasts and plasma membrane vesicles from diverse plant sources. The model assumes ion binding to a negatively charged site (R⁻ = 0.3074 μmol m⁻²) and to a neutral site (P⁰ = 2.4 μmol m⁻²) according to the reactions R⁻ + I^Ζ ⇌ RI^{Ζ−1 and} P⁰ + I^Ζ ⇌ PI^Ζ, where I^Ζ represents an ion of charge Ζ. Binding constants for the negative site are 21,500 m⁻¹ for H⁺, 20,000 m⁻¹ for Al³⁺, 2,200 m⁻¹ for La³⁺, 30 m⁻¹ for Ca²⁺ and Mg²⁺, and 1 m⁻¹ for Na⁺ and K⁺. Binding constants for the neutral site are 1/180 the value for binding to the negative site. Ion activities at the membrane surface, computed on the basis of ψ₀, appear to determine many aspects of plant-mineral interactions, including mineral nutrition and the induction and alleviation of mineral toxicities, according to previous and ongoing studies. A computer program with instructions for the computation of ψ₀, ion binding, ion concentrations, and ion activities at membrane surfaces may be requested from the authors.

PM electrical phenomena play an important role in plant physiology, especially plant-mineral interactions. Two global electrical properties of the PM are commonly recognized. The first is ψ_m, which may be measured relatively easily by the insertion of a microelectrode into cells in situ (Nobel, 1991). ψ_m is responsive to many factors, including the composition of the bathing medium, the activity of ion pumps, and the state of ion channels, which are themselves responsive to ψ_m (Hille, 1992). ψ_m is usually described in introductory plant physiology textbooks and in general treatments of plant-mineral interactions (Marschner, 1995).

The second global electrical feature of the PM is ψ₀, the measurement of which is more difficult than the measurement of ψ_m. The procedure entails the preparation of protoplasts or PM vesicles whose electrophoretic mobility is then measured to obtain a ζ potential (for refs., see Table I), which reflects the electrical potential at the hydrodynamic plane of shear at a small distance from the PM surface (McLaughlin, 1989; Morel and Hering, 1993). Consequently, the ζ potential has a somewhat lower magnitude than the ψ₀. Only a few ζ potential measurements have been reported for the plant PM (for refs., see Table I), and ψ₀ is rarely mentioned in introductory plant physiology textbooks or in general treatments of plant-mineral interactions.

Table I.

Published ζ potentials of plant protoplasts or PM vesicles in various media compared with calculated surface potentials

Species and Ref.	Solution No.	pH	CaCl2	MgSO4	NaCl	KCl	LaCl3	ζ Potential	ψ0,Y	ψ0,A
			mm					mV
Barley leaf protoplasts (Abe and Takeda, 1988)
	1	6.7	0.1		0.5			−48	−66	−65
	2	6.7	0.1		6.5			−39	−59	−58
	3	6.7	0.1		0.5	6.0		−39	−59	−58
	4	6.7	0.1		0.5		0.1	−17	−7	−6
	5	6.7	0.1		0.5		0.3	0	+2	+3
	6	6.7	0.1		0.5		1	+14	+12	+13
	7	3.6	0.1		6			−2	−14	+5
	8	3.6	0.1				1	+23	+10	+17
Barley leaf protoplasts (Obi et al., 1989a)
	9	7.6			3			−29	−100	−100
	10	7.6			7.5			−20	−82	−82
	11	7.6			15			−13	−67	−66
	12	7.6			1	14		−13	−67	−66
	13	7.6			30			−10	−52	−51
	14	7.6	1		4.5			−10	−36	−36
	15	7.6		1a	4.5			−11	−36	−36
	16	7.2			0.5		0.17	0	−2	−2
	17	6.5			5		1	+3	+12	+12
Corn root PM vesicles (Gibrat et al., 1985)
	18	6.5			15			−24	−64	−64
	19	6.5			65			−14	−36	−36
	20	6.5		6	15			−8	−16	−15
	21	6.5		6	65			−6	−13	−12
Tobacco leaf protoplasts (Nagata and Melchers, 1978)
	22	5.8			6.7	10		−28	−56	−55
	23	5.8	1		6.7	10		−25	−32	−32
	24	5.8	10		6.7	10		−9	−10	−9
	25	5.8	100		6.7	10		0	+7	+8
Rauwolfia serpentina cultured cell protoplasts (Obi et al., 1989b)
	26	7.3	0.02		6	1		−18	−72	−71
	27	6.3	0.02		6	1		−18	−69	−69
	28	5.3	0.02		6	1		−17	−56	−54
	29	4.2	0.02		6	1		−12	−27	−18
	30	3.0	0.02		6	1		+2	−5	+29
Barley leaf protoplasts (Obi et al., 1990)
	31	7.3	0.02		6	1		−18	−72	−71
	32	6.3	0.02		6	1		−16	−69	−69
	33	5.2	0.02		6	1		−12	−54	−51
	34	4.0	0.02		6	1		+1	−22	−10
	35	2.7	0.02		6	1		+18	−2	+40

ψ_0,Y is the potential computed by the model of Yermiyahu et al. (1997c) and ψ_0,A is the potential computed by the adjusted model of the present study. For each of the two models, a single set of parameter values was used throughout. The column designated NaCl includes NaCl plus any monovalent buffer ions.

^aMgCl₂. 

Both ψ_m and ψ₀ are physiologically important because electrical potential gradients influence the distribution of ions. ψ₀ is often sufficiently negative to enrich the concentration of cations or to deplete the concentration of anions at the PM surface by more than 10-fold relative to the bathing medium (Barber, 1980). ψ₀ is influenced by the composition of the medium. High ionic strength reduces the negativity of ψ₀, and some ions can convert ψ₀ to positive values (Table I). The common neglect of ψ₀ is inconsistent with its importance, so the neglect probably reflects the difficulty of measuring ψ₀ and the absence of verified model parameters for its computation in biological membranes. Nevertheless, investigators have considered physiological phenomena in terms of ψ₀ (Barber, 1980; Theuvenet and Borst-Pauwels, 1983; Gibrat et al., 1985, 1989; Abe and Takeda, 1988; Wagatsuma and Akiba, 1989; Suhayda et al., 1990; Kinraide et al., 1992; Kinraide, 1994, 1998; Yermiyahu et al., 1997a, 1997b, 1997c; for references to the animal literature, see McLaughlin, 1989; Hille, 1992; for extensive references and tabulations of relevant data, including constants for the binding of ions to artificial and biological membranes, see Tatulian, 1998).

The objective of the present study was to determine the suitability of a computational approach to the estimation of ψ₀ for the PM from diverse plant sources. In particular, the study assesses the suitability of a Gouy-Chapman-Stern model using a single set of model parameters derived principally by Yermiyahu et al. (1997c) for the computation of ψ₀ in response to ionic solutes. Additionally, our goal was to make available an easy-to-use computer program for the computation of ψ₀, ion binding, ion concentrations, and ion activities at plant PM surfaces. This program, together with a manual for its use, may be obtained from T.B.K. A more comprehensive program that integrates numerically the ion content of the diffuse layer from the membrane surface to any desired distance from the surface and that is suitable for the computation of total sorbed ions (ions bound and accumulated in the diffuse layer) may be obtained from G.R.

MATERIALS AND METHODS

The Gouy-Chapman-Stern Model

ζ potential measurements (Table I) were taken for PM vesicles or protoplasts, and the Gouy-Chapman-Stern model was applied to the PM of root cells as if the cell wall did not exist. This may be justified if the cell wall can be considered an independent phase interposed between the PM and the bulk-phase medium, achieving near ionic equilibrium with the medium and presenting an insignificant barrier to the flux of ions. Experimental justification for our treatment is provided by Gage et al. (1985, 1986), who compared ordinary yeast cells with cells that were plasmolyzed or enzymatically stripped of their walls. Rb⁺ uptake was dependent on the estimated ψ₀ of the PM and on the intracellular [K⁺], not on the cell wall Donnan potential. Furthermore, the ψ₀ of the PM did not appear to be influenced by the cell wall.

The Gouy-Chapman-Stern model may be expressed in a few equations (Lau et al., 1981; Kinraide, 1994; Nir et al., 1994; Yermiyahu et al., 1997c). The Gouy-Chapman portion of the model is expressed in the Müller equation (derivation presented by Barber [1980] and Tatulian [1998], the latter noting that the Müller equation has erroneously come to be known as the Grahame equation):

1

where ς is the charge density on the membrane surface expressed in coulombs per square meter (C m⁻²); 2ε_rε_oRT = 0.00345 at 25°C for concentrations expressed in molarity (ε_r is the dielectric constant for water, ε₀ is the permittivity of a vacuum, R is the gas constant, and T is temperature); [I^Ζ]_∞ is the concentration of ion I^Ζ (the i^th ion) in the bulk-phase medium; Ζ_i is the charge on ion I^Ζ; F is the Faraday constant; and ψ₀ is the electrical potential at the membrane surface measured with respect to the bulk-phase medium a long way from the surface; −Ζ_iFψ₀/RT = −Ζ_iψ₀/25.7 at 25°C for ψ₀ expressed in millivolts.

ς depends in part on ς_intrinsic, the surface charge density in the absence of any bound solute ions. ς also includes those solute ions that bind to the membrane surface, and the Stern modification of the model takes this binding into account. The present model can accommodate 1:1 binding of ions to a negatively charged site (R⁻) and a neutral site (P⁰) as expressed in the following reactions:

2

and

3

Binding constants, specific for ion and site, may be expressed as:

4

and

5

where [R⁻], [P⁰], [RI^Ζ−1], and [PI^Ζ] denote membrane surface densities in moles per square meter (mol m⁻²), and [I^Ζ]₀ denotes the concentration of the unbound ion at the membrane surface. [I^Ζ]₀ may be computed from bulk-phase concentrations by the Boltzmann equation:

6

To compute ς, it is necessary to know [R_T] (R⁻ + Σ_i[RI^Ζ−1]) and [P_T] (P⁰ + Σ_i[PI^Ζ]), which are the total surface densities of binding sites whether or not solute ions are bound. (ς_intrinsic = [R_T]F; multiplication by F is needed to convert units to coulombs per square meter.) It is also necessary to know the binding constants, K_R,I and K_P,I, and the concentrations of all of the ions in the bulk-phase medium. With that knowledge, the surface charge density of each membrane species (including R⁻, RI^Ζ−1, and PI^Ζ) can be computed from Equations 4, 5, and 6, contingent upon the value of ψ₀, and appropriately summed:

7

Thus, trial values for ψ₀ may be used to compute ς using Equations 1 and 7. When the values for ς converge, then the value of ψ₀ will have been found. With that value, we assume that the Nernst equation may be used to compute surface activities of ions (the issue of the simultaneous validity of Eqs. 6 and 8 is discussed, but not resolved, by Kinraide [1994]):

8

Selection of Initial Values for the Model Parameters

To use the Gouy-Chapman-Stern model, values for the adjustable model parameters must be selected. To begin, we adopted the following values assumed or estimated by Yermiyahu et al. (1997c) for wheat (Triticum aestivum L. cv Scout 66) root PM: ς_intrinsic = 29.7 mC m⁻² (corresponding to [R_T] = 0.3074 μmol m⁻² or 540 Ų intrinsic charge⁻¹), K_R,H = 21,500 m⁻¹, K_R,Al = 20,000 m⁻¹, K_R,La = 2,200 m⁻¹, K_R,Ca = K_R,Mg = 30 m⁻¹, K_R,Na = K_R,K = 1 m⁻¹, and K_R,anions = 0 (subscript Al refers to Al³⁺; AlOH²⁺ and Al(OH)₂⁺ almost certainly have trivial electrical effects because of very low concentrations, but their binding constants were assumed to equal K_R,Ca and K_R,Na, respectively). The binding constants were based on literature values (K⁺ and Ca²⁺) and on the measured sorption of H⁺, Al³⁺, and La³⁺ (sorption refers to ions bound and accumulated in the diffuse layer). Values for [P_T] and K_P,I were not assigned because binding to neutral sites was not assumed. The estimation of ς_intrinsic was based on the quenching of 9-aminoacridine fluorescence.

Selection of Published ζ Potentials

ζ potential measurements of the PM from several plant species and tissues were compiled from six publications (Table I). To be included, each publication had to present at least four measurements taken in solutions of variable solute compositions. For each solution, ψ₀ was computed according to the model and parameter values of Yermiyahu et al. (1997c) and will henceforth be denoted ψ_0,Y.

Improvement of the Model

The model was improved by altering adjustable parameters on the basis of two criteria. The first was the increase in the correlation:

9

for each of the six studies presented in Table I. (According to theory, Eq. 9 is approximate; a is expected to equal 0 because ζ potential = 0 when ψ₀ = 0; b is a constant only if the ionic strength, the distance of the plane of shear from the PM surface, and the temperature are constant [see below]. b is expected to be <1 if ψ₀ is accurately computed, and b may be even lower if the assumed value of ς_intrinsic is too high or if there are other model deficiencies.)

The second criterion was the avoidance of a significant degradation of the very high correspondence between measured and computed ion sorption observed in the study by Yermiyahu et al. (1997c).

RESULTS AND DISCUSSION

Correlations between Computed ψ_0,Y and Published ζ Potentials

Figure 1a presents a plot of ζ potential versus ψ_0,Y for all 35 points in Table I taken together. Table II presents statistics for regressions of the six individual studies. An examination of the data reveals the following points of interest.

Figure 1.

ζ potentials of plant protoplasts or PM vesicles plotted against calculated ψ₀. ζ potentials are the published values presented in Table I. Symbol diameters are proportional to pH, and shaded symbols refer to the first study in Table I. a, ψ_0,Y calculated by the Gouy-Chapman-Stern model presented by Yermiyahu et al. (1997c) in which ions bind only to negative sites. b, ψ_0,A calculated by the same model, except that ions were also allowed to bind to neutral sites. c, ψ_2,A calculated by the model of (b) for a region 2 nm from the PM surface. d, ψ_0,B calculated using a model in which parameters were adjusted without reference to the model of Yermiyahuet et al. (1997c). The binding constants_B in Figure 4 are from this model for ψ_0,B.

Table II.

Regression statistics for ζ potential compared with ψ₀

r² for ζ potential = a + bψ₀, where ψ₀ was computed by two models: ψ_0,Y was computed according to the model presented by Yermiyahu et al. (1997c), and ψ_0,A was computed according to the adjusted model of the present study. Each study is numbered according to its appearance in Table I.

Study No.	ζ Potential
	a + bψ0,Y			a + bψ0,A
	r2	a	b	r2	a	b
1	0.911	3.7a	0.761	0.929	0.0a	0.723
2	0.923	0.6a	0.252	0.929	0.6a	0.254
3	0.997	−1.9a	0.343	0.997	−1.9a	0.343
4	0.927	−4.9a	0.468	0.935	−5.2a	0.468
5	0.866	0.0a	0.274	0.945	−5.4	0.196
6	0.963	15.7	0.481	0.997	4.8	0.318

^aNot statistically different from 0. 

Within-study correlations for ζ potential versus ψ_0,Y are high; r² ≥ 0.866 in all cases. NaCl and KCl have similar effects on the ζ potential (see solutions 2 and 3 and 11 and 12 in Table I), which justifies the assignment K_R,Na = K_R,K. CaCl₂ and MgCl₂ appear to have similar effects on the ζ potentials (see solutions 14 and 15), thereby providing some justification for the assignment K_R,Ca = K_R,Mg. Furthermore, the fact that points corresponding to Ca²⁺- and Mg²⁺-containing solutions all lie essentially on their respective regression lines indicates that K_R,Ca and K_R,Mg are reasonable relative to other binding constants. The coefficient a is not statistically different from 0 generally, and the regression line in Figure 1a essentially passes through the origin. Most of the values for ψ_0,Y are more negative than the corresponding ζ potentials, and b < 1. Part of the explanation is that the potential at the hydrodynamic plane of shear is of lower magnitude than the potential at the membrane surface. The ζ potentials appear to vary from study to study even for similar solutions, perhaps reflecting differences in ς_intrinsic. This is reflected in the values for b in Table II. The first study shown in Table I is different from the others because of its high value for b. Points corresponding to that study are denoted by shaded symbols in Figure 1, where it can be seen that those points constitute the principal outliers. A known model deficiency (Yermiyahu et al., 1997c) is revealed in the last two studies shown in Table I. Those studies and others indicate that at low pH the ζ potential becomes positive, but declining pH by itself cannot drive ψ_0,Y to positive values (see Table II).

Adjusting the Model to Improve the Correlations between ψ₀ and ζ Potentials

The inability of low pH alone to increase ψ_0,Y above 0 follows from the fact that the 1:1 binding of monovalent cations to negative sites can only neutralize the PM surface. Consequently, our first step toward the improvement of the model was to assume ion binding to the neutral sites, an assumption with some experimental justification (Akeson et al., 1989). To implement the change, a value of 2.4 μmol m⁻² was assigned to [P_T] on the basis of the space occupied by phosphatidic acids in biological membranes (Akeson et al., 1989). Next, all binding constants for the neutral site were assigned values proportional to the binding constants for the negative site. Trial values for the proportionality were assigned until the sum of the r² values for ζ potential versus ψ₀ for the six studies reached a maximum at K_P,I = K_R,I/180.

These limited changes (assignments: [P_T] = 2.4 μmol m⁻² and K_P,I = K_R,I/180) increased r² for ζ potential versus ψ₀ for all but one study, in which r² did not change (Table II). Figure 2 presents plots of ζ potential versus ψ_0,A for each study (ψ_0,A refers to ψ₀ computed by the adjusted model). The biggest increases in r² were in the problematical last two studies. Now the model predicts positive values for ψ₀ at low pH; note the left-to-right shift in the low-pH points in Figure 1b. The changes also generally reduced the values of a. The first criterion for the improvement of the model, an increase in the within-study correlations for ζ potential versus ψ₀, has been met. The second criterion, the avoidance of a significant degradation of the correspondence between measured and computed ion sorption observed in the study by Yermiyahu et al. (1997c), needs to be considered.

Figure 2.

ζ potentials of plant protoplasts or PM vesicles plotted against ψ_0,A for the individual studies presented in Table I. See legend to Figure 1 for details.

Agreement between the Original and the Adjusted Models Relative to Ion Binding

The original model of Yermiyahu et al. (1997c) was based on ion sorption, not ζ potentials, and the original model and the adjusted model disagree significantly in the computed values for ψ₀ at low pH (Table I). Does that mean that the models disagree with respect to the computed sorption of ions at low pH? To make the determination, ψ₀ and ion binding were computed by the original and the adjusted models for solutions factorial in pH (3.7 and 4.7), [AlCl₃] (1, 10, and 100 μm), and [LaCl₃] (10, 100, and 1000 μm). Figure 3 illustrates the excellent agreement of the models for ion binding despite disagreement for ψ_0,Y and ψ_0,A at pH 3.7. (In the case of ions with high binding constants, sorption is virtually equivalent to binding.) Thus, both models predict ion binding accurately, but the adjusted model predicts ψ₀ more accurately. Differences in ψ₀ between the two models may have little effect on ion binding, but a greater effect on [I^Ζ]₀ and {I^Ζ}₀ is expected and was observed (bottom two panels of Fig. 3).

Figure 3.

Surface densities of PM-bound ions, ψ₀, [Al³⁺]₀, and {Al³⁺}₀ calculated by the model of Yermiyahu et al. (1997c) (subscript Y) and the adjusted model of the present study (subscript A). Responses were for solutions factorial in pH (3.7, small symbols; 4.7, large symbols), [AlCl₃] (1, 10, and 100 μm), and [LaCl₃] (10, 100, and 1000 μm).

Accounting for the Differences between ζ Potential and ψ₀

Despite good within-study correlations between ζ potential and ψ₀, some large differences between these values persist (Table I). The values for b are all less than 1 (Table II), and the sum of squares for the differences between ζ potential and ψ_0,A for all 35 values is 37,133. Some of the differences may be attributable to the fact that the ζ potential measures ψ_s, the electrical potential at the plane of shear, at distance s from the PM surface (McLaughlin, 1989). ψ_s and ψ₀ are related according to the approximation ψ_s ≈ ψ₀exp(−κs), where κ ≈ 3.29I^1/2 and I is the ionic strength in the bulk-phase medium (Morel and Herring, 1993). Greater precision for ψ_s can be achieved using the second program mentioned in the introduction. In our study the sum of squares for ζ potential versus ψ_s,A decreases as s increases, reaches a minimum of 3176 at s = 3.8 nm, and then increases again. For phospholipid vesicles, the plane of shear is <1 nm from the PM surface (McLaughlin, 1989), but may be much higher for the PM, which incorporates glycolipids, proteins, and other constituents that project from the surface of the lipid bilayer. Figure 1c presents a plot of ζ potential versus ψ_2,A; note the increases in slope and fit.

A second possible source of the difference between ζ potential and ψ_0,A is that ς_intrinsic may genuinely have been more negative for the PM studied by Yermiyahu et al. (1997c) than for the PM of the other studies, assuming a less negative ς_intrinsic did reduce the sum of squares, but that adjustment did not improve the model according to the criteria used above. A third source of error is the apparent differences in ς_intrinsic among the studies as indicated by differences in b in Table II. Consequently, we recommend only the adjustments [P_T] = 2.4 μmol m⁻² and K_P,I = K_R,I/180 and make the assumption that most of the differences between ζ potential and ψ_0,A can be accounted for by the fact that the ζ potential is measured at some distance from the PM surface. Whatever the differences between ζ potential and ψ_0,A, the within-study proportionality between the two is very precise, and the adjusted model accurately predicts measured ion sorption in wheat.

Further Confirmation of the Binding Constants

In the exercise that follows we attempt to derive values for binding constants solely from the ζ potentials shown in Table I. The optimization criterion of close agreement with the study of Yermiyahu (1997c) with respect to sorption was abandoned. Instead, parameters were varied to minimize (sum of squares)/r² using all 35 values. The only constraints were K_R,Ca = K_R,Mg, K_R,Na = K_R,K, and constant K_R,I/K_P,I. Thus, these seven parameters were evaluated: [R_T], [P_T], K_R,H, K_R,La, K_R,Ca, K_R,K, and K_R,I/K_P,I. The procedure was to successively hold six parameters constant and optimize the seventh. Stable values were achieved, irrespective of starting values. Figure 1d presents ζ potential versus ψ₀ obtained using the new parameters. Despite a better fit and a unit slope for the combined data, the within-study correlations were poorer, and agreement with Yermiyahu et al. (1997c) was lower than for the previous adjusted model.

The optimization yielded these values for binding at the negative site: K_R,H = 15,000, K_R,La = 2,000, K_R,Ca = K_R,Mg = 12, and K_R,Na = K_R,K = 0.07. These values may be compared with the former values 21,500, 2,200, 30, and 1, respectively. Figure 4 presents a plot of the two sets of binding constants and their log₁₀ values. The high correlations in Figure 4 provide independent verification of the relative values of the binding constants.

Figure 4.

Binding constants determined on the basis of ion sorption (Yermiyahu et al., 1997c) (subscript Y) plotted against constants determined on the basis of published ζ potentials (subscript B). (See Fig. 1d for details.)

Binding Constants for Al³⁺ and for Anions

Al³⁺ does not appear in Table I, so no confirmation of K_R,Al (Yermiyahu et al., 1997c) can be obtained from the ζ potentials there, but some additional data indicate the suitability of the binding affinities presented here. Akeson et al. (1989) calculated that the binding of Al³⁺ to phosphatidylcholine liposomes was 560 times greater than the binding of Ca²⁺, and Jones and Kochian (1997) estimated that Al³⁺ binding to wheat microsomes was 300 times greater than Ca²⁺ binding. Our ratio of binding affinities for Al³⁺ and Ca²⁺ is 667. Wilkinson et al. (1993) observed a crossover (ψ₀ = 0) from negative to positive potentials in fish gill cells when [AlCl₃] = 16 μm in a background of 0.1 m NaCl at pH 4.5. Our model predicts a crossover when [AlCl₃] = 21.6 μm under similar conditions. Thus, two studies indicate lower constants and a third study indicates higher constants for Al³⁺. Those studies, together with some confidence inspired by the verified suitability of the constants for the other ions, indicate that the original value for K_R,Al presented by Yermiyahu et al. (1997c) may also be suitable.

In all of our studies we assumed no binding of anions to negative sites, but the assignment of cation binding to neutral sites would seem to warrant a similar binding of anions to neutral sites. In the adjusted model we made the assignments K_P,Cl = K_P,K and K_{P,SO4 = KP,Ca, but these assignments had little effect on ψ0.}

Induction of Positive ζ Potentials

Experimental evidence indicates that H⁺, Al³⁺, and La³⁺ can induce ζ potentials to shift from negative to positive values (Wilkinson et al., 1993; refs. in Table I), but limited evidence indicates that Ca²⁺, Mg²⁺, Na⁺, and K⁺ cannot cause this shift (Gibrat et al., 1985; Obi et al., 1989b; refs. in Table I). Our adjusted model is in practical agreement with the experimental data because H⁺, Al³⁺, and La³⁺ do cause ψ_0,A to shift to positive values at appropriate concentrations, but Ca²⁺, Mg²⁺, Na⁺, and K⁺ do not cause this shift at physiologically reasonable concentrations. The model predicts crossover at the following concentrations: [Al³⁺] = 20.1 μm, [La³⁺] = 183 μm, [H⁺] = 199 μm, [Ca²⁺] = 24.7 mm, and [K⁺] = 4.0 m in a background of 0.5 mm CaCl₂ at pH 4.5, unless the latter two were varied.

CONCLUSIONS

Although generally neglected, ψ₀ appears to play a significant role in plant-mineral interactions. The neglect may reflect the difficulty of measuring ψ₀ and the previous absence of verified model parameters for its computation in biological membranes. In particular, binding constants for the PM have been determined infrequently. The study of Yermiyahu et al. (1997c) provided the first constants for trivalent cations, despite the great importance of the environmental toxicant Al³⁺. The present study achieves the goal of providing a model, using a single set of model parameters, suitable for the computation of ψ₀ (or at least a value that is proportional to ψ₀) for PM from diverse plant sources. In addition, the model allows the computation of ion binding, concentrations, and activities (or at least values that are proportional to them) at the PM surface.

Some limitations of the model are apparent, including some questions about the suitability of the Gouy-Chapman theory for biological membranes discussed by Barber (1980) and McLaughlin (1989). Some questions concerning the use of [I^Ζ]₀ versus {I^Ζ}₀ in the derivation of the Gouy-Chapman-Stern model and in the interpretation of physiological responses remain unresolved to our knowledge (Kinraide, 1994). It is recognized that the PM contains more than two ion-binding sites; R⁻ and P⁰ merely represent composites of many sites. We assume that the PM expresses much spatial variability with respect to charge density. Specialized structures such as the outer orifice of ion channels may have exceptional distributions of charges and binding sites (Hille, 1992). Therefore, our model computes global properties only, yet these properties have been remarkably helpful in the interpretation of plant-mineral interactions. A worthwhile goal for future research will be the extension of the model to other important nutrients and toxicants.

Abbreviations:

ψ_m

transmembrane electrical potential difference

ψ₀

electrical potential at the PM surfaceψ_0,

_A

ψ₀ computed according to the adjusted Gouy-Chapman-Stern model of the present studyψ_0,

_Y

ψ₀ computed according to a Gouy-Chapman-Stern model presented by Yermiyahu et al. (1997c)

ς

charge density on the PM surface

{I^Z}₀ and {I^Ζ}_∞

activity of ion I with charge Ζ at the PM surface and in the bulk-phase medium, respectively

[I^Ζ]₀ and [I^Ζ]_∞

concentration of ion I with charge Ζ at the PM surface and in the bulk-phase medium, respectivelyK_P,

_I

binding constant for ion I to the PM site P⁰K_R,

_I

binding constant for ion I to the PM site R⁻

PM

plasma membrane(s)

ζ potential

electrical potential of particles at the hydrodynamic plane of shear measured by electrophoresis