Plant Physiol. (1998) 118: 505-512

Computation of Surface Electrical Potentials of Plant Cell Membranes
Correspondence to Published Zeta Potentials from Diverse Plant Sources

Thomas B. Kinraide^*, Uri Yermiyahu, and Giora Rytwo

Appalachian Soil and Water Conservation Research Laboratory, Agricultural Research Service, United States Department of Agriculture, Beaver, West Virginia 25813-0400 (T.B.K.); Institute of Soils, Water and Environmental Sciences, Agricultural Research Organization, The Volcani Center, P.O. Box 6, Bet-Dagan 50250, Israel (U.Y.); and Tel Hai College, Upper Galilee 12210, Israel (G.R.)

	ABSTRACT

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A Gouy-Chapman-Stern model has been developed for the computation of surface electrical potential (₀) 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 ₀ that correlate highly with published  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 m²) and to a neutral site (P⁰ = 2.4 µmol m²) according to the reactions R + I^Z  RI^{Z1 and} P⁰ + I^Z  PI^Z, where I^Z represents an ion of charge Z. Binding constants for the negative site are 21,500 M¹ for H⁺, 20,000 M¹ for Al³⁺, 2,200 M¹ for La³⁺, 30 M¹ for Ca²⁺ and Mg²⁺, and 1 M¹ 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 ₀, 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 ₀, ion binding, ion concentrations, and ion activities at membrane surfaces may be requested from the authors.

	INTRODUCTION

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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 _m, which may be measured relatively easily by the insertion of a microelectrode into cells in situ (Nobel, 1991). _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 _m (Hille, 1992). _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 ₀, the measurement of which is more difficult than the measurement of _m. The procedure entails the preparation of protoplasts or PM vesicles whose electrophoretic mobility is then measured to obtain a 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  potential has a somewhat lower magnitude than the ₀. Only a few  potential measurements have been reported for the plant PM (for refs., see Table I), and ₀ is rarely mentioned in introductory plant physiology textbooks or in general treatments of plant-mineral interactions.

View this table: [in this window] [in a new window]   Table I. Published  potentials of plant protoplasts or PM vesicles in various media compared with calculated surface potentials 0,Y is the potential computed by the model of Yermiyahu et al. (1997c) and 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.

Both _m and ₀ are physiologically important because electrical potential gradients influence the distribution of ions. ₀ 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). ₀ is influenced by the composition of the medium. High ionic strength reduces the negativity of ₀, and some ions can convert ₀ to positive values (Table I). The common neglect of ₀ is inconsistent with its importance, so the neglect probably reflects the difficulty of measuring ₀ and the absence of verified model parameters for its computation in biological membranes. Nevertheless, investigators have considered physiological phenomena in terms of ₀ (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 ₀ 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 ₀ in response to ionic solutes. Additionally, our goal was to make available an easy-to-use computer program for the computation of ₀, 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

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The Gouy-Chapman-Stern Model

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  is the charge density on the membrane surface expressed in coulombs per square meter (C m²); 2_roRT = 0.00345 at 25°C for concentrations expressed in molarity (_r is the dielectric constant for water, ₀ is the permittivity of a vacuum, R is the gas constant, and T is temperature); [I^Z] is the concentration of ion I^Z (the i^th ion) in the bulk-phase medium; Z_i is the charge on ion I^Z; F is the Faraday constant; and ₀ is the electrical potential at the membrane surface measured with respect to the bulk-phase medium a long way from the surface; Z_iF₀/RT = Z_i0/25.7 at 25°C for ₀ expressed in millivolts.

depends in part on _intrinsic, the surface charge density in the absence of any bound solute ions.  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^Z1], and [PI^Z] denote membrane surface densities in moles per square meter (mol m²), and [I^Z]₀ denotes the concentration of the unbound ion at the membrane surface. [I^Z]₀ may be computed from bulk-phase concentrations by the Boltzmann equation:

(6)

To compute , it is necessary to know [R_T] (R + _i[RI^Z1]) and [P_T] (P⁰ + _i[PI^Z]), which are the total surface densities of binding sites whether or not solute ions are bound. (_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^Z1, and PI^Z) can be computed from Equations 4, 5, and 6, contingent upon the value of ₀, and appropriately summed:

(7)

Thus, trial values for ₀ may be used to compute  using Equations 1 and 7. When the values for  converge, then the value of ₀ 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

2

2

1

1

1

1

1

1

Selection of Published  Potentials

Improvement of the Model

(9)

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

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Correlations between Computed _0,Y and Published  Potentials

View larger version (35K): [in this window] [in a new window]   Figure 1. potentials of plant protoplasts or PM vesicles plotted against calculated 0.  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, 0,Y calculated by the Gouy-Chapman-Stern model presented by Yermiyahu et al. (1997c) in which ions bind only to negative sites. b, 0,A calculated by the same model, except that ions were also allowed to bind to neutral sites. c, 2,A calculated by the model of (b) for a region 2 nm from the PM surface. d, 0,B calculated using a model in which parameters were adjusted without reference to the model of Yermiyahuet et al. (1997c). The binding constantsB in Figure 4 are from this model for 0,B.

View this table: [in this window] [in a new window]   Table II. Regression statistics for  potential compared with 0 r2 for  potential = a + b0, where 0 was computed by two models: 0,Y was computed according to the model presented by Yermiyahu et al. (1997c), and 0,A was computed according to the adjusted model of the present study. Each study is numbered according to its appearance in Table I.

Within-study correlations for  potential versus _0,Y are high; r²  0.866 in all cases. NaCl and KCl have similar effects on the  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  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 _0,Y are more negative than the corresponding  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  potentials appear to vary from study to study even for similar solutions, perhaps reflecting differences in _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  potential becomes positive, but declining pH by itself cannot drive _0,Y to positive values (see Table II).

Adjusting the Model to Improve the Correlations between ₀ and  Potentials

2

These limited changes (assignments: [P_T] = 2.4 µmol m² and K_P,I = K_R,I/180) increased r² for  potential versus ₀ for all but one study, in which r² did not change (Table II). Figure 2 presents plots of  potential versus _0,A for each study (_0,A refers to ₀ computed by the adjusted model). The biggest increases in r² were in the problematical last two studies. Now the model predicts positive values for ₀ 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  potential versus ₀, 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.

View larger version (28K): [in this window] [in a new window]   Figure 2. potentials of plant protoplasts or PM vesicles plotted against 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

View larger version (27K): [in this window] [in a new window]   Figure 3. Surface densities of PM-bound ions, 0, [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  Potential and ₀

s

A second possible source of the difference between  potential and _0,A is that _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 _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 _intrinsic among the studies as indicated by differences in b in Table II. Consequently, we recommend only the adjustments [P_T] = 2.4 µmol m² and K_P,I = K_R,I/180 and make the assumption that most of the differences between  potential and _0,A can be accounted for by the fact that the  potential is measured at some distance from the PM surface. Whatever the differences between  potential and _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

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.

View larger version (18K): [in this window] [in a new window]   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  potentials (subscript B). (See Fig. 1d for details.)

Binding Constants for Al³⁺ and for Anions

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 0.}

Induction of Positive  Potentials

	CONCLUSIONS

Although generally neglected, ₀ appears to play a significant role in plant-mineral interactions. The neglect may reflect the difficulty of measuring ₀ 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 ₀ (or at least a value that is proportional to ₀) 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^Z]₀ versus {I^Z}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.

	FOOTNOTES

   Received March 9, 1998; accepted June 24, 1998.

	ABBREVIATIONS

Abbreviations: _m, transmembrane electrical potential difference. ₀, electrical potential at the PM surfacepsgr_0,. _A, ₀ computed according to the adjusted Gouy-Chapman-Stern model of the present studypsgr_0,. _Y, ₀ computed according to a Gouy-Chapman-Stern model presented by Yermiyahu et al. (1997c). , charge density on the PM surface. {I^Z}₀ and {I^Z}, activity of ion I with charge Z at the PM surface and in the bulk-phase medium, respectively . [I^Z]₀ and [I^Z], concentration of ion I with charge Z 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). potential, electrical potential of particles at the hydrodynamic plane of shear measured by electrophoresis .

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