Analysis of the diffusion theory of negative capacitance: the role of K+ and the unstirred layer thickness

J. theor. Biol. (1988) 131, 183-197

Analysis of the Diffusion Theory of Negative Capacitance: the Role of K+ and the Unstirred Layer Thickness FABRICE HOMBLi~t

Department of Botany, University of Toronto, Toronto, Ontario M 5 S 1A1, Canada AND JACK M . FERR1ER

M R C Group in Periodontal Physiology, 4384 Medical Sciences Building, University of Toronto, Toronto, Ontario M 5 S 1A8, Canada (Received 30 March 1987, and in revised form 10 September 1987) The diffusion theory of negative capacitance is extended to take into account potassium transport as well as proton or hydroxyl transport. It is shown that both the capacitance spectrum and the frequency at which the capacitance is zero can be used to experimentally test the theory. The effects of the fraction of potassium current, membrane conductance, NaCI concentration, and unstirred layer thickness on these two characteristics is investigated. Maximum negative capacitance can be obtained when the current flowing through the membrane is mainly carried by protons, the membrane conductance is high, the solution conductivity is low, and the unstirred layer thickness is large. The effect of a dominant hydroxyl transport in place of a proton transport is also discussed. We suggest simple experiments to test the theory on Characeaen plant cells. Introduction

Biophysical investigations of the m e m b r a n e conductance and capacitance of plant cell m e m b r a n e s have been mainly carried out on the large internodal cells of Characeae species by means of the alternating-current method (Cole, 1968; Coster & Smith, 1977; Kishimoto, 1974; Beilby & Beilby 1983), the direct-current method (Williams et al., 1964; Kishimoto et al., 1982, Hombl6 & Jenard, 1986) or white noise analysis (Ross et al., 1985). With each of these methods, it has been shown that a negative capacitance (pseudo-inductive) response can be observed in some conditions. For instance, Coster and Smith (1977) and Ross et al. (1985) reported that the m e m b r a n e s of Chara cells sometimes showed a negative capacitance at frequencies lower than 1 Hz. Pseudo inductance was also observed when the m e m brane of Chara was voltage-clamped near the punchthrough potential (Beilby & Beilby, 1983), and a pseudo-inductive voltage response occurred when a constant current was injected through the m e m b r a n e of Chara (Hombl6 & Jenard, 1984; Hombl6, 1985) and through that of Hydrodictyon (Findlay and Coleman, 1983). t t Permanent address: Laboratoire de Thermodynamique Electrochimique et laboratoire de Physiologic V6g6tale, Universit6 Libre de Bruxelles, Facult~ des Sciences-CP 160, 1050 Brussels, Belgium. 183 0022-5193/88/060183 + 15 $03.00/0

(~) 1988 Academic Press Limited

184

F. H O M B L I ~

AND

J. M. FERRIER

Coster and Smith (1977) and Beilby and Beilby (1983) have suggested that the negative capacitance observed using the alternating-current method could arise from an electro-osmotic effect. Ferrier et al. (1985) have shown that a negative capacitance can arise from the diffusion of ions in the unstirred layer if the proton (or hydroxyl) is the only ionic species carried through the membrane. That a pseudo inductance can be related to the time- and voltage-dependent K+-channels has also been shown (Findlay and Coleman, 1983; Hombl6 & Jenard, 1984; Hombl6, 1985). It has been shown that the membrane of Chara can be placed in three distinct states (Bisson and Walker, 1980, 1981, 1982; Bisson, 1986): the H+-pump state, the K÷-passive state and the H+-passive state. In the H+-pump state, the resting membrane potential and conductance mainly reflect the activity o f the electrogenic H+-pump whereas in the K+-passive and H+-passive states, the membrane behaves like a K +- or H+-selective electrode respectively. It is not certain that membrane transport of H ÷ can be distinguished from membrane transport of O H - or of HCO~ (e.g., Walker & Smith, 1977; Lucas et al., 1977; Ferrier & Lucas, 1979; Lucas & Ferrier, 1980; Ferrier, 1980). The aim of the present investigation is to study the relative contribution of proton (or hydroxyl) and potassium transport to the negative-capacitance produced across the unstirred layer by diffusion, and to provide theoretical results that will allow simple experimental tests of the theory. We will present our derivations in terms of proton transport across the unstirred layer, but will also point out modifications required to describe hydroxyl transport. Though the theory is applied to the case of plant cells, it is valid for any membrane (biological or synthetic) permeable to protons and potassium ions. A negative capacitance will occur when the membrane conductance will be higher for protons than for potassium ions. For instance, this could be the case in bipolar membranes or in synthetic membranes containing negative fixed charges.

Theory The measured phase shift between an intracellular and an extracellular electrode depends on the phase shift between the total applied alternating current and the resulting alternating electrical potential difference. At any time, the total electrical potential difference (A V,) can be written as: AV, = A V~°sin (tot+d#,)

= A V ° s i n ( w t + 4 ~ . , ) + A V ° sin(wt+ck.)+AV° sin(wt+ckb),

(1)

where subscripts m. u and b refer to the membrane, unstirred layer and bulk phase respectively, and w = 2n-v where v is the frequency. Equation (1) can be expanded using trigonometrical relations to obtain an expression for the total phase shift ($,): A V °, sin ~b.. + A V ° sin 4~,,+ A V ° sin ~bh tan (~b,) = A V o cos ~bm+ A V° cos ~b,,+ A V ° cos O~"

(2)

NEGATIVE

CAPACITANCE

DIFFUSION

THEORY

185

In the frequency domain of interest in plant cell biophysics, the external solution appears as purely resistive, so that ~bh = 0, and eqn (2) can be simplified to: tan

A V,°. sin ~b., + A V ° sin ~b,, (~')-

a v ° cos ,bin + ~ V° COS ,¢,. + A V °"

(3)

Since $., and $ . are small, the denominator of eqn (3) will be close to the total potential measured, so the denominator can be equated to AV ° and eqn (3) can be rewritten as: tan ~b, =

A V,°,, sin ~b.. + A V ° sin ~b,, A V,°

(4)

Using a parallel R - C electrical equivalent circuit it can be shown that tan ~ , - - - w R , C , , where R, and (7, are the measured resistance and capacitance, and since a similar expression can be written for the m e m b r a n e with tan ~b.~= sin ~bm, since ~b., is small, we can rewrite eqn (4):

R.,C.,A V,°, - ((sin ~b,,AV°)/oJ) c,-

n, a v o

(5)

It can be shown (see Appendix) that: A V ° sin ~b. = (Ioj, Bh + IokBk)/O"

(6)

where Ior, and Iok are, respectively, the proton and the potassium currents flowing through the m e m b r a n e ; cr is the total ionic electrical conductivity in the unstirred layer, and Bh and B k a r e given by:

Bh

1 - (Do~ Dh ) --

Bk --

2Kh 1 -- (Do~ Dk)

2Kk

{1 - ~ / 2 sin (Kr, L + zr/4) exp ( - K h L ) } ,

(7)

{1 - - 4 2 sin ( K k L + rr/4) exp ( - K k L ) } .

(8)

Do is a weighted diffusion coefficient for the ionic species not transported across the m e m b r a n e (Ferrier, 1981), Dk and Dh are the diffusion coefficients of the potassium ion and of the proton respectively, L is the unstirred layer thickness. Kr, = ( w l 2 D h ) 112 and Kk = (og12Dk) 112. If we assume that R, is close to R., and that A V, is close to A V,. = I,R.,. where I, is the total current through the m e m b r a n e , combining eqns (5) and (6) gives: lo,,Br, + IokBk

C, = C,,,

tol, R ~ t r

'

(9)

which is the expression for the capacitance measured by electrophysiological methods. It is the sum of the m e m b r a n e capacitance and a term arising from the diffusion of ions in the unstirred layer surrounding the cell membrane. Results The results reported here were calculated for a biological m e m b r a n e permeable to proton and potassium ions, surrounded by artificial fresh water. This is generally

186

F. HOMBLE AND J. M. FERRIER

the case for aquatic plant cells of the Characeae for which a number of biophysical data has now been collected. In order to investigate the effect of various physico-chemical factors on the total

membrane capacitance spectrum (the change of the capacitance with frequency), the values in Table 1 were used for the parameters of eqns (7), (8) and (9). Some of the parameters were varied over the range of biological interest. TABLE 1

Values of the different theoretical parameters Parameter

Value

Units

C,,,

10-"

F m -2

Do DI, Da

1.62 x 10 - ° 9 . 3 4 x 10 - ° 1 "98 x l 0 - 9

m 2 s -t m 2 s -I m 2 s -I

loh/I , lok/l~ L R,,,

0"7 0'3 10 -4 10 - I

m S -t m z

o"

3 - 0 8 5 x 10--"

Sm -t

EFFECT

OF

THE

FRACTION

H + AND

K+-CURRENT

In the present treatment, it is assumed that the total current flowing through the membrane (/,) is carried by protons (Ioh) and potassium (Iok) ions. This assumption is straightforward for various physiological conditions (see Discussion). When the fraction of potassium current decreases, both the slope of the low frequency spectrum and the frequency at which the capacitance is zero increase (Figs 1 and 2). EFFECT

OF

THE

MEMBRANE

CONDUCTANCE

Alternating bands of acid and alkaline pH develop around some plant cells in the light when the external solution is alkaline (Lucas, 1983). It has been shown that the biophysical properties of the membrane measured in the acid band are different from those in the alkaline band (Smith & Walker, 1985), and that the specific membrane conductance of the alkaline band is much greater than for the cell as a whole with a maximum value of 8 S m -2 (Smith & Walker, 1983). Published values of negative-capacitance range from about - 0 . 2 F m -2 for the plasmalemma alone (Coster & Smith, 1977) to about - 4 F m -2 for the tonoplast and the plasmalemma in series (Ross et aL, 1985) when measured at 0.01 Hz. To obtain such values of negative-capacitance at this frequency we must consider the membrane conductance to be greater than 10 S m -" (Fig. 3). When only the membrane conductance increases, the other parameters being constant, the low frequency slope of the capacitance spectrum increases (Fig. 3) and the frequency of zero total capacitance is shifted towards greater values (Fig. 4).

NEGATIVE

CAPACITANCE

DIFFUSION

187

THEORY

0

-0-1

c

8

==

-0.2

-0"3~/r-° .,I

o.oi

. . . . . . .

0'.,

.

. . . . . .

Frequency

.

.

.

.

.

.

.

Jo

i

(Hz)

FIG. 1. Effect of the fraction of proton current on the capacitance spectrum. Curves a, b, c refer to a fraction of proton current of 1 , 0 . 8 , a n d 0, respectively. The remaining fraction is that for potassium current.

I

0ol

I

I

0.2

I

I

I

0-4

I

0.6

I

I

0-8

I

Fraction of H+current F I ~ . 2. Effect of the fraction of proton current on the frequency of zero capacitance. The remaining

fraction is that For potassium current. EFFECT

OF

NaCI

It has been shown previously that a change in the NaC! concentration of less than 10 mM in the external solution surounding Chara corallina cells has little effect on the biophysical properties of the membrane (Kitasato, 1968, Spanswick, 1972; Beilby, 1985; Hombl6, 1985). However, increasing the NaCI concentration in the external solution will increase the solution conductivity (o-) which, in turn, will affect the phase shift between the current and the electical potential difference in the unstirred layer adjacent to the membrane [eqn (6)]. NaCl is thus a good salt to test the negative-capacitance diffusion theory. When the NaCl concentration is increased the low frequency slope of the total capacitance spectrum decreases (Fig. 5) and the zero capacitance frequency is shifted

188

E. H O M B L E

AND

J. M . F E R R I E R

_o t.)

-5

O.OI

,

,

,

,,

, , , i

O'l

'

'

'

'

'

'

" i

'

'

'

' '''

IO

Frequency (Hz) FIG. 3. Effect o f the m e m b r a n e r e s i s t a n c e o n the c a p a c i t a n c e s p e c t r u m . C u r v e s a, b, c r e f e r to a m e m b r a n e r e s i s t a n c e o f 0,03, 0.05 a n d 0.1 S -~ m +2, r e s p e c t i v e l y .

towards lower frequencies (Fig. 6). Although this change is less than that observed for variation of the other parameters, it is still measurable by classical electrophysiological methods. EFFECT

OF THE

UNSTIRRED

LAYER

WIDTH

It is now well established that all cells are surrounded to some extent by an unstirred layer of solution (Barry & Diamond, 1984). The external side of the plasmalemma of plant cells is surrounded by a cell wall of 5 to 10 Ixm which sets the lower limit of an unstirred layer. With usual rates of stirring used in biological experiments, an effective unstirred layer of between 20 and 500 p,m has been estimated, the actual thickness depending partly on the size of the biological material used and partly on the rates of stirring (Dainty & Hope, 1959; Dainty, 1963). Figure 7 shows that the low frequency slope of the capacitance spectrum becomes steeper when the width of the unstirred layer rises. The frequency of zero capacitance is constant for an unstirred layer thickness higher than 200 ~m, but is drastically shifted towards lower frequencies for values lower than 100 ~m (Fig. 8). Discussion

Since Kitasato (1968) showed that the electrophysiological properties of Nitella cells are strongly dependent on the pH, many experimental studies have been devoted to the role of proton in plant physiology (e.g. Smith & Raven, 1979; Tazawa & Shimmen, 1982). At pH close to neutrality, in low potassium medium provided with calcium and in saturating light, Characean cells are in their H*-pump state. At low pH (<6), or in the absence of calcium, or at K ÷ concentrations above 1 mM, or in dark conditions, the passive potassium permeability increases (Hope & Walker, 1975, Spanswick et

NEGATIVE

CAPACITANCE

DIFFUSION

THEORY

189

10r

u~

0-

0-011 0

I

I

I

I

I

0-1

0-2

0.3

0.4

0.5

Membrone resislonce (S-Ira 2)

FIG. 4. Effect of the membrane resistance on the frequency of zero capacitance.

aL, 1967; Richards & H o p e , 1974). At high p H ( > 10) and in the light, cells are in their H÷-passive state (Bisson, 1986). Moreover, even in the H * - p u m p state there is a leak conductance mainly associated with the passive potassium transport. However, because this leaky conductance is much lower than the conductance of the electrogenic pump, it is neglected to a first approximation. In Chara, the passive m e m b r a n e conductance accounts for about 50% of the total m e m b r a n e conductance when the cells are in the dark at pH 7 (Kishimoto et al., 1980). At pH 5.5, the p u m p conductance for illuminated cells is about 20% of the total resting m e m b r a n e conductance (Smith & Beilby, 1983) and the potassium channels contribute to the m e m b r a n e conductance (Hombl6 & Jenard, 1984; Hombl6, 1985; Hombl~ & Jenard, 1986). Thus, there are various physiological conditions in which a passive potassium conductance occurs in parallel with a proton conductance. It is interesting to point out that the negative-capacitance theory makes no distinction between the active and the passive transport of an ionic species. Hence the term lol, in eqn (9) stands for the proton current through the m e m b r a n e no matter if it is passive or active.

190

F. H O M B L I ~

AND

J. M. FERRIER

c

-0"1

g -0.2

-0.3

o-o,

.......

6'-~

........

I

........

,o

Frequency (Hz) FIG. 5. Effect o f NaCI concentration on the capacitance spectrum. Curves a, b, c refer to a N a C I concentration of 1, 5 a n d 10 m M , respectively.

hi "r

IX.

o.~

i

~

~

~

~

~

~

~

~

Io

NaCI concentration (mM) FIG. 6. Effect of the NaCI concentration on the frequency of zero capacitance. EFFECT

OF THE

FRACTION

OF

POTASSIUM

CURRENT

As any change in the proton or potassium conductance will modify the fraction of proton and potassium current through the membrane and, at the same time, the membrane conductance, Figs 1 to 4 must be analyzed simultaneously. If we adopt the model of a voltage-dependent pump conductance (Rapoport, 1970; Spanswick, 1972), two limiting cases can be easily distinguished. Firstly, an increase in the pump activity increases the membrane conductance and the fraction of proton current through the membrane. In this case, the zero capacitance frequency will shift towards higher frequencies and the slope of the low frequency capacitance spectrum will become steeper (Figs 1 to 4). Secondly, blocking the pump activity by means of a metabolic inhibitor will decrease the membrane conductance and will increase the effect of potassium transport. The fraction of proton current will

NEGATIVE

CAPACITANCE

DIFFUSION

191

THEORY

Oi

o

== t,J

O'Ot

O-I

I

IO

Frequency (Hz) FIG. 7. Effect o f the unstirred layer thickness on the capacitance spectrum. Curves a, b, c refer to an unstirred layer o f 5 x ]0 -4, 2-5 x 10 -4 and 1 x 10 -4 m, respectively

"Iv

g o~

O.O

0-00111

o

',

i

~

~

~

I

~

Unstirred Ioyer width (IO-4 m) FiG. 8. Effect o f the unstirred layer thickness on the frequency o f zero capacitance.

192

F. H O M B L t ~

AND

J. M.

FERRIER

then decrease and the negative-capacitance diffusion theory predicts a shift of the zero capacitance frequency towards lower frequencies and a decrease of the slope of the low frequency capacitance spectrum. EFFECT

OF THE

UNSTIRRED

LATER

WIDTH

Because of the cell wall, there is an obligatory unstirred layer adjoining the external side of the plasmalemma. Its effective thickness will depend both on the cell wall width and on the ionic diffusion coefficients in the cell wall (Dainty & Hope, 1959). Moreover, even with "perfect" stirring of the solution there is always an additional unstirred layer of solution at the solution-cell wall interface. In consequence, the minimum unstirred layer thickness is of the order of magnitude of 20 ~m, close to 50 p.m as a minimum (Dainty, 1963). The negative-capacitance diffusion theory predicts a large change in the slope of the low frequency capacitance spectrum when the unstirred layer thickness is changed (Fig. 7). Controlling, for instance, the speed of stirring of the solution thus provides an experimental test to check the theory.

EFFECT

OF

NaCI

As shown in Figs 5 and 6, a change of NaCl concentration will affect significantly both the low frequency slope of the capacitance spectrum and the zero capacitance frequency. This is so because an increase of the NaCI concentration will increase the total solution conductivity outside the cell and of the unstirred layer in particular. NEGATIVE-CAPACITANCE

AT PUNCHTHROUGH

In their study of the potential dependence of the admittance of Chara corallina, Beilby and Beilby (1983) showed that the membrane reactance is inductive at the punchthrough. However, they were not able to give any explanation for the negativecapacitance response because the double fixed charge model generally used to describe the punchthrough (Coster, 1965) does not predict this effect. The derivations leading to eqn (9) show that, if there is a large increase in membrane conductance for ions that have a diffusion coefficient close to Do, as would be expected at punchthrough, while the conductance for the high mobility species is not reduced, then there should be a large increase in the negative capacitance term in eqn (9). That is, since I, is proportional to R,~ ~, I,R~,, should be greatly decreased. Thus, the negative capacitance at punchthrough can be explained by the present theory. DISTRIBUTION

AND

IMPORTANCE

OF THE

H ~ TRANSPORT

SYSTEM

From Fig. 1 it can be concluded that the major part of the negative capacitance arises from transport of protons (active or passive) across the membrane. In order to obtain negative-capacitance values in the range of those published for the whole cell we must calculate eqn (9) using values of R,, lower than those

NEGATIVE CAPACITANCE DIFFUSION THEORY

193

measured on the whole cell. However, it has been shown that the membrane conductance is not homogeneous along the cell surface. For instance, in the case of Chara corallina, which has alternating acid and alkaline bands (Lucas, 1983), the membrane conductance is larger in the alkaline band than in the acid band. According to Smith and Walker (1983), the alkaline band specific conductance is at least 5 times greater than the average specific conductance of the whole cell. Furthermore, according to Lucas et al. (1977) the ion tranport zone of the alkaline band is at least one-sixth of the total cell surface area. This means that the conductance of the alkaline band would account for at least 80% of the cells' total conductance. The calculations of Smith and Walker (1983) and Lucas et al. (1977) assume that the transport is homogeneous over the transport zone. According to our results, to have as high a negative capacitance as that found by Ross et al. (1985), the specific membrane conductance must be even higher than that given by Smith & Walker (1983). This might occur if the ion carriers are not homogeneously distributed in the zone, but are clustered together, producing smaller areas of higher specific conductance.

EFFECT

OF

OH-

VERSUS

H ÷ TRANSPORT

The pH gradient measurements of Lucas (1975) clearly show that current in the alkaline band will be carried across the unstirred layer by diffusion of hydroxyl (Lucas et al., 1977; Ferrier & Lucas, 1979). Taken together, these results would imply that an injected current would be carried across the unstirred layer by hydroxyl diffusion. We can apply our derivations to hydroxyl ions by using the hydroxyl diffusion coefficient which would mean that K h increases by a factor o f ( D h / D o h )1/2 = 1.3. It can easily be calculated from eqn (7) that this would decrease B h at 1 Hz by about 30%. This change in B h becomes smaller at lower frequencies, so that at 0-1 Hz, B h is decreased by 6%, and at 0.01 Hz by less than 2%. Thus, the zero capacitance frequency would be shifted to slightly lower frequencies for hydroxyl transport in the unstirred layer, as can be seen from eqn (4), while the magnitude of negative capacitance would be changed only slightly at lower frequencies.

ORIGIN

OF

THE

PSEUDO-INDUCTIVE

RESPONSE

IN

PLANT

CELLS

The negative capacitance diffusion theory may account for the data obtained by means of the alternating current method (Coster & Smith, 1977) and with the white noise analysis (Ross et al., 1985). However, it has been shown that the pseudoinductive voltage response to a current step can be eliminated by TEA, which blocks the time-and voltage-dependent K+-channei in plant cells (Tazawa & Shimmen, 1980; Sokolik & Yurin, 1981; Findlay & Coleman, 1983; Hombl6, 1985; Beilby, 1985). In this case, the pseudo-inductive transient voltage response has a time constant of the order of 100 ms, which implies that there is a long lag time before channel activation, or that some other cellular feedback system controls the activation (Hombl6 & Jenard, 1984). However, a time constant of a few seconds would be

194

F. HOMBLI~ A N D J. M. F E R R I E R

r e q u i r e d f o r t h e l a r g e n e g a t i v e c a p a c i t a n c e o b s e r v e d a t 0 . 0 5 H z b y R o s s et al. ( 1 9 8 5 ) . This discrepancy suggests that the pseudo-inductive response to a constant current step might have a different origin from the negative capacitance observed at low frequencies. In physiological conditions both of these phenomena can occur simultaneously and their relative importance remains to be investigated.

F. H o m b l 6 is a S e n i o r R e s e a r c h Assistant o f the N a t i o n a l F u n d for Scientific R e s e a r c h (Belgium) w h i c h is kindly a c k n o w l e d g e d for its financial support. We also a c k n o w l e d g e the s u p p o r t o f the Medical R e s e a r c h Council ( M R C ) o f C a n a d a . We are also i n d e b t e d to D r Jack D a i n t y for critically r e a d i n g this m a n u s c r i p t a n d for financial s u p p o r t from his o p e r a t i n g g r a n t from the N a t i o n a l Sciences a n d E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a .

REFERENCES BARRY, P. H. & DIAMOND, J. M. (1984). Effects of unstirred layers on membrane phenomena. Physiol. ReD. 64, 763. BEILaY, M. J. & BEILBY, B. N. (1983). Potential dependence of the admitance of Chara plasmalemma. J. Membrane Biol. 74, 229. BEILBY, M. J. (1985a). Potassium channels at Chara plasmalemma. J. Exp. Bot. 36, 228. BEILBY, M. J. (1985b). Potassium channels and different states of Chara plasmalemma. J. Membrane Biol. 89, 241. BISSON, M. A. (1986). The effect of darkness on active and passive transport in Chara corallina. J. Exp. Bot. 37, 8. BISSON, M. A. 8I. WALKER, N. A. (1980). The Chara plasmalemma at high pH. Electrical measurements show rapid specific passive uniport of H + or OH-. J. Membrane Biol. 56, 1. BISSON, M. A. t~ WALKER,N. A. (1981). The hyperpolarization of the Chara membrane at high pH: effects of external potassium, internal pH, and DCCD. J. Exp. Bot. 32, 951. BISSON,M. A. • WALKER,N. A. (1982). Control of the passive permeability in the Chara plasmalemma. J. Exp. Bot. 33, 520. COLE, K. S. (1968). Membrane, Ions, and Impulses. Berkeley: University of California Press. COSTER, H. G. L. (1965). A quantitative analysis of the voltage-current relationships of fixed charged membranes and the associated property of punchthrough. Biophys. J. 5, 669. COSTER, H. G. L. & SMITH, J. R. (1977). Low frequency impedance of Chara corallina: simuultaneous measurements of separate plasmalemma and tonoplast capacitance and conductance. Aust. J. Plant Physiol. 4, 667. DAINTY, J. & HOPE, A. B. (1959). Ionic relations of cells of Chara australis. I. Ion exchange in the cell wall. Aust. J. Biol. Sci. 12, 395. DAINTY, J. (1963). Water relations of plant cells. Adv. Bot. Res. I, 279. FERRIER, J. M. & LUCAS, W. J. (1979). Plasmalemma transport of OH- in Chara corallina II. Further analysis of the diffusion system associated with OH- efflux. Z Exp. Bot. 30, 705. FERRIER, J. M. (1980). Apparent bicarbonate uptake and possible plasmalemma proton efflux in Chara corallina. Plant Physiol. 66, 1198. FERRIER, J. M. (1981). Time-dependent extracellular ion transport. J. Theor. Biol. 92, 363. FERRIER, J. M., DAINTY, J. & ROSS, S. M. (1985). Theory of negative capacitance in membrane impedance measurements. J. Membrane Biol. 85, 245. FINDLAY, G. P. ,~' COLEMAN, H. A. (1983). Potassium channels in the membrane of Hydrodictyon africanum. J. Membrane Biol. 75, 241. HOMaL[, F. & JENARD, A. (1984). Pseudo-inductive behaviour of the membrane potential of Chara corallina under galvanostatic condtions. J. Exp. Bot. 35, 1309. HOMaLE, F. (1985). Effect of sodium, potassium, calcium, magnesium and tetraethyammonium on the transient voltage response to a galvanostatic step and of the temperature on the steady membrane conductance of Chara corallina: a further evidence for the involvement of potassium channels in the fast time variant conductance. J. Exp. Bot. 36, 1603. HOMBLI~, E. & JENARD, A. (1986). Membrane operational impedance spectra in Chara corallina estimated by Laplace transform analysis. Plant PhysioL 81,919.

NEGATIVE

CAPACITANCE

DIFFUSION

THEORY

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HOPE, A. B. & WALKER, N. A. (1975). The Physiology of Giant Algal Cells. Cambridge: Cambridge University Press. KISHIMOTO, U. (1974). Transmembrane impedance of the Chara cell. Jap. J. Physiol. 24, 403. KISHIMOTO, O., KAMI-IKE, N. & TAKEUCHt, Y. (1980). The role of the electrogenic pump in Chara corallina. J. Membrane Biol. 55, 149. KISHIMOTO, U., KAMHKE, N., TAKEUCHI, Y. & OHKAWA, T. (1982). An improved method for determining the ionic conductance and capacitance of the membrane of Chara corallina. Plant Cell Physiol. 23, 1041. KITASATO, H. (1968). The influence of H + on the membrane potential and ion fluxes of Nitella. J. Gen. Physiol. 52, 60. LUCAS, W. J. (1975). Analysis of the diffusion symmetry developed by the alkaline and acid bands which form at the surface of Chara corallina cells../. Exp. Bot. 26, 271. LUCAS, W. J., FERRIER, J. M. & DAINTY, J. (1977). Plasmalemma transport of OH- in Chara corallina. Dynamic of activation deactivation. J. Membrane Biol. 32, 49. LUCAS, W. J. & FERRIER, J. M. (1980). Plasmalemma transport of OH- in Chara corallina III. Further studies on transport substrate and directionality. Plant Physiol. 66, 46. LUCAS, W. J., KEIFER, D. W. • SANDERS, D. (1983). Bicarbonate transport substrate in Chara corallina: Evidence for cotransport of HCO~- with H +. J. Membrane Biol. 73, 263. LUCAS, W. J. (1983). Photosynthetic assimilations of exogenous HCO~- by aquatic plants. Ann. Reo. Plant Physiol. 34, 71. RAr'OPORT, S. I. (1970). The sodium-potassium pump: relation of metabolism to electrical properties of the cell. I. Theory. Biophys. J. 10, 246. RICHARDS, J. L. & HOPE, A. B. (1974). The role of protons in determining membrane electrical characteristics in Chara corallina. J. Membrane Biol. 16, 121. Ross, S. M., FERRIER, J. & DAINTY, J. (1985). Frequency-dependent membrane impedance in Charo corallina estimated by Fourier analysis. J. Membrane Biol. 85, 233. SMITH, F. A. & RAVEN, J. A. (1979). Intracellular pH and regulation. Ann. Reu. Plant Physiol. 30, 289. SMITH, J. R. & BEILBY, M. J. (1983). Inhibition of the electrogenic transport associated with the action potential in Chara. J. Membrane Biol. 71, 131. SMITH, J. R. & WALKER, N. A. (1983). Membrane conductance of Chara measured in the acid and basic zones. J. Membrane Biol. 73, 193. SMITH, J. R. & WALKER, N. A. (1985). Effects of pH and light on the membrane conductance measured in acid and basic zones of Chara. J. Membrane Biol. 83, 193. SOKOLIK, A. I. & YURIN, V. M. (1981). Transport properties of potassium channels of the plasmalemma in Nitella cells at rest. Soviet Plant Physiol. 28, 206. SPANSWICK, R. M., STOLAREK, J. & WILLIAMS, E. J. (1967). The membrane potential of Nitella translucens. J. Exp. Bot. 18, 1. SPANSWlCK, R. M. (1972). Evidence for an electrogenic ion pump in Nitella translucens. I. The effects of pH, K +, Na +, light and temperature on the membrane potential and resistance. Biochim. Biophys. Acta 288, 73. TAZAWA, M. & SHIMMEN, T. (1980). Demonstration of the K+-channel in the plasmalemma of tonoplast-free cells of Chara australis. Plant Cell Physiol. 21, 1535. TAZAWA,M. 8£ SHIMMEN, T. (1982). Control of electrogenesis by ATP, Mg 2+, H +, and light in perfused cell of Chara. In: Current Topics in Membranes and Transport. (Slayman, C. L. ed.). pp. 49-67. New York: Academic Press. WALKER, N. A. & SMITH, F. A. (1977). Circulating electric currents between acid and alkaline zones associated with HCO~- assimilation in Chara. J. Exp. Bot. 28, 1190. WILLIAMS, E. J., JOHNSTON, R. J. & DAINTY, J. (1964). The electrical resistance and capacitance of the membranes of Nitella translucens. J. Exp. Bot. 15, 1.

APPENDIX T h e n e t flux o f a n i o n i c s p e c i e s i i n t h e e x t r a c e l l u l a r m e d i u m o-i J, = - D , V Ci - ~ F V~b + vC,,

is g i v e n b y : (A1)

w h e r e ~b is t h e e l e c t r i c a l p o t e n t i a l , F is t h e F a r a d a y c o n s t a n t , v is t h e f l u i d v e l o c i t y ,

196

F. H O M B L ~ .

AND

J. M . F E R R I E R

and C~, z~, D~ and cr~ are the concentration, valency, diffusion coefficient and electrical conductivity attributable to the ith ionic species, respectively Summing the currents for each ion and assuming electroneutrality, the net total current is given by: I = Z z, FJ~ = - E i

z,FD, V C , -

o-V~b

(A2)

i

where cr is the extracellular solution conductivity ( o - = ~ o-i). Let us define the diffusion-driven current density, ld, as: ld = -- ~ ziFDiV Ci.

(A3)

i

We can solve (A2) for V0 and substituting in (A1) results in: J ~ = - D , VC~+

o', ( l - I a ) + v C ~ . o'ziF

(A4)

Considering that the m e m b r a n e is mainly permeable to H + and K ÷, we can rewrite eqn (A3) as: ld = -- ~

z~FD, V Ci - FDh V Ch -- FDk V Ck.

(AS)

i~hk

Following the same procedure as that described by Ferrier et al. (1985), it can be shown that: - o ' V 0 = I - (1 - Rh)lh --(1 -- Rk)Ik,

(A6)

where Rh = ( D o / D h ) and Rk = ( D o / D k ) where Do is a weighted diffusion coefficient for the ionic species not transported across the m e m b r a n e (Ferrier, 1981), I is the net total current flowing through the m e m b r a n e and Ih and lk are the net currents flowing through the m e m b r a n e carried by the proton and the potassium ions, respectively ( I = Ih + lk ). The equation governing Ci can be calculated using the continuity relation: dCi dt

-- - V J i = DiV2Ci.

(A7)

This equation can be solved analytically for the c o m p o n e n t of current carried by the proton diffusion and by the potassium diffusion. Using the b o u n d a r y condition: L = Io~ sin wt at the membrane-solution interface (x = 0) gives: lh = lob exp (--KhX) sin ( w t - Khx)

Ik = lok exp ( - - K k x ) sin ( w t -- KkX)

(A8) where Kh = (~o/2Dh) w2 and Kk = ( c o / 2 D k ) '/2

NEGATIVE

CAPACITANCE

DIFFUSION

THEORY

197

Substituting eqn (AS) in (A6) and then integrating give an expression for the difference in electrical potential across the unstirred layer: ira ~ sin(tot + ~b.) =(Ioh + Iok) L sin tot

+ [( 1 - Do/Dh )/(x/2 rh ) ] Io~{[ 1 -- COS (KhL) exp ( - KhL)] + s i n (tot + 31r/4) +sin (KhL) exp (-KhL) sin (tot-37r/4)} +[(1 - Do/ Dk)/X/2 Kk)]lok X {[1 --COS (KkL) exp (-KkL)] sin (tot + 3zr/4) +sin (KkL) exp (-KkL) sin (tot-37r/4)}.

(A9)

The phase shift can be obtained by putting t = 0 in eqn (A9): o-A ~ sin ~b. = [1 - (Do/Dh)/2Kh)]Ioh[1 -,/2 sin (KhL+ zr/4) exp (-KhL)] +[1 - ( D o / D k ) / 2 K k ) ] I o k [ 1 --,/2 sin (KkL+ 1r/4) exp (-KkL)] (A10) where L is the unstirred layer thickness. This is eqn (6) in the main text.