Dielectric properties of mouse lymphocytes and erythrocytes

Biochimica et Biophysica Acta, 1010 (1989) 49-55

49

Elsevier BBA 12375

Dielectric properties of mouse lymphocytes and erythrocytes K. Asami, Y. Takahashi * and S. Takashima Department of Bioengineering, University of Pennsylvania. Philadelphia, PA (U.S.A.)

(Received 15 June 1988)

Key words: Dielectricbehavior; Membrane capacitance; (Mouse lymphocyte); (Mouse erythrocyte)

In order to study the effect of the nucleus on dielectric behavior of the whole cell, permittivity (dielectric constant) and conductivity of mouse lymphocytes and erythrocytes were measured over a frequency range from 0.1 to 250 Mttz. Erythrocytes (spherocytes) showed a single dielectric dispersion, which was explained by a single-shell model that is a conducting sphere covered with a thin insulating shell. On the other hand, iymphocytes showed a broad dielectric dispersion curve which was composed of two subdispersions. The high-frequency subdispersion, which was not found for erythrocytes, was assigned to the Maxwell-Wagner dispersion of the nucleus occupying about 65% of the totaJ cell volume. Analysis of the lymphocyte dispersion was carried out by a double-shell model, in which a shelled sphere, i.e., nucleus, is incorporated into the single-shell model. The following electrical parameters were consequently estimated; the capacitance of the plasma membrane, 0.86 #F-cm-2; the conductivity of the cytoplasm, 3.2 m S . c m - i ; the capacitance and conductance of the nuclear envelope are, respectively, 0.62 #F-era -2 and 15 S .era -2, and the permittivity and conductivity of the nucleoplasm are 52 and 13.5 m S . era-!.

Introduction Dielectric analysis of cell suspensions is a rather classical technique providing the electrical capacitance of the plasma membrane and the conductivity of the cell interior (for reviews, see Refs. 1-3). Fricke [4] first applied il to erythrocytes in the 1920's and estimated the membrane capacitance of 0.8 # F / c m 2 from his dielectric theory based on an electrical model in which a conducting homogeneous sphere (or ellipsoid) is covered with a thin shell less able to conduct (Fig. 1). This model, termed the single-shell model, is appropriate for characterizing electrical properties of mammalian erythrocytes which include neither nucleus nor cytoplasmic organelle. However, it is an over-simplified model for cells with an in:racellular structure. Actually, with such cells, we found ~ partial disagreement between experiments and theoretical calculations from this model (for example, see R,~.f. 5). Instead of the single-shell model, Irimajiri [6,7] pr ~posed a double-shell model in order to explain this disagreement. In this

* Permanent address: Olympus Optical Company, Hachioji, Tokyo, Japan. Abbreviation: PBS, phosphate-buffered saline. Correspondence (permanent address): K. Asami, Institute for Chemical Research, Kyoto University, Uji, Kyoto, Japan.

model, a shelled sphere, a nucleus in this case, is incorporated into the single-shell model (Fig. lb). Although its application is limited to special cells which have a high nucleocytoplasmic ratio and contain a few cytoplasmic organelles, the new model has the advantage of providing electrical properties of the nuclear envelope and nucleoplasm. In spite of this advantage, no application of the double-shell model has been reported except for in cultured murine lymphoblast (L5178Y) [7]. Erythrocytes and lymphocytes are of similar size, but their intrat.ellular structures differ; erythrocytes have no

(a)

(b)

Fig. 1. Two electrical models of a cell. (a) Single-shell model. (b) Double-shell model. The phase parameters are represented by complex permittivity defined by e* = e - j~/torv: r, relative permittivity; K, conductivity, to = 2~r/: f, frequency; ~v, permittivity of vacuum; j, imaginary unity. The subscripts refer to the following phases: m, plasma membrane; i, inner phase of cell; cp, cytoplasm; ne, nuclear envelope; np, nucleoplasm. The morphological parameters are: R, outer radius of cell; R n, o u t e r radius of nucleus; d, thickness of plasma membrane; d n, thickness of nuclear envelope.

0167-4889/89/$03.50 © 1989 Elsevier Science Publishers B.V. (Biomedical Division)

50 subcellular organelle, whereas lymphocytes have a sizable nucleus occupying 40-70% of the total cell volume. The morphological difference enables us to examine the effect of i~acleus on the dielectric properties of cells. In addition, the morphology of lymphocytes is appropriate for testing the double-shell model. In this study, we measure the dielectric properties of erythrocytes and lymphocytes and analyze them using the single- and double-shell models. M a t e r i a l s and M e t h o d s

Cell preparation. Erythrocytes were obtained from mouse blood by removing the buffy coat. After washing three times with phosphate-buffered saline (PBS), spherical erythrocytes (spherocytes) were prepared by swelling in 675 PBS, prior to dielectric measurements. Spherocytes can eliminate morphological complication in theoretical analysis. The mean radius of spherocytes was 2.5 # m . Lymphocytes isolated from mouse spleen were purified by a 'two-step density gradient'. The spleen cell suspension in PBS was layered over a sodium diatrizoate-Ficoll solution (the density was 1.077-1.080 g/ml at 20 "C) obtained from Organon Teknika Co. and was centrifuged at 3000 x g for 15 rain. The cells accumulated on the interface were collected and suspended in PBS. The cells were almost spherical and the size distribution was narrow. The mean radius was 2.9 /~m. The nucleus was easily visualized under a phasecontrast microscope, but accurate determination of i.~s size was difficult because of the high nucleocytoplasnfic ratio. A rough estimate of the nuclear radius was 2.3 /tm. Dielectric measurements. Capacitance and conductance of cell suspensions were measured with HP Impedance Analyzers (model 4191A and 4192A) between 0.01 and 250 MHz. The measuring cell used was a type of parallel capacitor, of which the cell constant and sample space were 0.03 pF and 100/tl, respectively. The data measured were corrected for residual inductance and stray capacitance arising from the measuring cell and its lead wire according to the method of Asami et al. [8], and then the permittivity and conductivity were calculat~ ~rom the corrected capacitance and conductance with the cell constant. The correction was, however, somewhat insufficient at higher frequencies. Judging from the corrected data of a salt solution (PBS) of which permittivity and conductivity curves are expected to be independent of frequency up to at least a few hundreds MHz, there was an unexpected slight increase in conductivity at frequencies higher than 30 MHz. Fortunately, the uncorrected increment of conductivity, As, was almost independent of sample conductivities in a limited range. Hence: we supplemented an empirical

correction by subtracting A~¢ of the suspending medium from the conductivity of the suspension. Calculation of permittivity and conductivity of cell. The permittivity and conductivity of the cell itself (ec and go) was calculated from those of the cell suspension (es and ss) according to Hanai's mixture equation [9] expressed in complex permittivities (defined as e * - e j~/CO~v: s, relative permittivity; ~¢, conductivity; to= 2~rf; f, frequency; s v, permittivity of vacuum, and j, imaginary unit), ~. _ ~. ( ~. ~1/3

~a* ~*~,~'~'1 --1-P

(1)

where P is volume fraction and ea and ga are permittivity and conductivity of the external phase, respectively. The values of e~ and sa are known from the measurements of the supernatant separated from the cell suspension by centrifugation. The value of P is calcualted from P ffi 1 -- ( g l / I r a )2/3

(2)

where ~q is the limiting conductivity of the suspension at low frequency. This equation is derived from Eqn. 1 on the assumption that the conductivity of the plasma membrane is negligibly small compared with that of the external phase. Curve-fitting procedure. Theories were fitted to observed data using a computer, according to the following procedure which is similar to that described by Irimajiri et al. [5-7]. The frequency dependence of calculated permittivity and conductivity was compared with measurements on a CRT display and the theoretical parameters were searched manually in order to get the best-fit curve. Details of the procedure is discussed by Irimajiri [5-7]. After a rough curve fit was attained, we searched the best-fit parameters more precisely so as to minimize the residual, R(e, g), between theoretical values (eti and ra) and the observed (eoi and Koi). R(e, g) is defined by: R(e, ~) - {~:(e,i - eoi)21~e2i + ~ ( ~ , , - ~oi) 2IE~oi2 } -2

(3)

Results

Comparison of dielectric dispersion curves between lymphocyte and erythrocyte Fig. 2 illustrates the frequency dependence of the relative permittivity, ec, and conductivity, •c, of lymphocytes and erythrocytes which was calculated from dielectric data of their suspensions using Hanai's mixture equation. Both cells showed dielectric dispersion phenomena between 0.1 and 100 MHz, but their dispersion curves were quite different from each other. The

51 4

i

'

,

,

12

TABLE I

I

Dielectric parameters of erythrocytes and lymphocytes determined by fitting Eqns. 4 or 5 to the dielectric dispersion curves shown in Fig. 3 "i

3

8E O9

._o 2

-4

15

6

7, Iog(f/Hz)

according to Hanai's mixture theory. The solid lines indicate the

curves calculated from Eqns. 4 and 5 with the parameters listed in Table I.

dispersion curve of lymphocytes (curve 1) was much broader than that of erythrocytes (curve 2). The difference is more clearly demonstrated in the complex plane plots of permittivity and conductivity (Fig. 3). The complex permittivity plots of erythrocytes traced a semicircle with a slightly depressed center, which is represented by the Cole-Cole empirical equation as: =

AE

%+

(4)

1 + (jf/fc) 1-"

where rh is the limiting permittivity at high frequencies, 1.5

I

='

I

l

I

'I

(o)

7:) 1.0 I.U

'~.5

I

I

2

I

I

(b)

-

O9

E

j

z

4

6

O~1

Erythrocyte

55

1950

5.30

0.02

Lymphocyte

55

2 300

1.45

0.08

Ae2 fc2 (MHz)

Or2

580

0.08

14.2

B

~e is the dielectric increment, a is the Cole-Cole parameter, fc is the characteristic frequency and f is the frequency. The best-fit parameters obtained by fitting Eqn. 4 to the data on erythrocytes are listed in Table I. The a-parameter is close to zero, indicating roughly that the dielectric dispersion has a single relaxation time. On the other hand, Eqn. 4 failed to characterize the dispersion curve of lvmphocytes, therefore we attempted to apply an equation which includes two ColeCole dispersion terms ec*= % +

Ael

+

1 + ( j f / f c l ) i - a,

Ae2

(5)

1 + ( Jf/fc2 )] - a2

where subscriots 1 and 2 refer to two separate dispersions at lower and higher frequencies, respectively. The dispersion curve of lymphocytes was well simulated using Eqn. 5 with the best-fit parameters listed in Table I. Eqn. 5 was found to account for the dispersion curves of several different kinds of cell (unpublished data). However, As 2 is, f:n" these cells, small compared to Ae!. Lymphocytes are ran unusual case in which ae2 is very large owing to the }ggh nucleocytoplasmic ratio.

2(1 - v)+(1 +2v)E e~=,e* 2 + v + ( 1 - v ) E

I

7 4 E

%

fcl (MHz)

Analysis based on ~ingle-shell model According to the single-shell model (Fig. la), in which an homogec~eous sphere (of the complex permittivity e*) is covered with a shell (e*), the complex permittivity of the shelled sphere, e*, is given by

@@

O(

AEI

90

8,

Fig. 2. Frequencydependence of relative permittivity r c and conductivity ~:c of lymphocytes(curve 1) and erythrocytes(curve 2). ec and ~c were calculated from the dielectric data of the cell suspensions

ec

Eh

io

Kc/rnScm -I Fig. 3. Complex permittivity (a) and conductivity (b) plane plots of the data of Fig. 2. (a) Ordinate (imaginary part): Ae~' = ( s ¢ Scl)/we,, where Kcl is the low frequency limit of sc (Kel = 0). Abscissa (real part): ec = rc. (b) Ordinate: ilK" --- (ec ,- rch)~0Ev, where r ech is the high-frequency limit of ec- Abscissa: ~ = ~¢.

(6)

where E = e i*/ e m, * v = (1 - d / R ) 3, R is outer radius and d is shell thickness. This equation allows us to calculate the frequency dependence of permittivity and conductivity of a cell from its phase parameters. Since spherical pre-swollen erythrocytes were used for measurements, Eqn. 6 is precisely applicable to the present data. Fig. 4 shows the best-fit curve for the erythrocyte dispersion calculated from Eqn. 6 with the phase parameters shown in Table II. The calculations were in excellent agreement with the measurements, suggesting that the single-shell model is appropriate to spherical erythrocytes. Similar results have already been

52 TABLE II

I

I

t2

I

Estimated dielectric phase parameters of erythrocytes and iymphocytes based on the single-shell model

T

Membrane capacitances are calculated from Cm = emev/d, where rv is permittivity of vacuum (e~=8.8541.10 -14 F ' c m -1) and d is membrane thickness. Assumed values: d = 7 nm and i¢m < 10-5.gi. Temperature, 24 ° C.

R

~m Cm

#m

Erythrocyte Lymphocyte curve I curve 2

2.5 2.9

4

~i

~i

/~F.cm -2 5.7 6.8 6.8

0.72

I

0.68

2.9

0.19

2.6

0,17

,

-8~

2

-4

5

6

7 log (f/Hz)

E

\e •e ee~...~__ eeoc o

-4

8

15

t

6

7 10g (f/Hz)

I

9

8

0

Fig. 5. Attempts to fit the single-shell model to the dielectric dispersion of lymphocytes. Solid curves were calculated from Eqn. 6 with the parameters listed in Table II.

12

3

I

03

~ , , QO 000 2 -

~i/~.,

6.2

810 49

I

o

J

-8~

mS'cm -t 59

0,86 0.86

\ _oe oe .eeoee -0%000

3

9

entire dispersion (curve 2). Thus, the single-shell model failed completely to simulate the dielectric dispersion of lymphocytes. This is because the dispersion has an additional subdispersion, due to a large nucleus at higher frequencies. Hence, we attempted to apply the doubleshell model to the lymphocyte dispersion. The complex permittivity of the whole cell (e*) is represented as a function of the phase parameters (e*, ecp, * e.~ * and enp *) defined as in Fig. 1.

T

D

Fig, 4. The best-fit curves for the erythrocyte dispersion calculated from Eqn. 6 based on the single-shell model. The phase parameters listed in Table II are used in this calculation,

obtained with pre-swollen human erythrocytes by Irimajiri et al. (personal communication).

Analysis based on double-shell model As shown in Fig, 5, experimental data obtained with lymphocyte could not be fitted to Eqn. 6 which predicts a dispersion with a single relaxation time. This figure shows two pairs of theoretical curves: one was obtained by fitting Eqn, 6 to the low frequency subdispersion only (curve 1) and the other by fitting Eqn. 6 to the

r*ffie*

2 ( 1 - v l ) + ( l + 2vl)E I (2+ vl)+tl_--~E!

(7)

where v t - (1 - d/R) 3. The intermediate parameter, E~, is given by El =ec*p 2 ( I - v2)+(l + 2v2)E 2 e--~" m ' (2~v2)+( I- v2)E 2

where v2 = ( R , / ( R

-

d ) ) 3.

(8)

Finally, E 2 is given by

E*ne 2 ( 1 - v 3 ) + ( I +2v3)E 3

(9)

E2 -- e"~,*p" ( 2 + v 3 ) + ( 1 - v3)E 3

where o3 "- (1 - d,L/Rn) 3 and E 3 -" Enp//Ene. * * Desp~*.e the .... complexity, Eqn. 7 can predict frequency dependence of

TABLE 111

Estimated dielectric pahse parameters of lymphocytes based on the double-shell model ~:m, 6.8; Cm, 0.86 t~F.cm-2; R, 2.9 #m. Assumed values: d, 7 nm; d , , 40 nnl; x m <10-s.Xcp, rcp; 60. The capacitance and conductance of the nuclear envelope are calculated from C,e -- ene. e v / d n and Gne -- xne/dn, respectively. Temperature, 24 ° C. Rn (Itm)

cue

Cne ttF. c m - 2

gne pS. cm - t

Gne S. cm - 2

gcp mS. cm - ]

enp

ICnp mS. cm - 1

Res/10 2

2.2 2,3 2,4 2,5 2.6 2,7

32,0 28.5 28.0 28.0 25.5 24.0

0.71 0,63 0.62 0.62 0.56 0.53

45 50 55 60 65 70

11.3 12.5 13.8 15.0 16.3 17.5

3.6 3.6 3.5 3.2 3.0 2.7

32 38 45 52 55 60

22.0 18.0 15.3 13.5 12.0 11.0

6.5 4.4 3.4 2.9 3.4 4.1

53 4

12

Discussion

/

-I

"

Effect of the nucleus on dielectric properties of cells I

3-

- 8 E L)

2

4

o

I

5

6

7

8

9

0

log (f/Hz) Fig. 6. The best-fit curves for the lymphocyte dispersion calculated from Eqn. 7 based on the double-shell model. The best-fit parameters with R n = 2.5 ~m listed in Table III were used in this calculation.

the permittivity and conductivity of the nucleated cells with the aid of a computer. Following the curve-fitting procedure described by Irimajiri et al. [6,7], we can determine the phase parameters of the double-shell model from the observed dispersion curve if the morphological cell parameters are available. For lymphocytes, however, there was some uncertainty in the determination of the nuclear radius, R n, so that we carried out the curve fitting by varying R , between 2.2 and 2.7 #m. The results are listed in Table III. Judging from the residual R(e, K), the best fit was obtained with R , - 2.5 #m, which is not far from the microscopically determined value ( R , = 2.3 #m). Agreement between theory and measurement was excellent over all the dispersion as shown in Figs. 6 and 7.

Comparison of dielectric behavior between erythrocytes and lymphocytes clearly demonstrates that the nucleus influences the dielectric property of the whole cell at high frequencies. Lymphocytes, which have an unusually high nucleocytoplasmic ratio, showed a wide spread dispersion which was composed of two subdispersions. The high frequency subdispersion, which was not found for erythrocytes, was assigned to the Maxwell-Wagner dispersion due to the nucleus. The effect of cytoplasmic organdies, such as mitochondria, may be ruled out for lymphocytes, because the cytoplasm contains relatively few organelles and their volume fraction is negligibly small.

Ervthrocyte (spherocyte) simulated by the single-shell model Pre-swollen erythrocytes showed a single dielectric dispersion which was explained by the single-shell model. A better agreement between observed and theoretical curves was obtained using the mixture equation of Hanai in the calculation than when using that of Wagner. The estimated phase parameters are almost consistent with those reported so far. The membrane capacitance was 0.72 # F . cm -2, similar to that of human erythrocyte (0.68 # F . cm -2) [18]. The value was only about 10% lower than that reported by Fricke [4]. The cytoplasmic electrical properties (e~ = 59 and x~ = 6.2 mS- cm -~) were in good agreement with those reported by Pauly and Schwan [10].

Lymphocyte simulated by the double-shell model 1.5

I

I

I

I

I

(a) 1.0
O0 _

I

3

2

/

I

I

0

2

4

I

I

6

8

I0

Ke/mScm-I Fig. 7. Complex permittivity (a) and conductivity (b) plane plots of the data of Fig. 6.

As discussed above in the Result section, the double-shell model gave a much better simulation for the lymphocyte dispersion than the single-shell model. The phase parameters relating to the nucleus are more valuable than the others because they cannot be obtained by any other currently available techniques. Although the microelectrode technique is an important tool for evaluating such parameters, its application is still limited to relatively large cells, such as saliverly gland cells and eggs [11]. In the double-shell model, the nuclear envelope, which is composed of double membranes, is represented by an homogeneous shell (capacitance, C,~, and conductance, G,e). The validity of this simplification, therefore, should be tested before we discuss the meanings of the phase parameters estimated from the model. A model faithful to the morphology of the nucleus is a double-shelled sphere (Fig. 8b) so that we can compare this model with the single-shelled sphere (Fig. 8a) numerically. Calculation of the permittivity (%) and conductivity (Kn) of the two nuclear models was carried

54 out in the same manner as that of the cell models (Fig. 1). The phase parameters of the double-shelled sphere were assumed as follows: the two shells have the same capacitance, C~m,and conductance, Gn., which are equal to 2C.e and 2G.e, respectively, and the permittivity and conductivity of the intermembrane space are the same as those of the nucleoplasm (Enp and grip)" The best-fit parameters of the nucleus shown in Table III were used in this calculation, that is, C.e ffi 0.62 p F - c m -2, Gn~ ffi 15 S. cm -2, e . p - 52 and ~Cnp- 13.5 mS. cm -1. Fig. 9 shows the numerical results obtained by varying the value of G.¢ as: 15 (the best-fit value), 2.5 and 0 S. cm -2. With Gne > 2.5 S' cm -2, there was no difference between the two models and only one dispersion was found, whereas another dispersion appeared for the double-shelled sphere at lower frequencies if G~ < 2.5 S.cm -2. Hence, it is concluded that the nuclear envelope (of G,.=15 S . c m -2) can be expressed by a single shell. The value of Cen (ffi 0.62/~F. cm -2) agreed closely with that reported by Irimajiri [7]. If the two nuclear membranes have the same electrical parameters, the capacitance of each membrane is 1.24 p F - cm -2, which is a reasonable value for biological membranes. The conductance of the nuclear envelop (G~) was 15 S. cm -2. This high conductance value is due to the nuclear pores existing in the envelope. If there are Np cylindrical pores (of diameter Dp) per unit area and the pore phase has a homogeneous conductivity (gpo~), the conductivity of the nuclear envelope ( s . . ) is given by the following equation on the assumption that the conductivity of the membrane phase is negligibly small compared with that cf the pore phase. s.e " Spo,~"Np.,n'(Dp/2) =

(10)

If Np and Dp are given, we can calculate ~por~ from g.,. The inner diameter of the pore (/~p) determined by electron microscopy, is about 80 ~m [12]. The pore density value (Np) of 3-4 pores//~m~ obtained for hu-

(o)

(b)

Fig. 8. Twoelectricalmodelsof the nucleus.(a) Single-shelledsphere. Co) Double-shelledsphere. The subscriptsrefer to differentphases as follows: ne, nuclear envelope; np, nucleoplasm; nm, nuclear membrane; hn, intermembranespace. Morphological parameters: R., outer radius of nucleus; d., thickness of nuclear envelope; dnm , thicknessof nuclearmembrane.

I

-

~2

3/~\\

I

"

_

I

I

\

I I

0 15

I

, I

I

I

~I0 0"}

E

2

6 7 8 9 Iog(f/Hz) Fig. 9. Frequencydependence of the permittivityand conductivity calculated, based on the two nuclear models.The solid lines indicate the calculations for the single-shelledsphere and the broken lines those for the double-shelledsphere. The numbe,'sbeside the curves refer to calculationswith different values of Ge.: 1, 2 and 3 refer to 15, 2.5 and 0 S.cm-2, respectively.

man lymphocytes [13] may be applicable to mouse lymphocytes. Hence, Kpore is estimated to be 3-4 mS. cm -! from ~.e (ffi 0.06 mS. cm-l). Ttds value is quite reasonable if the pore is fill:A with the cytoplasmic fluid and a part of the pore is occupied by large protein granules (the nuclear pore complex). The condu~tivities of the cytoplasm (~cp) and nucleoplasm (~.p) were 3.2 and 13.5 mS. cm -1, respectively. That the value of ~np is higher than that of gcp is roughly explainable by the following facts. The nucleoplasm, in general, has a higher water content than the cytoplasm (for example, the nuclei and cytoplasm of dmphibian oocytes have 74-85% and 35-50% water, respectively [14]) and several ions, such as K +, are more concentrated in the nucleoplasm than in the cytoplasm [14,151. The value of enp is determined from the permittivity of the cell at frequencies higher than 100 MHz. In such a frequency range, protein and DNA solutions show permittivity lower than water (~ ffi 80) [16,17] so that the present value of e,p ( = 52) seems to be reasonable. Acknowled,~ments This work was supported by grants from the Olympus Optical Company and from the Yamada Science Foundation.

55

References 1 Schwan, H.P. (1957) in Advances in Medical and Biological Physics Vol. 5 (Lawrence, S.H. and Tobias, C.A., eds.) Academic Press, New York. 2 Cole, K.S. (1968) in Membranes, Ions and Impulses. University of California Press, Berkeley. 3 Schanne, O.F. and P.-Ceretti, E.R. (1978) in Impedance Measuremerits in Biological Cells. John Wiley & Sons, New York. 4 Frick¢, H. (1924) Phys. Rev. 26, 682-687. 5 Irimajiri, A., Asami, K., lchinowatad, 1"., KJnoshita, Y. (1987) Bioehim. Biophys. Acta 896, 203-213. 6 Irimajiri, A., Hanai, T. and Inouye, A. (1979) -L Theor. Biol. 78, 251-269. 7 Irimajiri, A., Doida, Y., Hanai, T. and Inouye,, A. (1978) J. Membr. Biol. 38, 209-232. 8 Asami, K., Irimajiri, A., Hanai, T., Shiraishi, N. and Utumi, K. (1984) Bioehim. Biophys. Acta 778, 559-569. 9 Hanai, T. (1960) Kolloid-Z. 171, 23-31. 10 Pauly, H. and $chwan, H.P. (1966) Biophys. J. 6, 621--638.

11 Loewenstein, W.R., Kanno, Y. and Ito, S. (1966) Ann. N.Y. Acad. Sci. 137, 708-716. 12 Wunderlich, F., Berezney, R. and Kleinig, H. (1976) in Biological Membranes. Vol. 3 (Chapman, D. and Wallach, D.F.H., eds.), Academic Press, New York. 13 Maul, G.G., Maul, H.M., Scogna, J.E., Lieberman, M.W., Stein, G.S., Hsu, B.Y. and Borun, T.W. (1972) J. Cell Biol. 55, 433-447. 14 Century, T.J., Fenickel, I.R. and Horowitz, S.B. (1970) J. Cell Sci. 7, 5-13. 15 Langendorf, H., Siebert, G., Kesselberg, K. and Hannover, R. (1~66) Nature 209, 1130-1131. 16 Oncley, J.L. (1943) in Proteins, Amino Acids and Peptides as Ions and Dipolar Ions (Cohn, E.J. and Edsall, J.T., eds.), ReinhoM Publishing Corporation, New York. 17 Gabriel, C., Grant, E.H. and Young, I.R. (1986) J. Phys. E. 19, 843-846. 18 Takasifima, S., Asami, K. and Takahashi, Y. (1988) Biophys. J., in press.