- Email: [email protected]
Properties of lipid bilayer membranes membrane thickness
J. theor. Biol. (1970) 26, 277-287
Properties of Lipid Bilayer Membranes Membrane Thickness SHINPEI O H K I
Department of Biophysics, School of Pharmacy and Center for Theoretical Biology State University of New York at Buffalo, Buffalo, New York 14226, U.S.A. (Received 10 March 1969, and in revised form 22 July 1969) With two independent experimental measurements (capacitance and optical measurements) on the same lipid bilayer and the knowledge of an asymmetric dielectric constant calculated from a model of the bilayer, the thickness, the average area per hydrocarbon chain and the dielectric constant of the bilayer, which are consistent with the data of two independent experiments, were obtained. By using various data of capacitance and optical reflectance on lecithin bilayers, the thickness (54 ~ 48 /~), area per hydrocarbon chain (23.5 ~ 25.0 /~2) and dielectric constant (ell = 2.32 ~ 2.29, e.t = 2.59 ~ 2"25) for the lecithin (egg) bilayer were deduced. 1. Introduction Determination of the thickness of artificial lipid bilayers has been made by capacitance or optical measurements (Hanai, Haydon & Taylor, 1964; Tien, 1968; Huang & Thompson, 1965, and personal communication; Tien & Dawidowicz, 1966; Cherry & Chapman, personal communication; and others), and the thickness has been calculated using the bulk values for dielectric constant or the refractive index determined by the Brewster angle measurements. Hanai et al. (1964) evaluate the thickness (48 A) of a lecithin bilayer from capacitance measurements (0.38 IxF/cm) assuming the parallel plate condensor formula and using for the dielectric constant the value (2.07) for bulk hydrocarbon mixture. Tien (1968) also has measured for the capacitance of the lecithin bilayer in 0.1 i NaCI solution. He obtained the capacitance value of 0.45 laF/cm. From optical measurements, Huang & Thompson (1965, 1966) found the thickness of a lecithin bilayer in 0.1 M NaCI to be 72 4-10 A. Tien & Dawidowiez (1966) obtained the thickness for glycerol distearate bilayer of T.B,
277
18
278
s. OHKI
50 ,~ by the similar method. However, this method depends greatly upon determination of the refractive index of the bilayer from the Brewster angle. The refractive index obtained by Huang & Thompson (1965) for a lecithin bilayer was 1.66-I-0.003. Tien & Dawidowicz (1966) obtained the slightly lower value of 1.60 _ 0.03. Since the bilayer may be considered as an oriented hydrocarbon molecular assembly, the dielectric constant and refractive index must be anisotropic. In the previous paper, a method (Ohki, 1969) to obtain correct values for the thickness, the area per hydrocarbon, the dielectric constant and the refractive index of the lipid bilayer was proposed, using the values for the dielectric constant and refractive index calculated, and two independent experimental results for capacitance and optical reflectance measurements simultaneously. In this paper, the correct range of the thickness, average area per hydrocarbon dielectric constant and refractive index of the lecithin bilayer are deduced by using the data of the capacitance and optical measurements which were made by different workers. 2. Determination of Thickness and Area per Hydrocarbon For the model of a lipid bilayer, I have calculated the dielectric constant and the refractive index in terms of the membrane thickness h and area per hydrocarbon chain a 2 (Ohki, 1968). According to this consideration, one is able to obtain capacitance or reflective intensity of light in terms of thickness h and area per hydrocarbon a z. If the capacitance is measured applying an electric field perpendicular to the bilayer, only the parallel (to the chain) component 811 of the dielectric constant of the bilayer pertains to the capacitance measured. On the other hand, in the optical measurements, reflective intensity of light is related to both refractive indexes, that is, parallel (to the chain) component nil and perpendicular component nj. of refractive index. Especially, if the incident ray is normal to the layer, only the perpendicular component n. of refractive index pertains to the reflectance. In optical measurements, all measurements of reflectivity were made at small angle of incidence. Thus, these two experiments are independent of each other. We have the capacitance of a lipid bilayer as a function F(c, h, a 2) of thickness and area per chain from theoretical calculation. Thus, if we know the capacitance of the bilayer, we can obtain one relation of the thickness and area per hydrocarbon chain. We also have the reflectivity R as a function G(R, h, a z) of thickness and area per hydrocarbon chain from theoretical calculation. If we know the reflectivity of the bilayer from experiment, wecan obtain another relation of the thickness and area per hydrocarbon chain. If the materials used in two different experiments are the same, the correct thickness h and
PROPERTIES O F
LIPID
BILAYER
MEMBRANES
279
area per h y d r o c a r b o n chain a 2 should be the values which satisfy both relations simultaneously. I f we use the f o r m u l a of a parallel plate capacitor for the capacitance o f a bilayer, we have c = &_l 4nh
(1)
where c is the capacitance, ell the parallel c o m p o n e n t of the dielectric constant, and h is the thickness o f h y d r o c a r b o n portion. 811 is given (Ohki, 1968) 1
811 =
4~Pall
(2)
1 1 + TII~II where Tll = A ' - ( B ' / h ) a n d p is the n u m b e r density o f C 2 H 4 units. A ' a n d B ' are the n u m e r i c a l c o n s t a n t s which d e p e n d on the a r e a p e r h y d r o c a r b o n chain. T h e n u m e r i c a l values o f A ' a n d B ' with respect to a r e a p e r h y d r o c a r b o n chain a 2 are s h o w n in T a b l e 1. TABLE 1
b = 2.514 A all = 5.72 A 3 [for C2H 4 unit, Ohki & F u k u d a (1967)] 1
ell = 1
47tPCell 1 + a l l TII
B' T u = A' - - h a 2 (A =)
p c = . , ( x lO=2[cma)
A" CA-a)
B" (h-~)
20 21 22 23 24 25 26
1"9888 1-8941 1"8080 1"7294 1-6572 1-5910 1.5298
0"2454 0"2353 0"2253 0"2152 0"2055 0"1951 0"1869
0"8954 0"8689 0"8426 0'8162 0.7900 0"7639 0-7310
a ~ (M)
e~
e, (h = 60 A)
20 21 22 23 24 25 26
4.0277 3.3976 2.9776 2.6765 2.4512 2.2761 2-1316
2.6079 2"5092 2"4297 2"3651 2"3128 2-2708
( h = 5 0 h) 2.6394 2.5374 2.3889 2.3889 2.3350 2.2915
280
s.
oaxl
By solving equations (1) and (2) with respect to h, we obtain 1 +A'~II +4xCB'~II + h= +~/(1 + A'~II +4nCB'~II)2-16nC(1 +cqlA'-4nP~ll)B'~ll. (3) 8rrC(1 + A'= II- 4npc~II) The relationship between thickness and area per hydrocarbon chain is shown in Fig. 1 and Table 2 for capacitance 0-38 lff/cm (Hanai et aL, 1964) and for 0.45 laF/cm (Tien, 1968), respectively. Different measured capacitances produce a family of similar curves. i
i
|
!
I
i
C
I00
O
9O 8O 7O 60
E
3O 2O I0
~'o 2', 2~ 2; 2'4 2; 2~ 02 (~,z) FIo. 1. The relation between the thickness of lecithin bilayer and the average area per hydrocarbon. The values (h, a2) on the solid line are to satisfy each experimental data. A, Hanai, Haydon & Taylor, 1964; B, Tien, 1968; C, Huang & Thompson, 1965; D, Tien & Dawidowicz, 1966; E, Cherry & Chapman, personal communication. The optical properties of lipid film in aqueous solution are fundamentally the same as those of thin transparent solid film. The interface phenomenon produced when monochromatic light is reflected from the two faces of a thin dielectric plate may be described as follows (Heavens, 1955; Landau & Lifshitz, 1960). The geometrical figure of the system is as follows: h is the thickness, no is the refractive index of the medium (aqueous solution), nil and n± are the parallel (perpendicular to the layer) and perpendicular
PROPERTIES
OF
LIPID
BILAYER
MEMBRANES
281
TABLE 2
Thickness of the bilayer using capacitance data h =
1 +A'~II + 4~CB'~II + + x/(1 + A'Cell+ 4nCB'Ctll) 2 - 1 6 n C ( 1 + A'ct ii - 4/tp%)B'ct ii 8nC(1 + A'all - 4nPall ) a2 (]k2)
C = 0"38 liF (Hanai et at., 1964) h (h)
C = 0"45 lit: (Tien, 1968) h (A)
20 21 22 23 24 25 26
60-09 58.17 56.34 54.97 53.93 52-97 51.94
51.82 49.93 48.39 47.19 46.15 45.33 44.10
(parallel to the layer) components of dielectric constant of the bilayer, 0 o is the angle of the incident ray to the layer, and 01 is the angle of the reflected ray in the layer. The two reflected rays, A and B, have undergone a relative phase shift which results in the observed interference phenomenon. The phase shift
6'=26=2
(;)+ 2nnbh . . . os01 .
+
where 2 is the wave length of the light used and n b is the refractive index of the bilayer. The amplitude E of the reflected ray gives Yl +Y2 exp ( - 2 i 6 ) E = ] + ~ - 2 - e - - ~ p - ( S ~ ) ~0
(4)
where Eo is the amplitude of the incident ray, ?1 and ?2 are the reflective coefficients at the front surface and the back surface: __ n°--n~ and Y 2 - - -H1 - - 170 ?x no+n 1 hi+no If we consider the lipid bilayer as being composed of a parallel oriented hydrocarbon assembly, the optical properties of a bilayer correspond to the case of a plane parallel plate of a uniaxial crystal cut with surfaces perpendicular to the optical axis. In our lipid bilayer model, the optical axis coincides with the molecular axes. We shall take it as the Z-axis in cartesian co-ordinates, denoting the corresponding principal values of dielectric constant 8 by 811. The directions of the other two principal axes, in a plane
282
s. OHKI
perpendicular to the optical axis, are arbitrary, and the corresponding principal values, which we denote e±, are equal. The reflectivity R, defined as the ratio of the reflected energy 1 ( = E 2 = E 2 + E 2) to the incidence energy Io = Eo2, is:
R= I E~2 + E v2 Io E~ yls+271sY2s COS 26 +7z2~ 2 2 EsZo 1 +2?lsY2s COS 26 +YlsY2~ E 2 = ylp+2ylv'~2v cos 26 .~_y2p ._A 2 2 E~o l + 2ylVy2V COS 26 + YlVY2V
(5) (6) (7)
where suffixes s and p refer to the perpendicular and parallel components to the plane of incidence, and y's are their reflective coefficients respectively. For the lipid bilayer in the solution, 1?11 = IT2[ ~ Y. Y is composed of components of Yll and y±. According to Blodgett & Langmuir (1937), the reflection coefficients ys and ?v of light polarized perpendicular and parallel to the plane of incidence are given by 2 sin 2 0o)~ n o c o s 0 o - ( n 2 - no Y~- no cos Oo+(n2,-n~ sin ~ 0o)~ (8) no(n~l - no2 sin 2 0o)~ - n . nil cos Oo
YP = no(n~ - n~ sin 2 0o)"~-]-n . n II cos 0o °
(9)
If the angle of incidence is small (less than 25°), ?s and yv are reduced by n & -- n 0
?~ = ~v = - = ~'-" n±Tno In this case, the reflectivity R is R~--
I Io
~
2 2 y12±+2yl.t.y2± cos 2 6 + y ~ l E±+EII E~ 1+2?11Y21 cos 26+y1±7212 z 4y 2 sin 2 6
(10)
(11)
1 - 2y2 + 4~2 sin 2 6 + ?~" For lipid films in aqueous solution (0.1 r~ NaCI) the Fresnel reflection coefficient ?± is quite small. Therefore, equation (11) reduces to
R = 4yZ sin2 6 = 472 sin2127rn" h c ° s O!].
(12)
Recently, Cherry & Chapman (personal communication) have measured absolute reflective intensity at incident angle 10°. They obtained the reflectivity of a lecithin bilayer in the solutions of various refractive index. In the medium of 0.1 M NaCI the reflectivity R was 5.90x 10 - s and refractive
PROPERTIES
OF L I P I D
BILAYER
MEMBRANES
283
index n o o f the solution was 1.334. Using the reflectivity o f the bilayer, the refractive index o f the solution and our refractive index calculated for a lipid bilayer model, we can obtain the thickness with respect to area per h y d r o c a r b o n chain. The relation between the thickness and area per molecule which satisfy Cherry's experimental result, is shown in Fig. 1. F o r thin layers showing interference fringes, the phase difference should satisfy the following relation: c5 -
2rcnb h cos 01 zt 2 -- k ~,
(13)
where k is an integer. W h e n k = 1 the film gives the m a x i m u m reflection before the transition to the black state. Therefore, equation (12) m a y reduce to R s i l v = 4ysilv 2 sin 2 cSsilv 2 (14) = 4ysilv for the film with the m a x i m u m reflection which is observed in the silvery film, where nsilv ~ n 0 ~)silv - - nsilv n - - - ~ _]_ O.
Since the silvery film is m o s t c o m p o s e d o f liquid h y d r o c a r b o n , we m a y use the refractive index 2.09 (Hanai et al., 1964) for liquid h y d r o c a r b o n mixture as the refractive index o f the silvery film. F o r the bilayer, equation (12) is Rb = @2 sin 2 t5b = @2 sin 2 tSb,
(15)
TABLE 3
Experimental data f o r capacitance and reflectivity measurements Lipid Lecithin (egg) n-decane Lecithin (egg) Lecithin (egg) + n-tetradecane in CH Cla + CHaOH Glycerol distearate in n-hexane Lecithin (egg)
Aqueous phase Various
Capacity 0-38 ~tF/cm2
0.1 N NaC1 0"45 Iff[crn2 0.1 N NaC1 0.1 N NaCI 0.1 N NaC1
Reflectivity
Ib 1, -- 0-0618 1b -~, = 0.0424
Hanai et al. (1964) Tien (1967) Huang et aL (1965)
Tien & Dawidowiez (1966) Ib Cherry & R = To = 5.90 x 10 -5 Chapman (1968)
284
s. OHKI
where suffaxes silv and b refer to the silvery state and the bilayer state. The ratio Rb[R.itv is expressed b y Rb = Y_~_~sin2 2nn~h cos 01 Rsltv
Ysil~
2
(16)
= ~_~ sin2 2nnih cos 01
Ysilv /~ for the case o f small incident angle. With the experimental results o f relative intensity measurements (in Table 3), we can obtain the thickness as a function o f area per hydrocarbon chain for each case ( H u a n g & T h o m p s o n , 1965; Tien et al., 1966). The results are s h o w n in Fig. 1 and Table 4. Relying u p o n two independent TABLE 4
Thickness of the bilayer using reflectivity data
I--b=(rb~2sin2( krj 2nnbhc°sOt n b = 1.334 (for 0-1 N NaCl) nb =
n s = ~/e "ix'"'e = x/2-707 = 1"4387 For Cherry & Chapman (personal communication)
a~ (A ~) n~ = V ~ 20 21 22 23 24 25 26
(/L =) 20 21 22 23 24 25
r~=
2"0068 1"8432 1"7254 1.6360 1"5656 1"5084 1 "4600 lib
2.0069 1-8432 1"7254 1"6360 1-5656 1"5084
nb--Ho
ha-l-no
0-2014 0"1602 0"1279 0"1016 0"0798 0"0614 0"0450 rb
cos0~ 0"9933 0"9920 0.9909 0"9899 0"9890 0"9881 0"9871
rib--no n+--no nbq-no rs=nsq-no 0"2014 0.1602 0"1279 0-1016 0-0798 0"0614
0"0377 0"0377 0-0377 0"0377 0.0377 0"0377
9"63 13"20 17"68 23-50 31-30 42"26 59"65 00=20 ° cos0z 0"9743 0"9695 0"9650 0"9611 0"9574 0"9540
h (for Huang & Thompson, 1965) 21-38 29"40 39-54 52"60 70-11 94"57
h (for Tien, 1966) 17.12 23"56 31"67 42"22 56"39 76"32
PROPERTIES
OF L I P I D
BILAYER
285
MEMBRANES
experimental data in Table 3, we can deduce the bilayer thickness and area per hydrocarbon chain from the intersection of two curves in Fig. 1. 3. R e s u l t s and D i s c u s s i o n
F r o m Fig. I, using the experimental results by H u a n g & T h o m p s o n (1965) and Hanai et aL (1964), we obtain the thickness h = 54 A and area per hydrocarbon chain 23.1 A 2. We can calculate the corresponding dielectric constant and the refractive index from these values. Using another pair of experimental data (Huang & Thompson, 1965; Tien & Dawidowicz, 1966), we can obtain another set of values for h, a 2, ~11,~±, nu and n±. These results are shown in Table 5. In capacitance measurements, only a parallel component of dielectric constant for lipid bilayers pertains to the capacitance of the membrane. It was shown by Hanai et al. (1965) that the capacitance of lecithin films TABLE 5
Thickness, area per hydrocarbon chain, dielectric constant and refractive index o f the lipid bilayer
h aa ~l e~ ntl n±
Ref. 1 and 2
Ref. 1 and 3
Ref. 1 and 4
Ref. 2 and 5
Ref. 3 and 5
Ref. 4 and 5
54.5 23.1 2"34 2'65 1"53 1"62
54 23.8 2.32 2"59 1.52 1"60
52 25.6 2"23 2"20 1"49 1"48
47.5 22.6 2"41 2'79 1"55 1"67
46.5 23.3 2.36 2"60 1"53 1"61
45 25.2 2"29 2"25 1.51 1'50
References: x Hanai et al., 1964, 1965; 2 Huang & Thompson, 1965; 3 Tien & Dawidowicz, 1966; ~ Cherry & Chapman, personal communication; 5 Tien, 1967, 1968. was independent of frequency, area and electrolyte concentration. I f the polar groups were holding the charge, the capacitance should be influenced by the amount of electrolyte in solution. Since this is not the case, it is deduced that only the hydrocarbon portion would be involved in the membrane capacitance. We can neglect the effect of polar groups in calculating the capacitance of the bilayer. On the other hand, in optical measurements, the thickness of the film (by H u a n g & Thompson, 1965; Tien, 1967) greatly depends on determining the refractive index of the membrane from the Brewster angle. The refractive index obtained by H u a n g & T h o m p s o n for a black film of egg lecithin was 1.66 _ 0.83, and Tien obtained the slightly
286
s.
OHKX
lower value of 1"60 4-0"03. The measurements of refractive index by the Brewster angle methods is quite uncertain. If we discuss the thickness of the film using only reflectivity of the film, we may deduce the results with more certainty. However, in this measurement, one must bear in mind that the effect of polar groups to the reflectivity of the film should not be disregarded. This is different from the case of capacitance measurements. The phospholipid which Huang & Thompson (1965) and Cherry & Chapman (personal communication) used was egg lecithin which has a fairly large polar group. On the other hand, Tien (1966) used a glycerol distearate which has the smaller polar groups than lecithin. Judging from this point of view, Tien's data might be the most accurate one for hydrocarbon portion of the film. In theoretical calculation for dielectric constant and refractive index (Ohki, 1968), several assumptions were made: the rigidity of hydrocarbon chain for simplicity of calculation; neglect±on of the effect of polar groups. But it is calculated that even if the hydrocarbon chain is flexible, the numerical values calculated are not so different from the case of the rigid chain. As a result, we may conclude that the dimension of the lecithin bilayer to be in the following range: Thickness of 54 ,-, 48/~ and area per molecule 51 ,~ 47/~,. Corresponding dielectric constant and refractive index are: 811 = 2"32 ,-, 2.29, 8± = 2.59 ,,, 2-25, nil = 1.52 N 1"51, n± = 1"60 ~ 1"50. The dielectric constant and refractive index are not so much anisotropic for the bilayer as we expected for the molecular model of bilayers. In other words, although the hydrocarbon chain is more or less oriented perpendicular to the layer, as a whole, the anisotropy of dielectric constant and refractive index is small. The author would like to express his thanks to Professor J. F. Dan±ell± for his valuable advice and encouragement through this work. This work was partly supported by the National Institute of Health, I ROI NB 08739-01. REFERENCES BLODGETT,K. B. & LANG~'m, I. (1937). Phy. Rev. 51, 964. HANAI,T., HAYDON,D. A. & TAYLOR,J. (1964). Proc. R. Soc. 281A, 377. HANAI,T., HAYDON,D. A. & TAYLOR,J. (1965). J. theor. Biol. 9, 278. H~VSNS, O. S. (1955). "Optical Properties of Thin Solid Films." London: Butterworth. HUANO, C. & THO~mSO~,T. E. (1965). 3". molec. Biol. 13, 183.
PROPERTIES OF LIPID BILAYER MEMBRANES
287
LANDAU, L. D. & LtFSHrrz, E. M. (1960). "Electrodynamics of Continuous Media." Reading, Mass.: Addison-Wesley. OHKI, S. & FUKUDA,N. (1967). J. theor. Biol. 15, 362. OHKI, S. (1968). J. theor. Biol. 19, 97. OILr~, S. (1969). J. theor. Biol. 23, 158. TnorcrPSO~4,T. E. & HUAIqG,C. (1966). J. molec. Biol. 16, 576. TIEN, H. T. & DAWIDOWICZ,E. A. (1966). jr. col. Sci. 22, 438. T~N, H. T. (1967). J. phys. Chem. 71, 3395. T ~ , H. T. (1968). Chem. Phys. Liplds, 2, 55.
Properties of Lipid Bilayer Membranes Membrane Thickness SHINPEI O H K I
Department of Biophysics, School of Pharmacy and Center for Theoretical Biology State University of New York at Buffalo, Buffalo, New York 14226, U.S.A. (Received 10 March 1969, and in revised form 22 July 1969) With two independent experimental measurements (capacitance and optical measurements) on the same lipid bilayer and the knowledge of an asymmetric dielectric constant calculated from a model of the bilayer, the thickness, the average area per hydrocarbon chain and the dielectric constant of the bilayer, which are consistent with the data of two independent experiments, were obtained. By using various data of capacitance and optical reflectance on lecithin bilayers, the thickness (54 ~ 48 /~), area per hydrocarbon chain (23.5 ~ 25.0 /~2) and dielectric constant (ell = 2.32 ~ 2.29, e.t = 2.59 ~ 2"25) for the lecithin (egg) bilayer were deduced. 1. Introduction Determination of the thickness of artificial lipid bilayers has been made by capacitance or optical measurements (Hanai, Haydon & Taylor, 1964; Tien, 1968; Huang & Thompson, 1965, and personal communication; Tien & Dawidowicz, 1966; Cherry & Chapman, personal communication; and others), and the thickness has been calculated using the bulk values for dielectric constant or the refractive index determined by the Brewster angle measurements. Hanai et al. (1964) evaluate the thickness (48 A) of a lecithin bilayer from capacitance measurements (0.38 IxF/cm) assuming the parallel plate condensor formula and using for the dielectric constant the value (2.07) for bulk hydrocarbon mixture. Tien (1968) also has measured for the capacitance of the lecithin bilayer in 0.1 i NaCI solution. He obtained the capacitance value of 0.45 laF/cm. From optical measurements, Huang & Thompson (1965, 1966) found the thickness of a lecithin bilayer in 0.1 M NaCI to be 72 4-10 A. Tien & Dawidowiez (1966) obtained the thickness for glycerol distearate bilayer of T.B,
277
18
278
s. OHKI
50 ,~ by the similar method. However, this method depends greatly upon determination of the refractive index of the bilayer from the Brewster angle. The refractive index obtained by Huang & Thompson (1965) for a lecithin bilayer was 1.66-I-0.003. Tien & Dawidowicz (1966) obtained the slightly lower value of 1.60 _ 0.03. Since the bilayer may be considered as an oriented hydrocarbon molecular assembly, the dielectric constant and refractive index must be anisotropic. In the previous paper, a method (Ohki, 1969) to obtain correct values for the thickness, the area per hydrocarbon, the dielectric constant and the refractive index of the lipid bilayer was proposed, using the values for the dielectric constant and refractive index calculated, and two independent experimental results for capacitance and optical reflectance measurements simultaneously. In this paper, the correct range of the thickness, average area per hydrocarbon dielectric constant and refractive index of the lecithin bilayer are deduced by using the data of the capacitance and optical measurements which were made by different workers. 2. Determination of Thickness and Area per Hydrocarbon For the model of a lipid bilayer, I have calculated the dielectric constant and the refractive index in terms of the membrane thickness h and area per hydrocarbon chain a 2 (Ohki, 1968). According to this consideration, one is able to obtain capacitance or reflective intensity of light in terms of thickness h and area per hydrocarbon a z. If the capacitance is measured applying an electric field perpendicular to the bilayer, only the parallel (to the chain) component 811 of the dielectric constant of the bilayer pertains to the capacitance measured. On the other hand, in the optical measurements, reflective intensity of light is related to both refractive indexes, that is, parallel (to the chain) component nil and perpendicular component nj. of refractive index. Especially, if the incident ray is normal to the layer, only the perpendicular component n. of refractive index pertains to the reflectance. In optical measurements, all measurements of reflectivity were made at small angle of incidence. Thus, these two experiments are independent of each other. We have the capacitance of a lipid bilayer as a function F(c, h, a 2) of thickness and area per chain from theoretical calculation. Thus, if we know the capacitance of the bilayer, we can obtain one relation of the thickness and area per hydrocarbon chain. We also have the reflectivity R as a function G(R, h, a z) of thickness and area per hydrocarbon chain from theoretical calculation. If we know the reflectivity of the bilayer from experiment, wecan obtain another relation of the thickness and area per hydrocarbon chain. If the materials used in two different experiments are the same, the correct thickness h and
PROPERTIES O F
LIPID
BILAYER
MEMBRANES
279
area per h y d r o c a r b o n chain a 2 should be the values which satisfy both relations simultaneously. I f we use the f o r m u l a of a parallel plate capacitor for the capacitance o f a bilayer, we have c = &_l 4nh
(1)
where c is the capacitance, ell the parallel c o m p o n e n t of the dielectric constant, and h is the thickness o f h y d r o c a r b o n portion. 811 is given (Ohki, 1968) 1
811 =
4~Pall
(2)
1 1 + TII~II where Tll = A ' - ( B ' / h ) a n d p is the n u m b e r density o f C 2 H 4 units. A ' a n d B ' are the n u m e r i c a l c o n s t a n t s which d e p e n d on the a r e a p e r h y d r o c a r b o n chain. T h e n u m e r i c a l values o f A ' a n d B ' with respect to a r e a p e r h y d r o c a r b o n chain a 2 are s h o w n in T a b l e 1. TABLE 1
b = 2.514 A all = 5.72 A 3 [for C2H 4 unit, Ohki & F u k u d a (1967)] 1
ell = 1
47tPCell 1 + a l l TII
B' T u = A' - - h a 2 (A =)
p c = . , ( x lO=2[cma)
A" CA-a)
B" (h-~)
20 21 22 23 24 25 26
1"9888 1-8941 1"8080 1"7294 1-6572 1-5910 1.5298
0"2454 0"2353 0"2253 0"2152 0"2055 0"1951 0"1869
0"8954 0"8689 0"8426 0'8162 0.7900 0"7639 0-7310
a ~ (M)
e~
e, (h = 60 A)
20 21 22 23 24 25 26
4.0277 3.3976 2.9776 2.6765 2.4512 2.2761 2-1316
2.6079 2"5092 2"4297 2"3651 2"3128 2-2708
( h = 5 0 h) 2.6394 2.5374 2.3889 2.3889 2.3350 2.2915
280
s.
oaxl
By solving equations (1) and (2) with respect to h, we obtain 1 +A'~II +4xCB'~II + h= +~/(1 + A'~II +4nCB'~II)2-16nC(1 +cqlA'-4nP~ll)B'~ll. (3) 8rrC(1 + A'= II- 4npc~II) The relationship between thickness and area per hydrocarbon chain is shown in Fig. 1 and Table 2 for capacitance 0-38 lff/cm (Hanai et aL, 1964) and for 0.45 laF/cm (Tien, 1968), respectively. Different measured capacitances produce a family of similar curves. i
i
|
!
I
i
C
I00
O
9O 8O 7O 60
E
3O 2O I0
~'o 2', 2~ 2; 2'4 2; 2~ 02 (~,z) FIo. 1. The relation between the thickness of lecithin bilayer and the average area per hydrocarbon. The values (h, a2) on the solid line are to satisfy each experimental data. A, Hanai, Haydon & Taylor, 1964; B, Tien, 1968; C, Huang & Thompson, 1965; D, Tien & Dawidowicz, 1966; E, Cherry & Chapman, personal communication. The optical properties of lipid film in aqueous solution are fundamentally the same as those of thin transparent solid film. The interface phenomenon produced when monochromatic light is reflected from the two faces of a thin dielectric plate may be described as follows (Heavens, 1955; Landau & Lifshitz, 1960). The geometrical figure of the system is as follows: h is the thickness, no is the refractive index of the medium (aqueous solution), nil and n± are the parallel (perpendicular to the layer) and perpendicular
PROPERTIES
OF
LIPID
BILAYER
MEMBRANES
281
TABLE 2
Thickness of the bilayer using capacitance data h =
1 +A'~II + 4~CB'~II + + x/(1 + A'Cell+ 4nCB'Ctll) 2 - 1 6 n C ( 1 + A'ct ii - 4/tp%)B'ct ii 8nC(1 + A'all - 4nPall ) a2 (]k2)
C = 0"38 liF (Hanai et at., 1964) h (h)
C = 0"45 lit: (Tien, 1968) h (A)
20 21 22 23 24 25 26
60-09 58.17 56.34 54.97 53.93 52-97 51.94
51.82 49.93 48.39 47.19 46.15 45.33 44.10
(parallel to the layer) components of dielectric constant of the bilayer, 0 o is the angle of the incident ray to the layer, and 01 is the angle of the reflected ray in the layer. The two reflected rays, A and B, have undergone a relative phase shift which results in the observed interference phenomenon. The phase shift
6'=26=2
(;)+ 2nnbh . . . os01 .
+
where 2 is the wave length of the light used and n b is the refractive index of the bilayer. The amplitude E of the reflected ray gives Yl +Y2 exp ( - 2 i 6 ) E = ] + ~ - 2 - e - - ~ p - ( S ~ ) ~0
(4)
where Eo is the amplitude of the incident ray, ?1 and ?2 are the reflective coefficients at the front surface and the back surface: __ n°--n~ and Y 2 - - -H1 - - 170 ?x no+n 1 hi+no If we consider the lipid bilayer as being composed of a parallel oriented hydrocarbon assembly, the optical properties of a bilayer correspond to the case of a plane parallel plate of a uniaxial crystal cut with surfaces perpendicular to the optical axis. In our lipid bilayer model, the optical axis coincides with the molecular axes. We shall take it as the Z-axis in cartesian co-ordinates, denoting the corresponding principal values of dielectric constant 8 by 811. The directions of the other two principal axes, in a plane
282
s. OHKI
perpendicular to the optical axis, are arbitrary, and the corresponding principal values, which we denote e±, are equal. The reflectivity R, defined as the ratio of the reflected energy 1 ( = E 2 = E 2 + E 2) to the incidence energy Io = Eo2, is:
R= I E~2 + E v2 Io E~ yls+271sY2s COS 26 +7z2~ 2 2 EsZo 1 +2?lsY2s COS 26 +YlsY2~ E 2 = ylp+2ylv'~2v cos 26 .~_y2p ._A 2 2 E~o l + 2ylVy2V COS 26 + YlVY2V
(5) (6) (7)
where suffixes s and p refer to the perpendicular and parallel components to the plane of incidence, and y's are their reflective coefficients respectively. For the lipid bilayer in the solution, 1?11 = IT2[ ~ Y. Y is composed of components of Yll and y±. According to Blodgett & Langmuir (1937), the reflection coefficients ys and ?v of light polarized perpendicular and parallel to the plane of incidence are given by 2 sin 2 0o)~ n o c o s 0 o - ( n 2 - no Y~- no cos Oo+(n2,-n~ sin ~ 0o)~ (8) no(n~l - no2 sin 2 0o)~ - n . nil cos Oo
YP = no(n~ - n~ sin 2 0o)"~-]-n . n II cos 0o °
(9)
If the angle of incidence is small (less than 25°), ?s and yv are reduced by n & -- n 0
?~ = ~v = - = ~'-" n±Tno In this case, the reflectivity R is R~--
I Io
~
2 2 y12±+2yl.t.y2± cos 2 6 + y ~ l E±+EII E~ 1+2?11Y21 cos 26+y1±7212 z 4y 2 sin 2 6
(10)
(11)
1 - 2y2 + 4~2 sin 2 6 + ?~" For lipid films in aqueous solution (0.1 r~ NaCI) the Fresnel reflection coefficient ?± is quite small. Therefore, equation (11) reduces to
R = 4yZ sin2 6 = 472 sin2127rn" h c ° s O!].
(12)
Recently, Cherry & Chapman (personal communication) have measured absolute reflective intensity at incident angle 10°. They obtained the reflectivity of a lecithin bilayer in the solutions of various refractive index. In the medium of 0.1 M NaCI the reflectivity R was 5.90x 10 - s and refractive
PROPERTIES
OF L I P I D
BILAYER
MEMBRANES
283
index n o o f the solution was 1.334. Using the reflectivity o f the bilayer, the refractive index o f the solution and our refractive index calculated for a lipid bilayer model, we can obtain the thickness with respect to area per h y d r o c a r b o n chain. The relation between the thickness and area per molecule which satisfy Cherry's experimental result, is shown in Fig. 1. F o r thin layers showing interference fringes, the phase difference should satisfy the following relation: c5 -
2rcnb h cos 01 zt 2 -- k ~,
(13)
where k is an integer. W h e n k = 1 the film gives the m a x i m u m reflection before the transition to the black state. Therefore, equation (12) m a y reduce to R s i l v = 4ysilv 2 sin 2 cSsilv 2 (14) = 4ysilv for the film with the m a x i m u m reflection which is observed in the silvery film, where nsilv ~ n 0 ~)silv - - nsilv n - - - ~ _]_ O.
Since the silvery film is m o s t c o m p o s e d o f liquid h y d r o c a r b o n , we m a y use the refractive index 2.09 (Hanai et al., 1964) for liquid h y d r o c a r b o n mixture as the refractive index o f the silvery film. F o r the bilayer, equation (12) is Rb = @2 sin 2 t5b = @2 sin 2 tSb,
(15)
TABLE 3
Experimental data f o r capacitance and reflectivity measurements Lipid Lecithin (egg) n-decane Lecithin (egg) Lecithin (egg) + n-tetradecane in CH Cla + CHaOH Glycerol distearate in n-hexane Lecithin (egg)
Aqueous phase Various
Capacity 0-38 ~tF/cm2
0.1 N NaC1 0"45 Iff[crn2 0.1 N NaC1 0.1 N NaCI 0.1 N NaC1
Reflectivity
Ib 1, -- 0-0618 1b -~, = 0.0424
Hanai et al. (1964) Tien (1967) Huang et aL (1965)
Tien & Dawidowiez (1966) Ib Cherry & R = To = 5.90 x 10 -5 Chapman (1968)
284
s. OHKI
where suffaxes silv and b refer to the silvery state and the bilayer state. The ratio Rb[R.itv is expressed b y Rb = Y_~_~sin2 2nn~h cos 01 Rsltv
Ysil~
2
(16)
= ~_~ sin2 2nnih cos 01
Ysilv /~ for the case o f small incident angle. With the experimental results o f relative intensity measurements (in Table 3), we can obtain the thickness as a function o f area per hydrocarbon chain for each case ( H u a n g & T h o m p s o n , 1965; Tien et al., 1966). The results are s h o w n in Fig. 1 and Table 4. Relying u p o n two independent TABLE 4
Thickness of the bilayer using reflectivity data
I--b=(rb~2sin2( krj 2nnbhc°sOt n b = 1.334 (for 0-1 N NaCl) nb =
n s = ~/e "ix'"'e = x/2-707 = 1"4387 For Cherry & Chapman (personal communication)
a~ (A ~) n~ = V ~ 20 21 22 23 24 25 26
(/L =) 20 21 22 23 24 25
r~=
2"0068 1"8432 1"7254 1.6360 1"5656 1"5084 1 "4600 lib
2.0069 1-8432 1"7254 1"6360 1-5656 1"5084
nb--Ho
ha-l-no
0-2014 0"1602 0"1279 0"1016 0"0798 0"0614 0"0450 rb
cos0~ 0"9933 0"9920 0.9909 0"9899 0"9890 0"9881 0"9871
rib--no n+--no nbq-no rs=nsq-no 0"2014 0.1602 0"1279 0-1016 0-0798 0"0614
0"0377 0"0377 0-0377 0"0377 0.0377 0"0377
9"63 13"20 17"68 23-50 31-30 42"26 59"65 00=20 ° cos0z 0"9743 0"9695 0"9650 0"9611 0"9574 0"9540
h (for Huang & Thompson, 1965) 21-38 29"40 39-54 52"60 70-11 94"57
h (for Tien, 1966) 17.12 23"56 31"67 42"22 56"39 76"32
PROPERTIES
OF L I P I D
BILAYER
285
MEMBRANES
experimental data in Table 3, we can deduce the bilayer thickness and area per hydrocarbon chain from the intersection of two curves in Fig. 1. 3. R e s u l t s and D i s c u s s i o n
F r o m Fig. I, using the experimental results by H u a n g & T h o m p s o n (1965) and Hanai et aL (1964), we obtain the thickness h = 54 A and area per hydrocarbon chain 23.1 A 2. We can calculate the corresponding dielectric constant and the refractive index from these values. Using another pair of experimental data (Huang & Thompson, 1965; Tien & Dawidowicz, 1966), we can obtain another set of values for h, a 2, ~11,~±, nu and n±. These results are shown in Table 5. In capacitance measurements, only a parallel component of dielectric constant for lipid bilayers pertains to the capacitance of the membrane. It was shown by Hanai et al. (1965) that the capacitance of lecithin films TABLE 5
Thickness, area per hydrocarbon chain, dielectric constant and refractive index o f the lipid bilayer
h aa ~l e~ ntl n±
Ref. 1 and 2
Ref. 1 and 3
Ref. 1 and 4
Ref. 2 and 5
Ref. 3 and 5
Ref. 4 and 5
54.5 23.1 2"34 2'65 1"53 1"62
54 23.8 2.32 2"59 1.52 1"60
52 25.6 2"23 2"20 1"49 1"48
47.5 22.6 2"41 2'79 1"55 1"67
46.5 23.3 2.36 2"60 1"53 1"61
45 25.2 2"29 2"25 1.51 1'50
References: x Hanai et al., 1964, 1965; 2 Huang & Thompson, 1965; 3 Tien & Dawidowicz, 1966; ~ Cherry & Chapman, personal communication; 5 Tien, 1967, 1968. was independent of frequency, area and electrolyte concentration. I f the polar groups were holding the charge, the capacitance should be influenced by the amount of electrolyte in solution. Since this is not the case, it is deduced that only the hydrocarbon portion would be involved in the membrane capacitance. We can neglect the effect of polar groups in calculating the capacitance of the bilayer. On the other hand, in optical measurements, the thickness of the film (by H u a n g & Thompson, 1965; Tien, 1967) greatly depends on determining the refractive index of the membrane from the Brewster angle. The refractive index obtained by H u a n g & T h o m p s o n for a black film of egg lecithin was 1.66 _ 0.83, and Tien obtained the slightly
286
s.
OHKX
lower value of 1"60 4-0"03. The measurements of refractive index by the Brewster angle methods is quite uncertain. If we discuss the thickness of the film using only reflectivity of the film, we may deduce the results with more certainty. However, in this measurement, one must bear in mind that the effect of polar groups to the reflectivity of the film should not be disregarded. This is different from the case of capacitance measurements. The phospholipid which Huang & Thompson (1965) and Cherry & Chapman (personal communication) used was egg lecithin which has a fairly large polar group. On the other hand, Tien (1966) used a glycerol distearate which has the smaller polar groups than lecithin. Judging from this point of view, Tien's data might be the most accurate one for hydrocarbon portion of the film. In theoretical calculation for dielectric constant and refractive index (Ohki, 1968), several assumptions were made: the rigidity of hydrocarbon chain for simplicity of calculation; neglect±on of the effect of polar groups. But it is calculated that even if the hydrocarbon chain is flexible, the numerical values calculated are not so different from the case of the rigid chain. As a result, we may conclude that the dimension of the lecithin bilayer to be in the following range: Thickness of 54 ,-, 48/~ and area per molecule 51 ,~ 47/~,. Corresponding dielectric constant and refractive index are: 811 = 2"32 ,-, 2.29, 8± = 2.59 ,,, 2-25, nil = 1.52 N 1"51, n± = 1"60 ~ 1"50. The dielectric constant and refractive index are not so much anisotropic for the bilayer as we expected for the molecular model of bilayers. In other words, although the hydrocarbon chain is more or less oriented perpendicular to the layer, as a whole, the anisotropy of dielectric constant and refractive index is small. The author would like to express his thanks to Professor J. F. Dan±ell± for his valuable advice and encouragement through this work. This work was partly supported by the National Institute of Health, I ROI NB 08739-01. REFERENCES BLODGETT,K. B. & LANG~'m, I. (1937). Phy. Rev. 51, 964. HANAI,T., HAYDON,D. A. & TAYLOR,J. (1964). Proc. R. Soc. 281A, 377. HANAI,T., HAYDON,D. A. & TAYLOR,J. (1965). J. theor. Biol. 9, 278. H~VSNS, O. S. (1955). "Optical Properties of Thin Solid Films." London: Butterworth. HUANO, C. & THO~mSO~,T. E. (1965). 3". molec. Biol. 13, 183.
PROPERTIES OF LIPID BILAYER MEMBRANES
287
LANDAU, L. D. & LtFSHrrz, E. M. (1960). "Electrodynamics of Continuous Media." Reading, Mass.: Addison-Wesley. OHKI, S. & FUKUDA,N. (1967). J. theor. Biol. 15, 362. OHKI, S. (1968). J. theor. Biol. 19, 97. OILr~, S. (1969). J. theor. Biol. 23, 158. TnorcrPSO~4,T. E. & HUAIqG,C. (1966). J. molec. Biol. 16, 576. TIEN, H. T. & DAWIDOWICZ,E. A. (1966). jr. col. Sci. 22, 438. T~N, H. T. (1967). J. phys. Chem. 71, 3395. T ~ , H. T. (1968). Chem. Phys. Liplds, 2, 55.