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electrophoretic and time of relaxation corrections. Many methods are available for the determination of ion association constants.28 Among them are emf, conductimetric, spectroscopic (including NMR), solubility, and sound absorption techniques. The mobilities of simple ions may be obtained from conductivity and transference experiments on nonassociating or weakly associating systems. While it is more difficult to obtain the mobilities of ion aggregates, under favorable circumstances they can be estimated by fitting binary diffusion data of the associating electrolyte to an equation containing the mobility of the ion aggregate as an adjustable parameter. In this manner, (28) Robinson and Stokes, ref 24, Chapter 14.
mobilities of molecular organic acidd5J6@and metal sulfate ion pairsl8J9 have been determined. Thus with a few well-chosen binary experiments and some thought it should be possible to make a first approximation to the diffusive properties of complex multicomponent systemsB for which direct measurement may be difficult or not feasible. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. (29) The approximation = 6&Qi is a practical starting point for the description of diffusion of associating nonelectrolyte mixtures. For example, see (a) V. Vitagliano, J. Phys. Chem., 74,2949 (1970); (b) H. Kim., J. Solution Chem., 3, 271 (1974).
Electrostatic Potential of Bilayer Lipid Membranes with the Structural Surface Charge Smeared Perpendicular to the Membrane-Solution Interface. An Extension of the Gouy-Chapman Diffuse Double Layer Theory Gregor Cevc, * Institute for Biophysics, Faculty of Medicine, and J. Stefan Institute, E. Kardelj University Ljubljana, 6 1000 Yugoslavia, and Max-Planck Institut fur biophysikalische Chemie, Spektroskopie, D-3400, Gottlngen-Nikolausberg,Federal Republic of Germany
Saga Svetina, and BoZtjan Zek6 Institute for Blophyslcs, Faculty of Medicine. and J. Stefan Institute, E. Karde! University Ljubljana. 6 1000 Yugoslavia (Received: January 6, 198 I; In Final Form: February 27, 198 1)
An extension of the Gouy-Chapman diffuse double layer theory for bilayer (lipid) membranes is proposed, which takes into account the distribution of the structural charge perpendicular to the surface of the membrane. The linearized form of the Poisson-Boltzmann equation is solved for an arbitrary distribution of the structural charge normal to the surface plane, but numerical solutions of the corresponding nonlinearized form show that this linearization does not greatly affect the conclusions drawn from the model introduced. For the case of the exponential charge density profile perpendicular to the membrane surface, the explicit expressions for the electrostatic potential and the electric field of the membrane are derived. It is demonstrated that the electrostatic properties of the membrane are little sensitive to the precise shape of the charge distribution function, but depend strongly on the average transverse displacement of the structural charge from the membrane surface plane, d. Because of the structural membrane charge smearing perpendicular to the membrane surface, the electrostatic potential profile becomes shallower as d increases, and a region of relatively small electric field is created near the membrane-solution interface. It is concluded that the distribution of the structural charge perpendicular to the membrane-solution interface markedly affects the electrostatic properties of the membrane either if the electrolyte solution is sufficiently concentrated or if the average transverse displacement of charge is sufficiently large. In the physiological ion concentrations range (c = 0.15 mol/L) the electrostatic surface potential decreases by a factor of 0.2 for d = 0.2 nm when compared to d = 0.
Introduction Some membranes carry structural surface charge which arises from the presence of ions on lipids, lipopolysaccharides, or proteins. When such membranes are in electrolyte solution, the surface charges attract oppositely charged counterions from the solution, while repelling co-ions of the same charge. Thus, a layer of unequal positive and negative ion concentrations in thermal equilibrium is formed at the membrane-solution interface. Gouy and Chapman, who first discussed the existence of such ion distribution, have termed this layer the diffuse double layer.lv2 The structural ions and the ions of the solution give rise to the electrostatic potential of the membrane which de(1) G. GOUY,J. PhyS., 9, 457-68 (1910). (2) D. L. Chapman, Philos. Mug., 25, 475-81 (1913). 0022-3654/8 1/2085-1762$01.25/0
pends on the number as well as on the spatial distribution of the charges. Calculations of mutual electrostatic interactions have usually assumed either that the structural ions form the surface of uniform charge density (uniform charge density model3) or that they are discrete and ordered in a perfect two-dimensional Recently, Tsien has further improved the discrete charge model by developing the virial expansion calculation.' Many experimental results have now been gathered which support the view that some properties of the bilayer membranes are influenced by the electrostatic potential at the membrane. Such is the case with regard to mem(3) S. McLaughlin, Curr. Top. Membr. Trunsp., 9, 71-144 (1977). (4) D. C. Grahame, 2.Elektrochem., 62, 264-74 (1958). (5) R. H. Brown, Prog. Biophys. Mol. Biol., 28, 341-70 (1974). (6) R. Sauv6 and S. Ohki, J . Theor. Biol., 81, 157-79 (1979). (7) R. Y. Tsien, Biophys. J., 24, 561-7 (1978).
0 1981 American Chemical Society
Electrostatic Potential of Bilayer Lipid Membranes
The Journal of Physical Chemistry, Vol. 85,No. 12, 1981 1763
brane excitability, permeability, adsorption, adhesion, and even membrane structure. The understanding and the theoretical description of these membrane properties, however, all depend on the detailed knowledge of the electrostatic potential and of the electric field a t and near the membrane ~ u r f a c e . ~ The measured electrostatic potentials are often lower than what the uniform charge density model predicts."12 In general, this discrepancy was attributed to the fact that the discreteness of charge was neglected. The average electrostatic potentials as estimated within the limits of the discrete charge model are namely lower than the values obtained using the uniform surface charge density modelas Therefore, the results from the surface electrostatic studies and the fact that the lateral intercharge spacing is large (>0.6 nm) both seem to support the view that the discreteness of charge may need to be considered in quantitative attempts to analyze the surface charge phenomena. As yet, aside from the discreteness of structural charge, the fact should not be overlooked that the structural charges do not all lie in one plane. Surfaces of artificial membranes, cells, and cell organelles are neither homogeneous nor rigid, as is assumed in the uniform charge density model and other classical models. We therefore wish to propose a model which treats the membrane surface as a zone of transversely smeared structural charge. Several arguments speak in favor of such a description of the membrane-solution interface. Supported by the current picture of the bilayer membrane, the conclusion emerges that the structural surface charges can be smeared perpendicular to the membrane surface because of the thermal motion of the charge-carrying groups: longitudinal molecular stretchings, group rotations, and even out-ofplane fluctuations of whole molecules or parts of the surface. Indeed, in addition to the out-of-plane excitations of the membrane surface which were observed by laser light scattering on model membranes,13 a rippled membrane surface was observed by X-ray diffraction,14J5 electron microscopy,16 and refle~tance.~' In addition, NMR experiments show that the presence of ions induces conformational changes of charge-carrying groupsl8 and/or increases their m0bi1ity.l~ These phenomena all give rise to the smearing of the structural charge perpendicular to the membrane which therefore depends on the ambient temperature, the elasticity of the membrane, the rigidity of the charge-carrying-group conformations, or the thermodynamical state of the membrane as a whole. In this work we will inspect the consequences on the electrostatic properties of bilayer membranes of the structural charge smearing perpendicular to the membrane surface plane. We will derive general formulas for the
electrostatic potential of the asymmetrically charged and screened impermeable bilayer membrane with an arbitrary distribution of the structural charge perpendicular to the membrane surface. These equations will subsequently be solved for the special case of the bilayer membrane with exponentially decreasing transverse profile of the structural charge density. Then, the restrictions of the approximations used and the limitations of our model will be discussed. Finally, the available structural data for lipids and bilayer membranes will be used to make an estimate of the average displacement of structural charge, perpendicular to the membrane-solution interface, and the implications of our model will be correlated with the results of electrostatic potential measurements.
(8) P. Fromherz and B. Masters, Biochim. Biophys. Acta, 356,270-5 (1974). (9) 0. S. Anderson, S. Feldberg, H. Nakadomari, S. Levy, and S. McLaughlin, Biophys. J., 21, 227-43 (1978). (10)W. L. C. Vaz, A. Niksch, and F. Jihnig, Eur. J. Biochem., 83, 299-305 (1977). (11) H:IJ. Apell, E. Bamberg, and P. Lauger, Biochim. Biophys. Acta, 552, 369-77 (1979). (12) M. Eisenberg, T. Gresalfi, T. Riccio, and S. McLaughlin, Biochemistry, 18, 5213-23 (1979). (13) N. M. Amer, Liq. Cryst. Conf., (Abstr.),3rd, 1979, C-3E (1979). (14) A. Tardieu, V. Luzzatti, and F. C. Reman, J. Mol. Biol., 75,711-33 (1973). (15) M. J. Janiak, D. M. Small, and G. G. Shipley, J. Biol. Chem., 254, 6068-78 (1979). (16) R. Krbecek, C. Gebhart, H. Gruler, and E. Sackmann, Biochim. Biophys. Acta, 554, 1-22 (1979). (17) D. Bach and I. R. Miller, Biophys. J., 29, 183-7 (1980). (18) J. Westman and L. E. G. Eriksson, Biochim. Biophys. Acta, 557, 62-78 (1979). (19) H. Hauser and M. C. Phillips, Prog. Surf. Membr. Sci., 13, 297-413 (1979).
where reciprocal Debye length K is a measure of the thickness of the diffuse layer. For asymmetrically screened bilayers K = K1 -a < X 5 -Io =0 - X o < X < xo = K2 Xo 5 X < (44 and ~i denotes
General Theory The variation of the electrostatic potential $ with the distance from the charged surface is given by the Poisson equation, eq 1,where p t ( x ) is the total charge density (the A $ h ) = -Pth)/(EEo)
(1)
sum of the net density of the solution ions p i ( x ) and of the structural charge density p ( x ) ) , E is the relative permittivity of the solution, and to is the permittivity of free space. In the Gouy-Chapman approximation, i.e., in the uniform charge density model, we consider a flat, charged, impermeable membrane immersed in the electrolyte solution. The membrane is presumed to be 2x0 thick, is of uniform relative permittivity tM, and is located in the yz plane. The left and the right surface have structural charge densities g1 and a2,respectively, and therefore p ( x ) = a16(x
+ X o ) + fJ2S(X - X o ) .
Because of the mutual interactions between the solution ions and the charged surface, a diffuse double layer exists. In this event, p i ( x ) values are given in terms of the bulk concentrations of each of the total n ion species, cij, j = 1, 2, ..., n (i = 1 x I -x0, i = 2 x 1 xo) and of the valencies of these ions, Zj (eq 2). In eq 2, h is Boltzmann's
-
-
Pi(X)
= efiA
2
]=1
cijzj
e x ~ [ - ~ j e o $ ( x ) / ( k ~ ) ~(2)
constant, T i s the absolute temperature, NA is Avogadro's constant, and eo is the electronic charge; cij values are given in moi/m3. If we assume that the electrostatic potential of the membrane is low, eoJ.(x) - p(x>/(%)
Cevc et al.
To calculate the electrostatic potential of the
Equation l l a applies for x 2 xo;for the left membrane side we get
while the electrostatic potential within the bilayer changes in our approximation linearly between both surface potential values
Electrostatic Potential of Bilayer Lipid Membranes
The Journal of Physical Chemistry, Vol. 85,
Electrostatic Potential of the Membrane with Exponential Distribution of the Structural Charge Perpendicular to the Membrane-Solution Interface The formulas derived in the previous section apply for an arbitrary distribution function of the structural surface charge. However, before we can make use of them, we must make assumptions regarding the actual distribution of charge perpendicular to the membrane-solution interface. Let us first suppose that the charge density of the structural charge declines exponentially with distance from the membrane surface. The transverse distribution function of the structural charge pl(x) and p 2 ( x ) must satisfy the conditions of eq 6. Therefore
where dl and d2 are the average transverse displacements of charge at the left and right membrane surfaces, respectively. Using eq 12, the integration of eq 11yields the following expression for the electrostatic potential on the right side of the asymmetrically charged and screened flat bilayer membrane: $(x) =
Correspondingly, the expression for electrostatic potential on the left side of the membrane surface reads =
(14)
whereas the intramembraneous potential varies linearly between $(-xo) and $(xo). In eq 13 and 14 one parameter emerges which is not customary in the ordinary diffuse double layer theory, namely, the average displacement of charge, d. In the uniform surface charge density model this parameter vanishes because all of the structural charge is assumed to lie exactly in the membrane surface plane. The results of the uniform surface charge density model are therefore the limiting case of our model for dl = d, = 0. The electrostatic potentials as calculated from eq 13 for the case of a symmetrically charged bilayer membrane immersed in a solution of monovalent ions are depicted in Figure 2a. The transverse smearing of the structural charge obviously lessens the surface potential and smoothens the potential profile. A relative decrease of the electrostatic potential is observed together with an increase further into the bulk phase. Naturally, the higher the average displacement of charge, the larger is the reach of
15
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I
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t
I
No. 12,
1981
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,
a i
x lnml Figure 2. (a) Calculated electrostatic potential and (b) electric field of bilayer membrane with transversely smeared structural charge. Membrane is assumed to be symmetrically charged (u, = u2 = A s m-2) and screened by the solutions of monovalent ions ( c , = c2 = 0.1 mol L-I). The Debye-Huckel approximation is supposed to be justified. d denotes the average transverse displacement of the structural charge, being (A) 0, (B) 0.25, and (C) 0.5 nm.
the electrostatic potential and the smaller its value at the surface. The electric field profiles are shown in Figure 2b. The membrane-solution interface is a transitional zone where the electric fields of the two dielectric regions merge. The discontinuity of charge distribution which is introduced in the uniform surface charge density model by assuming p ( x ) = u18(x + xo) + u2S(x - xo) gives rise to large discontinuity of the electric field a t x = f x o for the case d = 0. The smearing of charge perpendicular to the membrane, however, creates near the membrane-solution interface a region of relatively small electric field and E ( x ) reaches its maximum value only a t a finite distance from the membrane surface. In general, the width of this intermediate zone increases with the increase of the average displacement of the structural charge. Transverse smearing of charge therefore withdraws the discontinuity of E(x) at x = fxO, which is a common feature of standard theories of membrane electrostatics. The effect of the transverse smearing of the structural charge on the electrostatic potential also depends on the bulk ion concentration. By inspecting the ratio plotted in Figure 3, we see that the deviation between the uniform surface charge density model and our model gradually increases either with the increase of the electrolyte solution concentration or with the increase of the average displacement of the structural charge. If the bathing solution is sufficiently concentrated, our model predicts that, even for small average displacements of the structural charge, the effect of the transverse smearing of charge on the electrostatic potential of the membrane will be considerable and also that the classical electrostatic theories will yield values for $(xo) which are too high.
Validity of the Model and Its Limitations The use of several approximations in our derivations imposes some limitations on the application of the results of previous sections to experimental systems.
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Cevc et al.
The Journal of Physical Chemistry, Vol. 85,No. 12, 198 1 1
00 1
io5
I
I
IO-^
I
I
I
01
IO
I
IO-*
I,
I
I IOO
c [ mole Iitei'I
Figure 3. Effect of the transverse smearing of the structural charge on the normalized electrostatic surface potential of bilayer membranes. Curves were calculated for symmetrically charged and screened bilayer membrane immersed in the monovalent electrolyte solution.
The transverse distribution function of the structural charge density was assumed ad hoc to decrease exponentially from the membrane surface. As yet, no direct evidence of such or other shape of p ( x ) is available. We therefore investigated whether the particular choice of one distribution function of the structural charge restricts the general validity of our conclusions. We have also calculated the electrostatic potential of the bilayer membrane with the assumption that the transverse distribution function of the structural charge is Gaussian. Similar suppositions were made in some previous theoretical works on membrane surface dynamics.20 Our results show that the electrostatic potentials of membranes calculated by assuming Gaussian and exponential transverse distribution functions of the structural charge agree to within 5% provided that the average displacement of charge remains the same. Throughout our calculations we were using the solutions of the Poisson-Boltzmann equation in the Debye-Huckel approximation. The requirement that the electrostatic equations be linearized, i.e., eo#
