Theoretical Study of Intermolecular Interaction at the LipidâWater

J. Phys. Chem. B 1997, 101, 4817-4825

4817

Theoretical Study of Intermolecular Interaction at the Lipid-Water Interface. 2. Analysis Based on the Poisson-Boltzmann Equation Hirohisa Tamagawa,† Minoru Sakurai,*,† Yoshio Inoue,† Katsuhiko Ariga,‡ and Toyoki Kunitake‡ Department of Biomolecular Engineering, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226, Japan, and Supermolecules Project, Japan Science and Technology Corporation (JST), Kurume Research Center Building, 2432 Aikawa-cho, Kurume, Fukuoka 839, Japan ReceiVed: December 26, 1996; In Final Form: March 25, 1997X

The interaction between a lipid monolayer with positive surface charges and anionic species dissolved in the aqueous phase is investigated using the Poisson-Boltzmann equation. The monolayer and the aqueous phase are approximated by dielectric continuums whose dielectric constants are 2 and 80, respectively. It is assumed that positive charges are periodically distributed on the interface formed by the two dielectrics. For such a system, the Poisson-Boltzmann equation is analytically solved with the Debye-Hu¨ckel approximation. It is indicated that the potential on the water side is significantly modulated by the presence of the lipid phase. This effectively contributes to a strengthening of intermolecular interaction near or on the interface. In addition, the interaction depends on the surface charge density on the monolayer. Combining these findings and the results of a previous quantum chemical study (Part 1 in a series of our studies), we discuss the reason why intermolecular binding is enhanced at the air-water interface.

Introduction Lipid monolayer assemblies expanded on water act as artificial receptors with a function of molecular recognition. A positively charged guanidinium-functionalized monolayer (1) specifically binds to negatively charged phosphate moieties of ATP and AMP through hydrogen bonding at the air-water interface. The structures of 1 and AMP are illustrated in Figure 1. The binding constant is 106-107 times larger than that between a guanidinium cation and phosphate anion in aqueous solution.1-4 Similar binding enhancement has been found between a neutral lipid monolayer and its substrates at the airwater interface.5 Thus, the interface behaves like an amplifier of intermolecular binding. In a previous study (Part 1 in a series of our studies),6 such interfacial phenomena were investigated using quantum chemical calculations based on a simple continuum model, where the lipid-water interface is approximated by a dielectric-dielectric double layer and a pair of interacting molecules is placed on the boundary. The calculations reproduced well the experimental binding energy for the 1-ATP system at the air-water interface and provided the evidence that such a hydrogenbonding complex can be formed, even when the reactive site is exposed to the aqueous medium. This means that the stability of the complex is assisted by the presence of the neighboring lipid phase. In other words, the lipid phase causes an appreciable magnitude of modification of the potential on the water side. However, the quantum chemical calculation did not provide information on to what extent the interfacial potential is influenced by the lipid layer. The Poisson-Boltzmann theory is a general one capable of calculating the potential in ionic solution and has been extensively applied to study the interaction between charged matters in the aqueous solution,7-10 for example, between the surfaces of some matters,11-19 between the charged surface of matter * Author to whom all correspondence should be addressed. † Tokyo Institute of Technology. ‡ Japan Science and Technology Corporation (JST). X Abstract published in AdVance ACS Abstracts, May 15, 1997.

S1089-5647(97)00060-6 CCC: $14.00

Figure 1. Molecular structures of guanidinium-functionalized lipid 1 and AMP.

and the charged particle,20 and between charged particles.21-27 Most of those studies have not taken into account effects of the inner region of matter on the interfacial potential, except for the case where ions in solution penetrate into the inner region of the matter. In addition, the surface charges of matter are usually assumed to be continuously distributed on the interface between a given matter and its surrounding aqueous medium. Of course, such an assumption may not cause serious errors in evaluating interactions between macroparticles.7 However, this may be inadequate when a charged macroparticle is interacting with small molecules. Kostogolou and Karaberas have already proposed a theory capable of treating this problem.28 In the continuum model assumed here, a lipid monolayerwater system is represented by a dielectric-dielectric double layer in which the lipid layer is explicitly treated as a lowdielectric medium, as described in Part 1.6 In addition, positive charges are periodically distributed on the boundary surface to simulate the actual surface charges that would be formed on © 1997 American Chemical Society

4818 J. Phys. Chem. B, Vol. 101, No. 24, 1997

(

∞

∑ m,n)0

4π2m2

a2

b2

+

∑ m,n)0

cos

dz2

(m,n)*(0,0)

( ) ( ) ( ) ( )

2πm 2πn x cos y + a b

Amn(z) cos

d2Amn(z)

∞

dz2

)

4π2n2 -

(m,n)*(0,0) d2A00(z)

Tamagawa et al.

2πm 2πn x cos y ) 0 (3) a b

When both sides of eq 3 are integrated over ranges of -a/2 e x e a/2 and -b/2 e y e b/2, the following equation is derived:

d2A00(z)

)0

dz2

(4)

The solution of eq 4 is given by

A00(z) ) C1z + C2

Figure 2. Schematic illustration of the continuum model. (a) The illustration of the whole system. Regions of z e 0 and z g 0 are the lipid and aqueous phases, respectively. The symbols O indicate the surface charges on the interface. They are located at every lattice point of (x, y, z) ) (ma, nb, 0), where m and n are integers and a and b indicate periods. (b) A view from the y-axis direction. The symbols + and - indicate the cation and anion species dissolved in the aqueous phase, respectively.

the guanidinium-functionalized lipid monolayer.1-3 For such a system, the Poisson-Boltzmann equation is analytically solved with the Debye-Hu¨ckel approximation. Although this approximation is not usually applied to the systems with high potential fields, we did adopt it here because it provides a clear physical picture that can be easily understood. On the basis of the potential functions obtained, we discuss the effects of the two factors, the low-dielctric property of the lipid phase and the surface charge density, on the interfacial potential, and finally refer to the reason why intermolecular binding is enhanced at the air-water interface.

where C1 and C2 are constants. Multiplying both sides of eq 3 by cos(2πsx/a) cos(2πty/b) and subsequently integrating the resultant equation over ranges of -a/2 e x e a/2 and -b/2 e y e b/2, one can obtain the following equation:

∞

φm(x,y,z) ) A00(z) +

∑

m,n)0 (m,n)*(0,0)

( ) ( )

2πm 2πn x cos y Amn(z) cos a b

)

m2 n2 + Amn(z) a2 b2

) 4π2

dz2

(6)


((

Amn(z) ) Rmn exp 2π

) )

m2 n2 + a2 b2

1/2

z +

( (

βmn exp -2π

) )

m2 n2 + a2 b2

1/2

z (7)

where Rmn and βmn are constants. By substituting eqs 5 and 7 into eq 1, the potential φm is given as follows:

φm(x,y,z) ) C1z + C2 +

∑

The system assumed here is composed of two phases, the lipid monolayer and aqueous phase. The coordinate systems are taken as shown in Figure 2, where a region of z e 0 is the lipid phase and that of z g 0 is the aqueous phase. Positive charges are located at every lattice point of (x, y, z) ) (ma, nb, 0), where m and n are integers and a and b indicate the periods of charge distribution along the x- and y-axes, respectively. Since the charges are located regularly and symmetrically with respect to the x- and y-axes, the potential in the lipid monolayer, φm, is written using the Fourier series29 as follows:

(

d2Amn(z)

∞

Derivation of the Interfacial Potential

(5)

m,n)0 (m,n)*(0,0)

( (( )) ( ( ) )) ( ) ( ) Rmn exp 2π

m2

n2

1/2

+

a2

b2

n2

z + βmn exp -2π

1/2

b2

z cos

m2 +

a2

2πm 2πn x cos y (8) a b

Since the potential φm(x,y,z) must decay on going away from the interface, C1 and βmn must vanish at the infinite distance along the -z-direction. Hence

φm(x,y,z) ) Θ00 + ∞

∑ m,n)0 (m,n)*(0,0)

(( )) ( ) ( )

Θmn exp 2π

m2

n2

1/2

+

a2

b2

z cos

2πm 2πn x cos y a b

(1)

(9)

Since there are no charges in the inner side of the lipid phase, the potential φm must satisfy the Laplace equation.

In eq 9 the two coefficients, Θ00 and Θmn, are introduced, and they are defined as follows:

∆φm ) 0

C2 ) Θ00

(10)

Rmn ) Θmn

(11)

The following equation must be thus fulfilled:

(2)

Lipid-Water Poisson-Boltzmann Electrostatics

J. Phys. Chem. B, Vol. 101, No. 24, 1997 4819

Next, the potential φs in the aqueous phase is derived. In a region of z g 0 the potential is also written using the Fourier series29 as follows: ∞

∑ m,n)0

φs(x,y,z) ) B00(z) +

Bmn(z) cos

( ) ( ) 2πm 2πn x cos y a b

(m,n)*(0,0)

(12)

The Poisson-Boltzmann equation must be satisfied in the aqueous phase. Here the Debye-Hu¨ckel approximation is used for simplicity, and thus the potential φs must satisfy the following equation:

∆φs ) κ2φs

(13)

where κ is the reciprocal of the Debye length. By substituting eq 12 into eq 13, one can obtain the following equation:

(

∞

∑

4π2m2

-

m,n)0 (m,n)*(0,0) d2B00(z)

a2

2

cos

dz

(m,n)*(0,0) ∞

κ2 B00(z) +

2πm 2πn x cos y + Bmn(z) cos a b

d2Bmn(z)

∑ m,n)0

+

(

b2 ∞

2

dz

) ( )( ) ( )( ) ( )( )

4π2n2

∑ m,n)0

2πm 2πn x cos y a b

Bmn(z) cos

(m,n)*(0,0)

2πm 2πn x cos y ) a b

dz

2

( (( ( )) ) ( ( ( )) )) ( ) ( ) ∞

∑

ξmn exp κ2 + 4π2

) κ2B00(z)

m2

m,n)0 (m,n)*(0,0)

ηmn exp - κ2 + 4π2

m2

n2

a2

n2

+

a2

1/2

z cos

+

b2

b2

1/2

z +

2πm 2πn x cos y a b (19)

Since eq 19 must decay on going away from the interface, γ00 and ξmn must vanish at the infinite distance along the +zdirection. The following equation is thus derived:

(( ( )) ) ( ) ( ) ∞

∑ m,n)0

φs(x,y,z) ) Γ00 exp(-κz) + m2

4π2

n2

a2

Γmn exp - κ2 +

(m,n)*(0,0) 1/2

z cos

+

b2


In eq 20 the two new coefficients, Γ00 and Γmn, are introduced, and they are defined as follows:

(14)

When both sides of eq 14 are integrated over ranges of -a/2 e x e a/2 and -b/2 e y e b/2, the following equation is obtained:

d2B00(z)

φs(x,y,z) ) γ00 exp(κz) + δ00 exp(-κz) +

δ00 ) Γ00

(21)

ηmn ) Γmn

(22)

Since the potential must be continuous at the interface between the lipid and aqueous phases, the following equation must be fulfilled:

(15) (23)

φm ) φs


B00(z) ) γ00 exp(κz) + δ00 exp(-κz)

(16)

Consequently, the following two equations are obtained:

where γ00 and δ00 are constants. Multiplying both sides of eq 14 by cos(2πsx/a) cos(2πty/b) and subsequently integrating the resultant equation over ranges of -a/2 e x e a/2 and -b/2 e y e b/2, the following equation is obtained:

d2Bmn(z) dz2

(

) κ2 + 4π2

(

n m + Bmn(z) a2 b2

((

(

(25)

ηmn exp - κ2 + 4π2

∞

φs(x,y,z) ) Θ00 exp(-κz) +

1/2

z + 2

(( ( )) ) ( ) ( )

(17)

)) ) ( ( ( )) ) m2 n2 + a2 b2

Θmn ) Γmn

2

The solution of eq 17 is given as follows:

Bmn(z) ) ξmn exp κ2 + 4π2

(24)

The potential φs can be written as follows:

))

2

Θ00 ) Γ00

2

n m + 2 2 a b

1/2

4π2

m2

n2

+

a2

b2

∑ m,n)0

(m,n)*(0,0) 1/2

z cos

Θmn exp - κ2 +


z (18)

where ξmn and ηmn are constants. By substituting eqs 16 and 18 into eq 12, the potential φs is given as follows:

The coefficients Θ00 and Θmn are determined by applying the Gauss theorem to a region involving the interface between the lipid and aqueous phases.30,31 The charges and the electric displacement on the interface must satisfy the following relation:


(

∞

e

Tamagawa et al.

∂φs(x,y,z) + δ(x - ia) δ(y - jb) ) -s ∂z i,j)-∞ ∞ ∂φm(x,y,z) m2 m |z)0 ) sκΘ00 + s κ2 + 4π2 + ∂z m,n)0 a2

∑

))

n2

)

1/2

b2

∑

(

( (

( ) ( ) ∑ ) ( ) ( ) (m,n)*(0,0)

2πm 2πn x cos y + m a b

Θmn cos

n2

1/2

b2

∞

2π

m,n)0 (m,n)*(0,0)

m2 +

a2

Results and Discussion

2πm 2πn x cos y (27) Θmn cos a b

where δ is Dirac’s delta function, s and m are the dielectric constants of the aqueous phase and the lipid phase, respectively, and e is elementary electric charge. By integrating both sides of eq 27 over ranges of -a/2 e x e a/2 and -b/2 e y e b/2, the coefficient Θ00 is given as follows:

e κsab

Θ00 )

(28)

Multiplying both sides of eq 27 by cos(2πsx/a) cos(2πty/b) and subsequently integrating the resultant equation over ranges of -a/2 e x e a/2 and -b/2 e y e b/2, the following equation is obtained:

(

e ) s κ2 + 4π2

(

))

m2 n2 + a2 b2

1/2

Θmn

These solutions have interesting features. Despite the fact that eq 31 indicates the potential in the lipid phase, it depends on the dielectric constant of the aqueous phase s and the Debye length κ-1. On the other hand, despite the fact that eq 32 indicates the potential in the aqueous phase, it depends on the dielectric constant of the lipid phase m. Therefore it is expected that intermolecular bindings at the lipid-water interface exhibit different properties from those in the pure lipid or pure aqueous phase.

( )

ab m2 + m2π 2 + 4 a 2 1/2 n ab Θmn (29) 4 b2

From eq 32 alone, it is impossible to evaluate the binding energy between the positively charged monolayer and the anionic species dissolved in the aqueous phase. However, one can obtain the anion concentration as a function of the z-coordinate, from which the strength of the interaction could be indirectly estimated. Namely, it is interpreted that the higher the anion concentration is in the vicinity of the interface, the stronger the interaction between the surface charges and the anionic species becomes. As an example, we consider the potential profile along the line (x, y, z) ) (0, 0, z). Then, eq 32 is given by

φs(0,0,z) )

e κsab

∞

(m,n)*(0,0)

4e s(κ2a2b2 + 4π2(m2b2 + n2a2))1/2 + 2πm(m2b2 + n2a2)1/2

((

exp - κ2 + 4π2

4e s(κ2a2b2 + 4π2(m2b2 + n2a2))1/2 + 2πm(m2b2 + n2a2)1/2 (30) By the use of eqs 9, 26, 28, and 30, the final expressions of φm(x,y,z) and φs(x,y,z) are given by

φm(x,y,z) )

∞

e

∑

+ κsab

m,n)0 (m,n)*(0,0)

4e 2 2 2

2

2 2

2 2 1/2

s(κ a b + 4π (m b + n a ))

×

(( )) ( ) ( )

exp 2π

m2

n2

1/2

z cos

+

2

a

φs(x,y,z) )

+ 2πm(m2b2 + n2a2)1/2

b

2


∞

e κsab

∑ m,n)0

exp(-κz) +

s(κ2a2b2 + 4π2(m2b2 + n2a2))1/2 + 2πm(m2b2 + n2a2)1/2 exp - κ2 + 4π2

( )) ) ( ) ( ) m2

n2

+

a2

b2

1/2

z cos

n2

+

a2

b2

eφs kT


z (33)

where [n0] is the anion concentration in the region infinitely apart from the interface. Next, we apply eq 33 for analysis of the actual system composed of the monolayer of 1 and the aqueous phase including the AMP anion. According to the experiments by Sasaki et al.,1 the occupied cross section of 1 in the monolayer is about 30 Å2 and the concentration of AMP in the aqueous solution is 0.001 M AMP. For comparison, we examine the following several cases where the values of a and b both are 5.0, 5.5, or 6.0 Å, and the anion concentration is 0.001, 0.01, or 0.1 M. In all the cases, the dielectric constant of the aqueous phase was taken to be 80. The Debye length κ-1 was determined in the following way. In the aqueous phase the AMP anions are present together with their counterions and the counterions of the monolayer surface charge. From the condition of the electric neutrality in the bulk region, the following Poisson-Boltzmann equation must be fulfilled:

( (

×

1/2

(34)

1 eφ eφ d2φ ) - ne exp - exp 2 kT kT dz

(m,n)*(0,0)

4e

( )) ) m2

( )

[n] ) [n0] exp

Θmn )

×

The anion concentration is expressed as follows:

Consequently, the coefficient Θmn is given by

((

∑ m,n)0

exp(-κz) +

)

( ))

(35)

where the valences of cation and anion both are assumed to be 1, n is the concentrations of cation and anion in the bulk region, and is the dielectric constant of solution. The following equation can be derived by applying the Debye-Hu¨ckel approximation to eq 35:



Figure 3. Distance dependence of the potential in the aqueous phase in the case of the parameters a and b ) 5.0 Å. (a), (b), and (c) indicate the data for solutions different in the anion concentration in bulk, that is, [n0] ) 0.001, 0.001, and 0.1 M, respectively. In each figure, symbols 0, O, and 4 indicate the data for the cases of m ) 1 and 2, 40, and 80, respectively.

d2φ 2ne2 φ ) κ2φ ≈ 2 kT dz

(36)

Consequently, the Debye length is given by κ-1 ) (kT/2ne2)1/2. Since the Debye length was obtained, we can calculate the potential and the anion concentration using eqs 33 and 34. The resulting potential profiles are shown in Figures 3, 4, and 5, which correspond to the cases of a and b ) 5.0, 5.5, 6.0 Å, respectively. In each figure, a, b, and c indicate the data for solutions different in the anion concentration in bulk, that is, [n0] ) 0.001, 0.01, and 0.1 M, respectively. Additionally, for each solution, the potential profiles were obtained for the four cases different in the dielectric constant of the lipid layer, that is, m ) 1, 2, 40, and 80. Similarly, the anion concentration profiles along the z-axis were obtained for the 36 cases different in parameters a, b, [n0], and m. The results are shown in Figures 6, 7, and 8. In all the cases of Figures 3-8, the data for m ) 1 and 2 almost overlap each other. As can be seen from Figures 3, 4, and 5, the difference in m

Figure 4. Distance dependence of the potential in the aqueous phase in the case of the parameters a and b ) 5.5 Å. The meanings of (a), (b), and (c) and of the symbols 0, O, and 4 are the same as those in Figure 3.

causes remarkable differences for the potential profiles in a region from 0 to 3 Å apart from the monolayer surface. It can be thus concluded that the potential in the extreme vicinity of the interface is significantly modified by the presence of the nearby lipid phase. As a result, the anion concentration in such a region is also remarkably influenced by m (Figures 6-8). Table 1 summarizes the ratio of the anion concentration at the position of z ) 1 Å to that in the bulk. Clearly, with lowering m, the anion becomes more concentrated in the vicinity of the interface, meaning that the low dielectric property of the lipid phase promotes the anion-surface charge binding. Table 2 summarizes the potentials and anion concentration at the position of z ) 1 Å. With an increase in the anion concentration in the bulk region and/or with an increase in the values of a and b, the effect of the dielectric constant of the lipid phase on the interfacial potential becomes more significant. This is indicated by comparing the interfacial potential in the case of m ) 2 with that in the case of m ) 80. Their ratio φs(2)/φs(80) is 1.52 when the values of a and b both are 6.0 Å and [n0] is 0.1 M. In accordance with this tendency, the low dielectricity property of the lipid phase tends to more largely


Tamagawa et al.

Figure 5. Distance dependence of the potential in the aqueous phase in the case of the parameters a and b ) 6.0 Å. The meanings of (a), (b), and (c) and of the symbols 0, O, and 4 are the same as those in Figure 3.

Figure 6. Distance dependence of the anion concentration in the aqueous phase in the case of a and b ) 5.0 Å. The meanings of (a), (b), and (c) and of the symbols 0, O, and 4 are the same as those in Figure 3.

affect the anion concentration near the interface with a lowering of the anion concentration in bulk and/or with an increase in the values of a and b. The present results from the Poisson-Boltzmann electrostatics predict that the modulation of the interfacial potential occurs in an extremely limited region no more than 3 Å apart from the interface. In Part 1,6 we have explicitly treated a pair of interacting molecules placed on/near the dielectric-dielectric interface and showed that the hydrogen bonding on the water side is stabilized only when the center of the interaction site is placed within 2-3 Å from the interface. Therefore, the results from the pure classical electrostatics are consistent with those from the previous quantum chemical calculation. As can be seen from Table 2, the interfacial potential becomes large as the periods a and b decrease. As a result, the anion concentration near the interface becomes increased with a decrease in the periods of a and b. In particular, when the anion concentration in the bulk region is low (0.001 M), the potential at z ) 1 Å changes several tens of millivolts by only a change of 0.5 Å in the periods of a and b, corresponding to a 4-5 times change of the anion concentration at this position. Thus,

it can be concluded that the surface charge density also largely influences the binding constant of intermolecular interactions near the interface, especially when the anion concentration in bulk is very low. Finally we check whether the above results are consistent with the experimental results by Sasaki et al.,1 who indicated that 1 binds to AMP in a molar ratio of 1:1. According to molecular modeling of AMP, its maximum diameter is about 10 Å (Figure 1). Here, AMP is approximated by a rectangular parallelepiped with a height of 10 Å (Figure 9a). The experiments have also indicated that most of the lipids forming the monolayer bind to AMPs when the AMP concentration is 0.001 M-1. Thus, the binding of AMP molecules to the monolayer surface would be schematically represented as shown in Figure 9b, where the whole surface of the monolayer is covered with regularly arranged AMP molecules. In this system, the average cross section of the rectangular parallelepiped could be assumed to be a × b Å2, because the periods of the surface charge distribution are equal to those of the lipid arrangement in the monolayer. In the actual system, the average values of a and b both are about 5.5 Å on the basis of the data





for the observed cross section of 1 in the monolayer. This indicates that near the interface there must exist one AMP molecule per a volume of 10 × 5.5 × 5.5 ) 302.5 Å3. On the other hand, from Figures 7-9 and Table 1, one can obtain the anion (AMP) concentration near the interface. It seems to be reasonable that the dielectric constant m of the lipid phase is taken to be 2 according to Part 1.6 Then, when the parameters a and b both are 5.5 Å and the anion concentration in bulk is 0.001 M-1, the anion concentration near the interface (at z ) 1 Å) is 2.131 M (Table 2). Considering the size of AMP, it could be said that this value represents the average concentration of AMP in a region of 0 Å e z e 10 Å. Consequently, from the present continuum model, the average number of AMP molecules included in a volume of 10 × a × b (302.5 Å3) is calculated as follows: 2.131(M) × 1000(l) × 6.02 × 1023(mol-1) × 10ab(Å3) ) 0.39. This is in fairly good agreement with the value estimated from the experimental data, indicating that the present continuum model reproduces well the essential aspect of the actual monolayer-AMP system. In many years ago, Nelson and McQuarrie provided an analytic solution of the linearized Poisson-Boltzmann equation

TABLE 1: Ratio of the Anion Concentration at z ) 1 Å to That in the Bulk Regiona mb 1

2

40

80

[n]z)1/[n0]c

a and b ) 5.0 Å 8164 8069

5967

5070

[n]z)1/[n0]c

a and b ) 5.5 Å 2157 2131

1544

1298

[n]z)1/[n0]c

a and b ) 6.0 Å 789 779

555

462

a

All the data are obtained on the assumption that the anion concentration in bulk is 0.001 M. b The dielectric constant of the lipid phase. c [n]z)1 and [n0] indicate the anion concentrations at the position of z ) 1 Å and that in the bulk region, respectively.

for a continnum model with a discrete array of surface charges,32 which is very similar to our present model. They revealed that the electrostatic potential differs significantly from results of a smeared-charge model. However, that report provided no direct information about effects of the dielectric constant of the lipid layer on the interfacial potential. Recently, Peitzsch et al. have


Tamagawa et al. We thus believe that the results provide a sufficiently reliable explanation for the effect of the lipid layer.

TABLE 2: Dependence of the Potential and the Anion Concentration at z ) 1 Å on the Parameters Used in the Continuum Model [n0]a mb

0.001 2

0.01 80

2

0.1 80

2

Conclusion 80

φsc [n]d φs(2)/φs(80)e [n](2)/[n](80) f

a and b ) 5.0 Å 0.233 0.221 0.085 0.073 8.069 5.070 0.263 0.167 1.05 1.16 1.59 1.57

0.037 0.026 0.414 0.273 1.42 1.52

φsc [n]d φs(2)/φs(80)e [n](2)/[n](80) f

a and b ) 5.5 Å 0.199 0.185 0.076 0.063 2.131 1.300 0.188 0.115 1.08 1.21 1.64 1.63

0.036 0.024 0.395 0.255 1.50 1.55

φsc [n]d φs(2)/φs(80)e [n](2)/[n](80) f

a and b ) 6.0 Å 0.172 0.159 0.069 0.056 0.779 0.245 0.146 0.087 1.08 1.23 3.18 1.68

0.035 0.023 0.383 0.243 1.52 1.58

a The molar concentration of anion in the bulk region. b The dielectric constant of the lipid phase. c The interfacial potential (in volts) calculated at the position of z ) 1 Å. d The anion concentration calculated at the position of z ) 1 Å. e The ratio of the interfacial potential in the case of m ) 2 to that in the case of m ) 80. f The ratio of the anion concentration in the case of m ) 2 to that in the case of m ) 80.

In this study, the equation of the interfacial potential for the monolayer-water system was derived under the conditions that the dielectric property of the lipid phase and the discrete distribution of the surface charges were explicitly considered. On the basis of this equation, mainly two conclusions are deduced. (1) The interfacial potential on the water side significantly depends on the dielectric constant of the lipid phase, and the interaction between the monolayer and the anion dissolved in water becomes stronger with lowering the dielectric constant of the lipid phase. (2) The monolayer surface charge density also affects the interaction between the monolayer and the anion, especially when the anion concentration in bulk is very low. Conclusion 1 supports the results obtained in the previous study (Part 1).6 It can be therefore concluded that the enhancement of intermolecular binding at the air-water interface is a universal phenomena. However, according to the interfacial potential obtained, the occurrence of such phenomena is limited to the case that the binding site is located in the extreme vicinity of the interface, at most a few angstroms apart from the interface. Beyond this threshold, the molecular behavior becomes almost similar to that in the bulk region. Acknowledgment. The authors thank the computer center of the Institute for Molecular Science, Okazaki, Japan, for the use of the super computer NEC SX-3. References and Notes

Figure 9. Schematic illustration for the binding of AMP to the monolayer. (a) The rectangular parallelpiped indicates one AMP molecule. (b) AMPs are binding to the monolayer surface.

provided a solution of the nonlinear Poisson-Boltzmann equation for a more sophisticated bilayer-water model.33 In their model, the following four factors were taken into account: the charges on each lipid were distributed over several atoms, these partial charges were non-coplanar, there was an ion-exclusion region adjacent to the polar head group, and the molecular surface was rough. However, they have not also examined effects of the lipid layer. As far as we know, the current study is the first one to explicitly refer to a role of the lipid layer in modifying the the potential in the aqueous phase. According to the study by Peitzsch et al.,33 the linearized Poisson-Boltzmann equation tends to overestimate the interfacial potential compared with the nonlinear Poisson-Boltzmann equation. However, as mentioned above, the present numerical results are consistent with the experimental values.

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Lipid-Water Poisson-Boltzmann Electrostatics (24) Anandarajah, A.; Chen, J. J. Colloid Interface Sci. 1994, 168, 111117. (25) Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1993, 155, 499505. (26) Ohshima, H.; Kondo, T. Colloid Polym. Sci. 1993, 271, 11911196. (27) Ohshima, H. J. Colloid Interface Sci. 1994, 168, 255-265. (28) Kostogolou, M.; Karaberas, A. J. J. Colloid Interface Sci. 1992, 151, 534-545.

J. Phys. Chem. B, Vol. 101, No. 24, 1997 4825 (29) Spiegel, M. R. Schaum’s Outline Series Theory and Problems of Fourier Analysis; McGraw-Hill: New York, 1974. (30) Panofsky, W. K. H.; Phillips, M. Classical Electricity and Magnetism; Addison-Wesley: Cambridge, 1961. (31) Feynman, R. P.; Leighton, R. B.; Sands, M. L. The Feynman Lectures on Physics; Addison-Wesley: Cambridge, 1965. (32) Nelson, N. D.; Mcquarrie, D. A. J. Theor. Biol. 1975, 55, 13-27. (33) Peitzsch, R. M.; Eisenberg, M.; Sharp, K. A.; McLaughlin, S. Biophys. J. 1995, 68, 729-738.

Theoretical Study of Intermolecular Interaction at the LipidâWater

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