Langmuir 2005, 21, 5677-5680
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Cationic DMPC/DMTAP Lipid Bilayers: Local Lateral Polarization of Phosphatidylcholine Headgroups Victor Levadny*,† and Masahito Yamazaki*,‡ Department of Physics, Faculty of Science, Shizuoka University, Shizuoka, 422-8529, Japan, and The Scientific Council for Cybernetics, Russian Academy of Sciences, Vavilov str. 34, 333117 Moscow, Russia Received December 9, 2004. In Final Form: May 8, 2005 We investigated the effect of dimyristoyltrimethylammonium propane (DMTAP) charge on area per molecule of mixed DMTAP/dimyristoylphosphatidylcholine (DMPC) bilayers in a simple model. Assuming that trimethylammonium (TAP) charge causes lateral polarization of neighboring PC molecules, we analyzed variation in area per molecule as the mole fraction of TAP increases. The theoretical predictions obtained in the present study are consistent with results of a recent molecular dynamics simulation study (Gurtovenko et al. Biophys. J. 2004, 86, 3461).
Introduction Lipid bilayers may have fixed electric charges (commonly named as surface charges) within their headgroup regions. These charges influence essentially on the physical state and polymorphism of the lipid membranes. It was shown that surface charges induce a lateral expansion of the lipid bilayer (see, e.g., ref 1 and references therein). This phenomenon agrees well with the intuitively obvious idea: because of their electrostatic repulsion, surface charges induce a lateral expansion of the lipid bilayer. But the data which contradict this idea were also obtained. Recently Gurtovenko et al.2 used molecular dynamics simulations in a detailed atomic study of lipid bilayers consisting of a mixture of cationic dimyristoyltrimethylammonium propane (DMTAP) and zwitterionic dimyristoylphosphatidylcholine (DMPC). These lipids have the same nonpolar hydrocarbon chains and differ only in their headgroups. By varying the molar fraction χ of trimethylammonium (TAP), they found that the area per molecule 〈A〉 ) S/N is markedly nonmonotonically dependent on the TAP concentration and has its minimum around the point of equimolar DMPC/DMTAP mixture, at which χ ) 0.5 (here, S is the total membrane area and N is the total number of lipid molecules). They also found that TAP causes significant reorientation of the P-N dipole vector of the PC headgroup. The angle between the PC headgroups and bilayer normal changes from R0 ≈ 80° at χ ) 0 to R0.75 ≈ 20° at χ ) 0.75. A similar result was obtained in an earlier Monte Carlo computer simulation.3 The physical cause of these variations and the P-N dipole reorientation is unclear. In the present Letter we consider a phenomenon which may exist in a charged lipid bilayer on a microscopic level and affect its mechanical state. Namely, the lateral polarization of the neutral lipid molecule heads by nearest * Address correspondence to Dr. Masahito Yamazaki, Department of Physics, Faculty of Science, Shizuoka University, 836 Oya, Shizuoka, 422-8529, Japan. TEL and FAX: 81-54-238-4741. E-mail:
[email protected]. † The Scientific Council for Cybernetics, Russian Academy of Sciences. ‡ Department of Physics, Faculty of Science, Shizuoka University. (1) Cevc, G. Biochim. Biophys. Acta 1990, 1031-1033, 11. (2) Gurtovenko, A. A.; Patra, M.; Karttunen, M.; Vattulainen, I. Biophys. J. 2004, 86, 3461. (3) Pink, D. A.; Quinn, B.; Moeller, J.; Merkel, R. Phys. Chem. Chem. Phys. 2000, 2, 4529.
Figure 1. View of lipid bilayer surface from above: (a) neutral bilayer, the lipid molecules are shown as circles with the arrows inside which mimic its lateral dipoles; (b) charged lipid bilayer, the polarized cluster is shown by a big dotted circle, a single charged molecule is mimicked as a positive charge in the center of cluster (see main text for details).
surface charge (Figure 1). Because the electrostatic interaction of the surface charge with the dipole of neighboring neutral lipid molecule tends to orient the latter in a certain direction (which results in a chargedipole attraction), this polarization can generate a local bilayer compression. This phenomenon becomes pro-
10.1021/la046976x CCC: $30.25 © 2005 American Chemical Society Published on Web 05/27/2005
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membrane head region at a depth of h from the hypothetical “lipid-solution” boundary (see Figure 2); in addition, a < h((w/in - 1)1/2 ≈ 1.5h. We estimated the energy for a ∼ 0.9 nm. Assuming p ∼ a/2, h ∼ 0.75a, and R ≈ 50°, we obtained a value of uq-d ∼ (5-10)kBT. This energy is sufficient to hold dipoles of neighboring PC molecules in an orientation in which they are exactly parallel to the TAP-PC line. In other words, TAP polarizes neighboring PC molecules, but this interaction uq-d is only sufficient to polarize the nearest PC molecules. Thus, each TAP forms the polarized cluster, which consists of a central TAP itself and n nearest neighboring PC molecules. Due to attraction between 〈dL〉 and the TAP charge, the polarization causes local membrane compression. As a result, the intracluster PC-TAP distance (which we refer to as the PT bond) becomes shorter than the intermolecular bonds outside the cluster. But the polarization also generates local membrane repulsion due to dipole-dipole interaction ud-d between neighboring PC heads. Thus, the additional lateral tension inside the cluster (compared with the region outside of the cluster) is calculated as follows Figure 2. Sketch of lipid bilayer cross section in different scales. q is charge of the charged lipid molecule; d is lateral component of the neutral lipid molecule neighboring the charged one.
nounced for lipid bilayer at the liquid-crystalline phase LR when the lipid molecules are in the act of doing significant thermal rotation around their vertical axis. Specifically we discuss the effect of TAP charge on PC dipole reorientation in the lateral direction and its effect on clustering of neighboring PC molecules. We propose a scenario that emphasizes reorientation of the PC headgroup dipole lateral components in the nearest TAP environment. There is no preferred orientation of lateral dipoles in the neutral PC membrane (Figure 1a). Due to thermal rotation, the average orientation of lateral dipoles 〈dL〉 is zero. In the DMPC/DMTAP membrane, owing to electrostatic interaction between the TAP charge and the dipole of the neighboring PC, the preferred orientation of the latter arises. It is oriented so that the PC dipole lateral component is exactly parallel to the TAP-PC line; consequently, 〈dL〉 * 0 (Figure 1b). Taking into account that the charge attracts these polarized dipoles, we believe that this interaction is the main factor responsible for the decrease of area per molecule in the DMPC/DMTAP mixture. Theory Consider the interaction between the TAP charge e and the headgroup dipole B d of a neighboring PC molecule. Generally, the thermal averaged energy uq-d of the interaction is calculated as follows:
〈d BLE uq-d B〉 〈d BE B〉 ))) kBT kBT kBT -Y(coth Y - 1/Y) ) -YLv(Y) (1) where kB is Boltzmann’s constant, T is the temperature, Y ) dLE/kBT, the electric field of the TAP charge at distance a from its center (where the PC center is located) is given as E ) 3eh2/2π0ina4, 0 the permeability of vacuum and in (∼5-20) the dielectric constant of the lipid headgroup region, dL )ep sin(a), where p is the effective size of the PC total dipole, and the Langevin function Lv(Y) ) coth(Y) - 1/Y is introduced. For simplicity, it is assumed that the TAP charge and PC dipoles are located inside the
[
] ( )
nkBT ∂ n (uq-d + ud-d) ≈ × ∂A n + 1 n+1 d ln(Lv) pg sin(R) 8YLv(Y) Lv(Y) (2) 1+ 12 d ln Y a πa
τad ≈ -
[
][
][
]
The geometrical factor g is determined by the specific two-dimensional structure formed by lipid molecules; e.g., assuming a hexagonal lattice, g ) 7/4. The entropic term in eq 2 is neglected because uq-d and ud-d are relatively weakly dependent on T at Y ) ∼5 to 10. Intracluster PT bonds are shorter than bonds outside of the cluster. This results in a decrease in the cluster area Scl, compared with the area of n +1 molecules outside of the cluster. Scl can be calculated from the number of shortened bonds kPT and the number of “normal” PC-PC (PP) bonds kPP. Taking into account that in any pure uniform membrane the proper fraction of area per molecule A per bond is 2A/n, we calculated the cluster area Scl as follows
2 Scl ) (kPTAsh + kPPAPP) n
(3)
Ash and APP are the contributions of shortened PT and intracluster PP bonds, respectively, to the membrane area. kPT ) n for χ < χ0 ) 1/(n + 1) when there are sufficient free PCs and each TAP is surrounded by only PC molecules; i.e., the probability of formation of TAP-PC-TAP is lower than that of TAP-PC-PC. Assuming that the contribution APP of PP bonds in eq 3 is the same as APP of similar bonds involving free PC molecules, and taking into account that the total number of the latter type of bonds is [1 - χ/χ0] N, we calculated the area per molecule of the total membrane as follows:
(
〈A〉 ) χScl + 1 -
)
χ A ) 2χAsh + (1 - 2χ)APP (4) χ0 PP
In eq 4, the total number of PP bonds in the membrane is as follows:
[
(
)]
n n χ mn ) mPP ) (1 - 2χ)N ) χkPP + 1 N 2 2 χ0
(5)
The area per molecule outside the cluster An ) APP is
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determined only by electrostatic interaction between TAP molecules and basic membrane interactions such as van der Waals forces and hydrophobic interactions. Following the well-known Israelachvilli’s opposing forces model,4 which interpolates of the internal energy of lipid bilayer by means of attraction and repulsion terms, we calculated An outside polarized cluster
An ) A0/(1 - τel/γ)1/2
(6)
where A0 is the average area per molecule in a pure PC bilayer; γ ) 0.05 N/m is the interfacial free energy per unit area of the bilayer-water interface. Using a uniform surface charge model,5 we estimated the electrically induced membrane tension τel as
τel ) 2χ
Q - 1 kBT R A0
(7)
where R ) 2χ(e /π0wkBT)/a κ and Q ) R + 1, and κ is the inverse Debye screening length of the solution. Ash is the contribution of PT bonds to the total mechanical tension of the membrane. In Israelachvilli’s approach4 mentioned above, it is calculated as 2
2
2
2
Ash ) A0/(1 - (τel + τad)/γ)1/2
(8)
where τad is calculated using eq 2. Ash can be treated as the area per molecule in some hypothetical membranes that have only TAP-PC bonds. Equation 4 is valid for cases in which χ < χ0 ) 1/(n + 1) when there are enough PC molecules so that all TAP molecules can form “correct” clusters consisting of 1 central TAP and n peripheral PC molecules. If the TAP fraction increases, χ0 clusters can become superimposed on each other. This situation can be modeled in the following way. We assume that the number of clusters is constant and equal to Nχ0 for any χ > χ0. But there is at least one TAP molecule among the peripheral molecules of those clusters, in addition to the central TAP. The apparent cluster area is determined by the number of shortened PC-TAP bonds, as described above. The distribution of bonds is random. The probability of finding specific bonds in specific places does not depend on the neighboring bonds. Therefore, to calculate the number of proper bonds, we can use the common lattice treatment of the mixed membrane, in which each cell is occupied by TAP or a PC molecule. The probability that a particular cell is occupied by a PC molecule is (1 - χ). The number of TAP molecules among its neighbors is nχ; i.e., averaged PC can form nχ PT bonds. The total number of these bonds in the membrane is mPT ) nχ(1 - χ)N, and the total number of normal bonds is mn ) (nN/2) - mPT ) {n[1 - 2χ(1 - χ)]N}/2. Averaged 〈A〉 is calculated as follows:
〈A〉 )
2 (m A + mnAn) ) 2χ(1 - χ)Ash + nN PT sh [1 - 2χ(1 - χ)]An (9)
An and Ash are calculated using eqs 6 and 8 as described above. Discussion Figure 3 shows 〈A〉 as a function of the TAP mole fraction χ (solid lines). The curve is the result of calculations using eqs 6-9. The results of ref 2 are also shown. Our theoretical curve exhibits good agreement with numerical simulations. (4) Israelachivili, J. N. Intermolecular and surface forces, 2nd ed.; Academic Press: New York, 1992. (5) May, S. J. Chem. Phys. 1996, 105, 8314.
Figure 3. Area per molecule 〈A〉 as a function of the DMTAP mole fraction χ. The curves are a result of the calculations in accordance with eqs 6-9 for a ) 0.9 nm, A0 ) 0.656 nm2, in ) 10, n ) 6, γ ) 0.05 N/m, p ) 0.25a, h ) a, and κ ) 3 nm-1. (The values An ) 0.71 nm2 for the pure DMTAP bilayer at χ )1 (from ref 1) correspond to κ ) 3 nm-1 in our model.) The dependence of P-N vector tilt on TAP fraction R(χ) was obtained from ref 2. Open squares with error bars represent the results of molecular dynamics simulations.2 The dashed line displays the calculations of 〈A〉 “without electrostatics”. The dotted line displays the calculations of 〈A〉 with fixed PC molecule P-N vector tilt R ) 20°.
The dashed line in Figure 3 shows the dependence of 〈A〉 on χ “without electrostatics” when τel ≡ 0 in eqs 6 and 8. The qualitative behavior of that curve is the same as the behavior of the main curve. The electrostatics affects the quantitative results, particularly by shifting the point χmin at which 〈A〉 is at minimum from χ ) 0.5, at which the number of shortened bonds mPT reaches its maximum value. However, this displacement is sufficiently small. Equation 9 shows that ∂(mPT)/∂χ is the main contributor to ∂〈A〉/∂χ. One can easily obtain the value χmin ) 0.5 + o(1) ≈ 0.5. It was obtained2 that the tilt of the P-N vector changes from 80° for fraction χ ) 0 to 20° for χ ) 0.75. To clarify the role of this reorientation, we calculated 〈A〉 using a fixed value of R ) 20° (dotted line in Figure 3). The difference between two lines is negligible. This indicates that the vertical reorientation of the P-N vector itself does not directly affect the mechanical state of the membrane. A similar conclusion follows from the comparison between Figure 3 and Figure 8 in ref 2. The present findings suggest the following scenario. TAP molecules induce polarization of neighboring PC molecules. This polarization causes the thermally averaged lateral component of the cluster PC P-N vector to be directed outward from the cluster center. Such an orientation results in attraction of PC to the central TAP and consequent local compression of the membrane inside the clusters. The area per molecule inside a cluster is less than the area per molecule outside the cluster. The apparent area of clusters is constant for any value of χ e 1/(n + 1). The average area per molecule of the total DMTAP/DMPC bilayer 〈A〉 decreases with χ growth due to the increase in the number of clusters. In the range χ > 1/(n + 1), the total number of clusters remains constant at N/(n + 1), but the clusters are deformed due to superimposition of clusters on each other. The mechanical state of the membrane (as well as its molar area) is determined by the number of shortened PC-TAP bonds. The length of this bond and other bonds depends on longrange TAP-TAP electrostatic interaction. However, longrange TAP-TAP electrostatic interaction affects the total
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mechanical state weakly compared with the effect of local compression due to short-range TAP-PC interaction. Thus, the role of TAP is to polarize neighboring PC molecules and to generate shortened TAP-PC separations. The total number of TAP-PC separations determines the mechanical state of the membrane and 〈A〉 of the charged membrane. In the range χ < χmin ≈ 0.5, the number of PC-TAP bonds increases as χ increases, and the increase in χ determines the decrease of 〈A〉. In the range χ > χmin, the number of PT bonds, which determines 〈A〉, decreases. In the framework of our model, the vertical reorientation of the P-N vector is a consequence of membrane compression, rather than its cause. The polarization of cluster PC molecules not only decreases the distance between these TAP and PC molecules but also decreases the separation between cluster PC molecules. Due to the increase in steric interaction between neighboring cluster PC molecules, the vertical tilt of the P-N vector increases. However, this only happens inside clusters. PC molecules outside of clusters have the same tilt as those in a neutral membrane. Because clusters do not interact, the tilt is constant in the range χ < 1/(n + 1). In the range χ > 1/(n + 1), intercluster interaction causes the P-N vector to become more vertical. For χ ) n/(n + 1), all neighbors of each PC are TAPs. The tilt P-N vector reaches its
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maximum value. A further increase in χ does not change the tilt. A closer inspection of Figure 8 in ref 2 reveals features that are consistent with this scenario. It is seen that the P-N vector tilt of PC molecules neighboring TAP is constant for χ < χ1 ≈ 0.16 and χ > χ2 ≈ 0.8, which correspond quite well to two characteristics points in our model, e.g., χ1 ) 1/(n + 1) ) 0.17 and χ2 ) n/(n + 1) ) 0.83 for n ) 5. We have considered here one phenomenon which may play the important role in positively charged lipid membrane. Of course, it is not a unique phenomenon which determines the mechanical state of this membrane. The other ones (such as variations of the surface hydrations, change of the hydrophobic chain tilt etc.) are also involved in the reaction of the lipid bilayer on the surface charges. The additional analysis is needed to clarify the conditions determining the leading role of the specific phenomenon. Acknowledgment. V.L. thanks JSPS (Japan) for the financial support through a grant for invited scientists (L03545). This research was partly supported by a Grantin-Aid for General Scientific Research from the Ministry of Education, Science, and Culture (Japan) to M.Y. We appreciate Miss Ryoko Sano for drawing Figures 1 and 2. LA046976X
