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Langmuir 1996, 12, 3498-3502
Possibility of Direct Experimental Check Up of the Theory of Repulsion Forces between Amphiphilic Surfaces via Neutron and X-ray Diffraction V. I. Gordeliy* Forschungszentrum Ju¨ lich, IBI-2: Biologische Strukturforschung, D-52425 Ju¨ lich, Germany Received May 25, 1995. In Final Form: March 22, 1996X The paper proposes a way to analyze the entropic contribution to the balance of intermembrane forces. The relationship between the amplitude of out-of-plane fluctuations of lipid molecules UT at different intermembrane distances and the decay length λ of entropic forces has been derived. It is shown that the value of these fluctuations UT, calculated in accordance with the theory of J. Israelachvili and H. Wennerstro¨m (Langmuir, 1990, 6, 1873-876), only depends on the intermembrane distance D and the decay length λ, the parameters determined in the experiment. Moreover, with the increase of the distance D the value of these fluctuations is rapidly approaching the value of the decay length λ (for instance, at D ) 8 Å and λ ) 1 Å, UT ) 0.97 Å). The paper is offering an analysis of the determination of the amplitude of lipid molecules UT from neutron and X-ray diffraction data. If entropic forces are dominating, the values of UT, determined from neutron and X-ray measurements and those predicted by the theory must be the same. The known neutron and X-ray diffraction data (J. Mol. Biol. 1979, 134, 673-691. Biophys. J. 1992, 61, 434-447) are being discussed. It is shown that they confirm the existence of out-of-plane thermal fluctuations of lipid molecules, which are, at least in the liquid phase, sufficient to contribute significantly to intermembrane short-range repulsion.
Introduction The so-called “hydration” (or structural) forces, acting between two hydrophilic surfaces of amphiphilic structures (for instance, lipid membranes), are dominating over the distance of up to 30 Å and decay exponentially with a characteristic decay length of 1-3 Å.3 Quite recently it has been established that these forces play an important role in the interaction and fusion of cells, biological membranes4 and macromolecules,5 as well as in the stability of colloids and wetting films.6 Until now these forces have been regarded as the result of the indirect interaction of surfaces via solvent caused by the fluctuations of an order parameter of a certain type (polarization, density, etc.), determined by the structure of the surface,1,7-9 and the last several years have seen some progress in the theoretic understanding of the possible mechanisms of these forces. * Present address: Forschungszentrum Ju¨lich, IBI-2: Biologische Strukturforschung, D-52425 Ju¨lich, Germany. Telephone: 49-2461-615874. Fax: 49-2461-612020. E-mail: valentin@ ibistr.dnet.kfa-juelich.de. Permanent address: Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Dubna, Moscow District, Russia, 141980. X Abstract published in Advance ACS Abstracts, June 1, 1996. (1) Kornyshev, A. A.; Leikin, S. Phys. Rev. A 1989, 40 (11), 64316437. (2) Israelachvili, J. N.; Wennerstro¨m, H. Langmuir 1990, 6, 873876. (3) Rand, R. P.; Parsegian, V. A. Biochem. Biophys. Acta 1989, 988, 351-376. (4) Gruen, D. W. R.; Marcelija, S.; Parsegian, V. A. In Cell Surface Dynamics; Perelson, A. S., Delisi, C., Wiegel, F. W., Eds.; Marcel Dekker: New York, 1984; p 59. (5) Rau, D. C.; Lee, B. K.; Parsegian, V. A. Proc. Natl. Acad. Sci. 1984, 81, 2621. (6) Churaev, N. V.; Derjaguin, B. V. J. Colloid Interface Sci. 1984, 103, 542. (7) Marcelia, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (8) Cevc, G. Proceedings of the Nobel Conference on Hydration Forces and Molecular Aspects of Solvatation, Ørena¨s, Sweden, 1984; Chem. Scr. 1985, 96. (9) Belaya, M.; Feigelman, M.; Levadny, V. G. Chem. Phys. Lett. 1986, 126, 361. (10) Gordeliy, V. I. JINR Comm., 1991, Dubna.
S0743-7463(95)00406-9 CCC: $12.00
A few years ago there was published an alternative theory of J. Israelachvili and H. Wennerstro¨m,2 where “hydration” forces are not caused by the surface disturbing the water structure but come as a result of entropic (osmotic) repulsion of molecule polar groups, belonging to the neighboring membranes, which are sufficiently agitated thermally to leave the rather liquid membrane surface and directly interact with each other. In fact, these forces are steric, resulting from direct interaction of amphiphilic molecules. This paper will show an experimental way to study the entropic contribution to short-range forces between membranes. Amplitude of Out-of-Plane Thermal Fluctuations of Lipid Molecules and Its Dependence on Intermembrane Distance Theory2 is based on the assumption of the existing fluctuation of lipid molecule positions in the direction normal to the bilayer plane and the existing linear dependence of the potential of these fluctuations on the value of the amplitude of these fluctuationssthat is a shift of a molecule a certain distance away from the membrane plane Z:
ν ) RZ
(1)
where R is a constant, characterizing the corresponding interaction (“interaction parameter”). It follows from eq 1 that the probability density of molecular fluctuation at the distance Z from the membrane plane equals
P(Z) ) A exp[-RZ/kT]
(2)
where A is a constant which can be determined from the condition of normalizing P(Z). Now, let us determine P2(Z)sthe probability density of the fluctuations at the distance z of the molecule, belonging to membrane 1, with the presence of a second membranes at the distance D from the first membrane. P2(Z) is equal to the probability density P(Z) multiplied by the probability © 1996 American Chemical Society
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of the opposite molecule belonging to membrane 2 being placed at the distance Z1 form membrane 2, which is smaller than D - Z. It follows then that
∫0D-Zexp[-RZ1/kT] dZ1 )
P2(Z) ) A2 exp[-RZ/kT]
KT 2 A exp[-RZ/kT]{1 - exp[-R(D - Z)/KT]} (3) R In theory2 KT/R is nothing more than the decay length λ of the forces
PH ) P0 exp[-D/λ]
(4)
Determined from the normalizing condition
∫0∞P(Z) dZ ) 1
(5)
coefficient A equals 1/λ. And then eq 3 can be written
{
[
P2(Z) ) 1/λ exp[-Z/λ] 1 - exp -
]}
D-Z λ
(6)
Equation 6 makes it possible to calculate the average value of fluctuations Z
∫0DP2(Z)Z dZ ) 〈Z〉 ) ∫0∞P2(Z) dZ λ2 - [Dλ + λ2 + D2/2] exp[-D/λ] (7) λ - (λ + D) exp[-D/λ] It follows from eq 7 that at D/λ . 1
〈Z〉 ) λ
(8)
Now let us determine the fluctuation value of lipid molecules with respect to their average position 〈Z〉. Dispersion, i.e. the root-mean-square displacement of Z from 〈Z〉, is equal to
UT ) {〈Z2〉 - 〈Z〉2}1/2
(9)
where
∫0DP2(Z)Z2 dZ ) 〈Z 〉 ) ∫0DP2(Z) dZ 2
2λ3 - (D2λ + 2Dλ2 + 2λ3 + D3/3) exp[-D/λ] (10) λ - (λ + D) exp[-D/λ] It should be noted that it follows from eq 10 that at D/λ .1
〈Z2〉 ) 2λ2
(11)
with 〈Z2〉 and 〈Z〉 in eq 9 from eqs 10 and 9 we get
{ ( (
)
D3λ -D/λ uT ) λ4 - 2λ4 + 2Dλ3 + e + 3 1/2 1 5 λ4 + λD3 + D2λ2 + 2λ3D + D4 e-2D/λ 3 12 [λ - (λ + D)e-D/λ] (12)
) }/
From the last equation it becomes evident that at large enough intermembrane distances D (at D/λ . 1) the following holds true:
uT ) λ
(13)
Figure 1 show 〈Z〉 and uT dependences on intermembrane distance D at λ equal to 1 and 2 Å. Most of the values, determined from the measurements of hydration forces, are between 1 and 2 Å.3 And already at D ) 6 Å for λ ) 1 Å and at D ) 15 Å for λ ) 2 Å both 〈Z〉 and uT differ from λ by not more than several percent. Possibility of Experimental Check Up of Prediction of Theory2 via Neutron and X-ray Diffraction The direct experimental check up of theory2 would be the measuring of lipid molecule position fluctuations in the direction normal to the bilayer plane. Indeed, it is from the existence of such fluctuations that the theory in question draws the dependence of intermembrane repulsion forces on the distance between membranes. More than that, as was demonstrated earlier in the previous part, there exists a direct connection, defined by eqs 7 and 12, between the average value of fluctuations 〈Z〉, dispersion uT, and parameters D and λ (and only these parameters) of theory,2 determined from the experiments on measuring hydration forces.3 Besides, at rather big D and 〈Z〉 uT is just equal to the hydration force decay length λ. Neutron diffraction is the most direct method of investigating the lipid membrane structure.11 As has been demonstrated in papers12,13 on the investigation of 1,2dipalmitoyl-sn-glycero-3-phosphatidylcholine (DPPC) membrane structure, with the help of deuterium labeling one can determine with high accuracy not only the label position but also the width of its distribution relative to its average position in the direction normal to membrane plane. Another method, allowing us to determine with high accuracy the molecular groups’ positions and their distribution width, was worked up in refs 17-19. It comprises the bilayer structure determination by the combined use of X-ray and neutron diffraction. The structural parameters of 1,2-dioleoyl-sn-glycero-3-phosphatidylcholine (DOPC) membranes in the liquid phase at relative humidity RH ) 66% and temperature T ) 23 °C were determined by fitting of the experimental structure factors by a quasimolecular model: multi-Gaussian representations of bilayers. The scattering density distribution of molecular groups in membranes in both cases (refs 12-13 and 17-19) was represented by the Gaussian function
[
Fmg(x) ∼ exp -
]
(x - 〈x〉)2 ν2
(14)
where 〈x〉 is the average position and ν is the width of the molecular group distribution. The distribution width of the deuterium label ν12,13 is determined both from its own size (hard-sphere radius) and the thermal interlayer (11) Worcester, D. L. In Biological Membranes; Chapman, D., Ed.; Academic Press: New York, 1976; Vol. 3, pp 1-46. (12) Bu¨ldt, G.; Gally, H. U.; Seeling, J. J. Mol. Biol. 1979, 134, 673691. (13) Zaccai, G.; Bu¨ldt, G.; Seelig, A.; Seelig, J. J. Mol. Biol. 1979, 134, 693-706. (14) Lis, L. J.; McAlister, M.; Fuller, N. L.; Rand, R. P.; Parsegian, V. Biophys. J. 1982, 37, 657-666. (15) Pearson, L.; Pasher, I. Nature (London) 1979, 281, 499. (16) Brosio, G. F.; Di Nola, A.; Koraev, A. L. J. Theor. Biol. 1977, 67 (2), 319-334. (17) Wiener, M. C.; White, S. H. Biophys. J. 1991, 59, 162-173. (18) Wiener, M. C.; White, S. H. Biophys. J. 1991, 59, 174-185. (19) Wiener, M. C.; White, S. H. Biophys. J. 1992, 61, 434-447.
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them not in excess water, as it may seem at first sight, but rather at smaller hydrations, and the trivial calculations of the intermembrane distance (hydration) dependence of uT given above were indispensable. It is the obtained equation that makes it possible to evaluate from the measured values of uT at small hydrations (in line with theory2) the amplitude of these fluctuations in excess water, and this value allows us to assess the contribution of local motions of the membrane surface to intermembrane interactions. It should also be noted that intermembrane distance D is the distance between thermally nonexcited membranes. The real distance between membrane surfaces equals D - 2〈Z〉 and it approximately corresponds to the dcγ distance between cγ labels minus the size of the choline group N(CH3)3 lCHOL in the direction normal to the membrane plane.12 Assuming that the polar head conformation in membranes does not differ much from that of lipid crystals,12,15 one can make use of structural parameters, determined via X-ray diffraction on lipid crystals15 and also in ref 16. The choline group size in this case may be evaluated as
lCHOL = 2{lNC cos(φ/2) + lCHx1 - 16/9 sin2(ψ/2) sin2 60 + rH} (17) Figure 1. (a) Dependence of the average value of lipid molecules’ fluctuations 〈Z〉 in the direction normal to the membrane plane on intermembrane distance D for the two decay lengths of hydration forces λ ) 1 and 2 Å. (b) Same kind of dependence for dispersions uT of these fluctuations.
motion of the label along the normal of the membrane plane as well.17-19 It would not be the case if the bilayer packing in a multilayer structure had a second kind of disorder.20 Yet, as demonstrated in ref 17 the contribution of this type of disorder of packing of a multilamellar structure is negligibly small in the case of phospholipid membranes in the liquid phase. Our study of the second kind of disorder in multilayers of egg lecithin in the liquid phase at relative humidities 85, 97, and 100% supports this statement (Gordeliy et al. To be published). Thus, in phospholipid multilayers from the distribution width of the deuterium label ν one can determine the amplitude of thermal fluctuations νT;17-19
νT ) xν2 - νH2
(15)
where νH is the hard-sphere radius. The root-mean-square amplitude of thermal motions uT is connected with νT by the equation19
uT ) νT/x2
(16)
In the general case uT is the total contribution of individual out-of-plane fluctuations and collective out-of-plane motions due to membrane undulations.21 In ref 19 it was shown that the amplitude of membrane undulations in multilayers at least at small hydration is small. Undulation forces are long-range by nature,21,22 and their contribution to intermembrane interaction will be rapidly decreasing with a decrease of intermembrane distance.12 Thus, to determine the amplitude of individual motions of the membrane surface, it is necessary to investigate (20) Blaurock, A. E. Biochim. Biophys. Acta 1982, 650, 167-207. (21) Helfrich, W. Z. Naturforsch. 1978, 33a, 305-315. (22) Evans, E. A.; Parsegian, V. A. Proc. Natl. Acad. Sci. USA 1986, 83, 7132-7135.
where lNC and lCH are the lengths of the N-C and C-H bonds, φ and ψ are the valence angles C12-N-CH3 and H-C-H, respectively, and rH is the molecular radius of hydrogen. Substituting from ref 16 lNC ) 1.5 Å, lCH ) 1.1 Å, φ ) 109.5°, ψ ) 109°, and rH ) 1.2 Å in eq 17, we get lCHOL ) 4.8 Å. The equation D - 2〈Z〉 ≈ dcj - lCHOL can be rewritten
D = dcγ - lCHOL + 2〈Z〉
(18)
Thus the values of the distribution widths (along the normal of the bilayer plane) of molecular groups of lipid molecules on the membrane surface ν, determined in refs 13, 14, and 19, make it possible to calculate with the help of eqs 15 and 16 the values of the root-mean-square amplitude of out-of-plane thermal fluctuations uT. The projection of “hard-sphere” radii of molecular groups on bilayer normal νH was calculated from the literature data.15,16 On the other hand, we use the derived equations (eqs 12 and 18) to calculate (from the experimental values of the decay length of hydration forces λ and the intermembrane distance (eqs 13 and 20) the root-mean-square amplitude of these fluctuations uτ. Comparison of the value calculated in this way uτ with uT, determined from experimental data,13,19 will provide a possibility of estimating the contribution of short-range (noncollective) outof-plane fluctuations of lipid molecules in the magnitude of short-range intermembrane forces.2 The “determined” uT and “calculated” uτ values for different parts of the lipid polar head for membranes from DPPC and DOPC in the liquid phase are collected in Table 1. Discussion Additional Out-of-Plane Thermal Fluctuations of the Lipid Polar Head Itself and Their Contribution to the Balance of Intermembrane Forces. As becomes evident from Table 1B the amplitude uT of the out-ofplane fluctuations of the membrane surface depends on the type of molecular group of the lipid polar head. The value uT is bigger in the choline group (2.2 Å) than that
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Table 1. Calculated Parameters of Lipid Membranes in the Liquid Phase molecular group of the lipid molecule CH2 group of polar head (β-position) choline group PO4 group glycerol group
νa (Å)
membrane parameters νHb (Å) uTc (Å) uτd (Å)
A. DPPC Membranesc 3.6 ( 0.6 0.9 2.5 ( 0.4 B. DOPC Membranesf 3.5 ( 0.5 1.5 2.2 ( 0.3 3.1 ( 0.1 1.8 1.8 ( 0.1 2.5 ( 0.3 1.8 1.2 ( 0.2
1.6 ( 0.2
1.4 ( 0.2
a 1/e - width of molecular group distribution. b Projection of hard-sphere radius (for neutron scattering). c Root-mean-square amplitude of out-of-plane fluctuations. d Predicted by ref 2 rootmean-square amplitude of out-of-plane fluctuations. e At 70 °C; 10% (w/w) water; the repeat distance d ) 50.8 Å, the intermembrane distance D ) 7.6 Å,13 and the decay length of short-range forces λ ) 2.1 Å (at 50 °C).3 f At 23 °C; 12% (w/w) water; d ) 49.1 Å; D ) 5 Å;19 and λ ) 2.1 Å.3
in the phosphate one. The same value for the CH2 group of the DPPC polar head, which finds itself close to the choline group, is equal to 2.5 Å. It confirms the previous conclusion. It can be accounted for by the freedom which the choline group has and, primarily, because of the small energetic rotation barrier of the choline group around the P-O bond (Eb ∼ 5 kcal/mol).23 It should be noted that these additional out-of-plane fluctuations of the polar head itself (i.e. in relation to the whole of the lipid molecule) have a different nature. For instance, they are obviously limited by the size of the polar head of the lipid molecule. It means that the contribution of the polar head to the repulsion of membranes will begin to manifest itself at intermembrane distances smaller than its size (∼8 Å). From the amplitude values for the choline and phosphate groups (Table 1A) one can evaluate the root-mean-square amplitude for the choline group out-of-plane fluctuations relative to the whole of the lipid molecule. This rootmean-square amplitude is equal to uChol ) 1.2 ( 0.3 Å. While we do not have any model of this kind of polar head motions, it is impossible to exactly evaluate the decay length of the very short-range force component due to the independent motions of the polar head. However, both the values uChol and λs must be close. Indeed, it was shown experimentally that upward breaks in the pressure-distance dependences occur at intermembrane distances =5-6 Å for bilayers having phosphatidylcholine head groups in both the liquid and gel phases.24-28 The calculated decay length λsh of the intermembrane forces at these short distances is estimated at ∼0.6-0.8 Å.26-29 Thus, the amplitude value of the out-of-plane motions of the choline group itself, calculated above, is in agreement with the decay length λs. It is direct evidence of the interpretation of the (very) short-range repulsive force component proposed earlier by McIntosh and Simon.26-29 Short-Range Entropic Forces due to Out-of-Plane Thermal Fluctuations of Lipid Molecules. It is the motion of the phosphate group that reflects the motions (23) Gupta, S. P.; Govie, G. FEBS Lett. 1972, 27 (1), 68-70. (24) Ko¨nig, S.; Pfeiffer, W.; Bayerl, T.; Richter, D.; Sackmann, E. J. Phys. II 1992, 2, 1589-1615. (25) Pfeiffer, W.; Henkel, Th.; Sackmann, E.; Knoll, W.; Richter, D. Europhys. Lett. 1987, 8 (2), 201-206. (26) McIntosh, T. J.; Simon, S. A. Biochemistry 1993, 32, 83748384. (27) McIntosh, T. J.; Magid, A. D.; Simon, S. A. Biochemistry 1987, 26, 7325-7332. (28) McIntosh, T. J.; Magid, A. D.; Simon, S. A. Biochemistry 1989, 28, 7904-7912. (29) McIntosh, T. J.; Simon, S. A.; Needham, D.; Shany, G. H. Biochemistry 1992, 31, 2020-2024.
of the lipid molecule, which manifest themselves on the membrane surface. The root-mean-square amplitude of the out-of-plane fluctuations of the phosphate group is equal to 1.8 ( 0.1 Å (Table 1B). This value is in excellent agreement with the prediction of theory2 (see Figure 1; uT ) 1.4 ( 0.2 Å for DOPC membranes and uT ) 1.6 ( 0.2 Å for DPPC membranes). Thus, out-of-plane thermal fluctuations of the membrane surface are large and contribute to short-range repulsive forces between lipid bilayers in the liquid phase. The decay length of this entropic component of the forces is close to that (λ ) 2.1 Å) which has been observed experimentally.3 The dramatic changes in the lamellar diffraction patterns (X-ray and neutron) of lipid membranes at the gel-liquid phase transition provide evidence of individual out-of-plane thermal fluctuations of lipid molecules.12-13,17-19,etc. Incoherent quasielastic neutron scattering measurements led to the same conclusion.24,25 The calculated amplitude of these motions strongly depends on the model applied for the treatment of IQENS data.24,25 Nevertheless, it is worth mentioning here that, if the interpretation of such data is based on the simultaneous occurrence of out-of-plane diffusive motions of the lipid molecule (all protons are assumed to be equivalent), the amplitude is equal to ∼2 Å for DPPC membranes at T ) 60 °C and 12% (w/w) water concentration.24,25 The measurements of isothermal compressibility χT of lipid membranes from dimyristoylphosphatidylcholine showed a considerable increase of lipid volume fluctuations at the gel-liquid phase transition.30 The χT values are equal to (4.7 ( 1.7) × 10-5 bar-1 and (13.1 ( 1.3) × 10-5 bar-1 for the gel and the liquidcrystalline phase, respectively.35 It means that the rootmean-square value of the lipid molecule volume VL fluctuations (〈∆V2L〉 ) KBTVLχT31) increases at the phase transition by a factor of 1.7. The mean field lattice theory32 and the Monte Carlo computer simulations33 also resulted in the conclusion about large-amplitude out-of-plane surface fluctuations of membranes. Two general questions also have to be discussed here: (i) Is it correct to extrapolate the data at low humidities over a higher range of hydrations? Of course it would be of primary importance to have experimental data about the amplitude of out-of-plane thermal fluctuations in a wide range of hydrations. Unfortunately, only the data which have been discussed here exist in the literature for the moment. However, it does not mean that the data at small hydrations are not useful at all for the analysis. There are a few arguments in favor of the extrapolation of the existing data to higher humidities. First and foremost, both DPPC and DOPC membranes at increased humidity (under the unchanged other external conditionsssee the paper) remain in the same phase state. It means the absence of qualitative (dramatic) changes in the dynamics of these membranes with increasing humidity. And then, with membrane hydration increasing, the amplitude value of the fluctuations will, of course, be changing quantitatively. How it will be changing is one (30) Bo¨ttner, M.; Ceh, D.; Jacobs, U.; Winter, R. Z. Phys. Chem. 1994, 184, 205-218. (31) Landau, L. D.; Lifshitz, E. Statistical Physics; Pergamon Press: London, 1959. (32) Leermarkes, F. A. M.; Scheujens, J. H. M. J. Chem. Phys. 1988, 89, 3264, 6912. (33) Egberts, E. Thesis, Chapter IV, University of Groningen, 1988. (34) Cevc, G. J. Chem. Soc., Faraday Trans. 1991, 87 (17), 27332739. (35) Bo¨ttner, M.; Winter, R. Biophys. J. 1993, 65, 2041-2046.
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of our concerns in the paper. It has been looked at within the formalism (model) of J. Israelachvili and H. Wennerstro¨m. But it is this model that is considered in the paper. The conclusion which has been drawn about the growing amplitude of the fluctuations with the intermembrane distance (hydration) increasing, for instance, is in agreement with a well-known experimental fact: the rapidly decreasing integral intensities of diffraction reflexes with a growing number of reflection order at increased membrane hydration. Last but not least, it is generally agreed that at higher hydrations undulation forces begin contributing considerably to the balance of intermembrane forces.22,36 Thus, it is not the region of higher hydrations which corresponds to the dominating role of short-range repulsive forces and which is under discussion.2,3,36 Recent neutron diffraction measurements with DPPC membranes at relative humidity RH ) 96% and temperature T ) 50 °C (i.e. at the hydration 12 water molecules per lipid, which corresponds to the upper limit where short-range repulsive forces can be considered dominating3) are in agreement with what we describe in this paper.37 (ii) Can one make a conclusion on the identity of the two parameters λ and uT based on the similarity of their values? The basis of J. Israelachvili’s and H. Wennerstro¨m’s theory is a hypothesis that out-of-plane fluctuations have sufficient amplitude to result in the experimentally observed value of short-range repulsive forces between membranes. A most important consequence of their model is that the values of the decay length of entropic forces and the amplitude of out-of-plane fluctuations are interdependent. A corresponding equation is derived in our paper. Thus, the decay length of the forces and the amplitude of the fluctuations are the basic parameters to be compared to evaluate the correspondence of the theory and experimental data. Moreover, to characterize the experimentally measured repulsive forces (Ph ) Po exp[-dw/λ], see elsewhere, for instance, ref 3), one generally has to know the value not only of λ but also of Po. However, in the framework of the model under discussion the values of Po and λ are connected by the equation
which is easily determined and usually known. Previously, it was shown that the value of Po calculated from eq 1 does not contradict the experimental data.2 We did not focus our attention in the paper on Po. This is not only because of what we have discussed above. It is well-known now that the interval of experimentally determined values of Po is too wide, in particular, because of the insufficient accuracy of the present methods used for the determination of intermembrane distance (see, for example, ref 26).
Po ≈ 2.7kTλ-1A-1
Acknowledgment. I am grateful to the Alexander von Humboldt Foundation for the support of this work.
(19)
where A is the area per lipid molecule, the parameter (36) Israelachvili, J. N.; Wennerstro¨m, H. J. Phys.Chem. 1992, 96, 520-531.
Conclusions The lipid membrane surface in the liquid phase undergoes out-of-plane thermal fluctuations with a large amplitude. The amplitudes are not the same for different parts of the polar head. They have a growing tendency in the direction from the phosphate to the choline group. The additional amplitude of the choline group, resulting from the motions of the polar head relative to the phosphate group, is equal to ∼1 Å. These additional motions contribute to the intermembrane forces at short intermembrane distances (smaller than ∼8 Å). The amplitude of the PO4 group is about 2 Å, and these motions result in entropic short-range intermembrane repulsive forces with a decay length of about 2 Å. Still another important conclusion can be made: the membrane surface in the liquid phase is not smooth but rough. This fact must be taken into account in theoretical treatments of membranes and in interpretations of experiments and can result in new conclusions about membrane properties.2,34 In particular, it can result in an increase in the liquid phase of the decay length of the hydration component of the forces, as is predicted by Cevc.34 Precise systematic measurements via neutron and X-ray diffraction of out-of-plane thermal fluctuations of lipid molecules at different hydrations are imperative. They seem to be a way to get more complete and exact information about the contribution of different kinds of forces (entropic, hydration, undulation) to the balance of intermembrane interactions. For instance, measurements at the highest hydrations could give additional information about the magnitude of undulation forces in membranes, predicted by W. Helfrich.21,22,26
LA9504063 (37) Gordeliy, V. I.; Bartels, K.; Hauss, T.; Papadopoulos, G.; Syrykh, A.; Watts, A. Bensc Experimental Reports; 1994; p 253.
