1.Phys. Chem. 1986, 90, 190-195
190
faster decay of eaq-at final pHs below 9 may be ascribed to the reaction (e22-)aq H 3 + 0 H2 OH(*) which is known to be fast.22,23With an activation energy of 3.5 kcal and WMconcentrations of H 3 + 0this reaction may become rate determining.
-
+
+
Acknowledgment. T. Johansen and H. Corfitzen are gratefully acknowledged for technical assistance, and E. Bjergbakke for assistance in the computer calculations. The Si analysis were carried out by Toshiaki Ohe. We are indebted to E. E. Larsen for making it possible to attain 300 OC by constructional changes of the cell. Thanks are also due to E. J. Hart and J. Holcman for valuable discussion contributions.
CONDENSED PHASES AND MACROMOLECULES Physical Characteristics of Membranes from Solubility Measurements of Noble Gases Yehuda Katz The National Physical Laboratory of Israel, Hebrew University Campus, Givat Ram, Jerusalem, Israel (Received: January 7, 1985; I n Final Form: May 29, 1985)
We deduce properties of the hydrophobic region of the dimyristoyllecithin membrane using adsorption data of noble gases to this region. The deduction is based on the use of a theory which we developed in a previous article. We determine the effect of nearest neighbors on the magnitude of the intermolecular interactions, and deduce the number of close neighbors. We also determine the dielectric constant in the medium and the magnitude of the fluctuations which the hydrocarbon chains of this region undergo. We show the relation between the attractive energy of the chains and the density of the region. We demonstrate the usefulness of solubility measurements to the understanding of phase transitions in membranes and discuss the factors which have to be considered when solubility measurements are employed in the investigation of membranes.
Introduction The interactions of simple nonpolar solute molecules with solvents are of interest because of the information they provide on the nature of liquids and solutions.’ Many investigations have been made about the interactions of these solute molecules with ~ a t e r , isotropic ~ - ~ organic solvents,68 detergents,”* polymer^,'^,'^ and liquid crystal^.^^ More recently, these studies have been extended to bilayers and biological membranes.*b22 The studies (1) Hildebrand, J. H.; Scott, R. L. “The Solubility of Nonelectrolytes”; Dover: New York, 1964. (2) Frank, H. S.; Evans, M. W. J . Chem. Phys. 1945, 13, 507. (3) Eisenberg, D.; Kauzmann, W. ‘The Structure and Properties of Water”; Oxford University Press: London, 1969. (4) Ben-Naim, A. “Water and Aqueous Solutions”; Plenum: New York, 1974. (5) Franks, F., Mathias, S., Eds. “Biophysics of Water”; Wiley: New York, 1982. (6) Battino, R.; Clever, H. Chem. Rev. 1966, 66, 395. (7) Leo, A,; Hansch, C.; Elkins, D. Chem. Rev. 1971, 71, 521. (8) Wilhelm, E.; Battino, R. Chem. Reo. 1973, 7 3 , I . (9) Mukerjee, P. In “Solution Chemistry of Surfactants”; Mittal, K. L., Ed.; Plenum: New York, 1978; Vol. I . (IO) Tanford, C. “The Hydrophobic Effect: Formation of Micelles and Biological Membranes”; 2nd ed.; Wiley: New York, 1980. ( 1 1 ) Klevens, H. B. Chem. Reu. 1951, 47, 1 . (12) Ben-Naim, A,; Wilf, J. J . Solution Chem. 1983, 12, 671. ( 1 3) Crank, J.; Park, G. S. “Diffusion in Polymers”;Academic Press: New York, 1968. (14) Felder, R. M.; Huvard, G.S. Merhods Exp. Phys. 1980, 16c, 315. (15) Martire, D. E. In “The Molecular Physics of Liquid Crystals”; Luckhurstand, G. R., Gray, G. W., Eds.; Academic Press: New York, 1979. (16) Katz, Y.; Diamond, J. M. J . Membr. Biol. 1974, 17, 101. (17) Requena, J.; Haydon, D. A. Proc. R . Soc. London, Ser. A 1975, 347, 161. (18) Simon, S. A,; Stone, W. L.; Bustc-Lattore, P. Biochim. Biophys. Acta 1977, 468, 378. (19) Miller, K. W.; Hammond, L.; Porter, E. G. Chem. Phys. Lipids 1977, 20, 229
0022-3654/86/2090-Ol90$01 S O / O
on bilayers are of special importance to anesthesiologists, toxicologists, and physiologists because most of the small nonpolar molecules are anesthetic^^^ and several are carcinogen^.^^ These molecules either act directly on the membrane or their action depends on their rate of permeation through membra ne^.^^-^' Two kinds of questions are usually asked when dealing with solubilities: (1) What are the solubilities and the thermodynamic functions of solution of the solutes in the solvent and (2) How does solvent structure influence the solubilities and the related thermodynamical functions. An additional question which is important when dealing with micelles and bilayers is a question about the location of the solutes in these heterogeneous solvent^.^^^^^^^^ Only partial answers have been given to these questions in the case of bilayers. The question which is least answered is the one about the relation between solubility and bilayer structure. This article describes the use of solubilities of noble gases in the dimyristoyllecithin (DMPC) membrane to obtain molecular features and physical properties of the hydrophobic region of the membrane. In this evaluation I use a theory which describes the behavior of noble gases in phospholipid membranes as well as in isotropic organic hydrophobic solvents.30 The characteristic of (20) McIntosh, T. J.; Simon, S. A,; MacDonald, R. C. Biochim. Biophys. Acta 1980, 597, 445. (21) Katz, Y. Biochim. Biophys. Acta 1981, 647, 119. (22) Simon, S. A,; McDaniel, R. V.; McIntosh, T.J. J . Phys. Chem. 1982, 86, 1449. (23) Seeman, P. Pharmacol. Reo. 1972, 24, 583. (24) Snyder, R.; Keusis, J . J. Crit. Reo. Toxicol. 1975, 3, 265. (25) Goldstein, A,; Aronow, L.; Kalman, S. M. “Principles of Drug Action”; Harper and Row: New York, 1969. (26) Diamond, J. M.; Wright, E. M. Annu. Rev. Physiol. 1969, 31, 581. (27) Diamond, J. M.; Katz, Y. J . Membr. Biol. 1974, 17, 121. (28) Rehfeld, S. J. J . Phys. Chem. 1970, 74, 117. (29) Katz, Y.; Hoffman, M. E.; Blumenthal, R. J . Theor. Biol. 1983, 105, 493. (30) Katz, Y. J . Chem. SOC.,Faraday Trans. 1 1985, 81, 579
0 1986 American Chemical Society
Physical Characteristics of Membranes the theory is that it has parameters which describe the behavior of the pure solvent and its influence on solubility. Several considerations guided me in choosing noble gases for this purpose: (1) Noble gases are the simplest of all solutes. They are spherically symmetrical and interact only through the weak well-defined London forces,31a fact which is well recognized by investigators who test the more fundamental aspects of the liquid-state theories by examining liquified noble gases and noble gas mixture^.^^,)^ (2) Theories have been formulated which correlate quantitatively between the physical properties of the solvent and the solvent influence on s o l ~ b i l i t y . ~ *One ~ ~of~ these ~ ~ - ~theories ~ was tested and shown to describe well the behavior of noble gases in DMPC.30 These theories consider intermolecular interactions between mixing entities and ignore other contributions to the partition function.37 (3) Close examination of solubilities to bilayers reveal that factors other than the intermolecular interactions have profound effect on the s o l ~ b i l i t y .The ~ ~ sensitivity of the solubility to such factors as solute size39 and the number of rotational degrees of freedom it has3* complicates analysis and limits the use of the above-mentioned theories to the description of noble gas solubility. (4) Noble gases dissolve preferentially into one region of the bilayer, the hydrophobic region.21 (5) The hydrophobic region is a bulk solvent when the solubility of the noble gases is dealt with, since the dimensions of the gases are small in comparison to the dimensions of the region. It has to be remembered, however, that the hydrophobic region is not isotropic and that the solubilities may vary along the region.29 Because of this we must refer to any property which is derived from noble gas solubility data as reflecting only the average behavior inside the hydrophobic region. In what follows we derive membrane properties from the enthalpies of solution of the noble gases. These include the cohesive energy inside the membrane, the number of CH2groups involved in each interaction, and the field of force which a solute entering the membrane will feel. This kind of information which is important, for example, to the analysis of phase transitions in bil a y e r ~ and ~ ~ to , ~the ~ understanding of transport phenomena in membranes42cannot be furnished by the common methods employed in membrane research.43 Although we carry our analysis on one phospholipid we are able to show that our conclusions are general which means that the derivation may be used for measurements of the mentioned properties in other bilayers as well. We consider our results to be applicable to the analysis of biomembranes because phospholipids are major building blocks of the biological membranes. They are arranged mostly in bilayer lamellae and form the basic structural element and also the permeability barrier of the membrane.44 The Description of the Bilayer We measured solubilities of noble gases into DMPC bilayerz1 and developed a theory which describes these solubilities in terms of solvent structure and solute proper tie^.^^ These provide the experimental and theoretical background for our analysis of the bilayer and for our quantitative evaluation of its physical properties which determine the cohesive energy. Here we summarize shortly the most relevant findings of our previous articles. (31) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. “Molecular Theory of Gases and Liquids”; Wiley: New York, 1965. (32) Stokes, R. H.; Marsh, K. N. Annu. Rev. Phys. Chem. 1972, 23, 65. (33) Reiss, H. Adv. Chem. Phys. 1966, 9, 1 . (34) Pierotti, R. A. J . Phys. Chem. 1963, 67, 1840. (35) Pierotti, R. A. J . Phys. Chem. 1965, 69, 218. (36) de Ligny, C. L.; van der Veen, N. G. J. Solution Chem. 1975.4, 841. (37). Fowler, R. H.; Guggenheim, E. A. “Statistical Thermodynamics”; Cambridge University Press: London, 1965. (38) White, S. H. Nuture 1976, 262, 421. (39) Seelig, J.; Seelig, A. Q.Rev. Biophys. 1980, 13, 19. (40) Marcelja, S. Biochim. Biophys. Acta 1974, 367, 165. (41) Nagle, J. F. J . Chem. Phys. 1973, 58, 252. (42) Trauble, H. J . Membr. Biol. 1971, 4 , 193. (43) Anderson, H . C. Annu. Rev.Biochem. 1978, 47, 359. (44) Martonosi, A. N., Ed. “Membrane and Transport”; Plenum: New York, 1982.
The Journal of Physical Chemistry, Vol. 90, No. I , 1986 191 TABLE I: Standard Enthalpies of Solution and the Characteristic Energy Parameters of the Noble Gases in the Dimyristoyllecithin Membrane”
-AHo,
solute heliumb neon argon krypton xenon
kJ/mol 26.20 14.32 1.80 -2.96 -8.82
N%l? kJ/mol 2.50 14.38 26.89 31.66 31.52
“The cohesive energy parameter for the DMPC membrane is Nq,= 31.21 kJ/mol. bThe results for helium which were derived by using the Barclay-Butler relation are not reliable. The reasons for this deviation has been explained p r e v i o ~ s l y . ~ ~
Adsorption of noble gases to suspensions of DMPC in water gave the standard free energy of solution of these gases in the hydrophobic region of the bilayer A p o = R T In (p/x)
(1)
where Apo is the standard free energy, p is the partial pressure of the gas, and x is its mole fraction inside the bilayer. From these results we obtained the standard enthalpies of solution by using either the general thermodynamical relation AHo = - p a ( A p o / T ) / d T or the empirical Barclay-Butler relation ASo = a bAHo ( a and b being constants). These enthalpies, which are listed in Table I, characterize the solution of the noble gases into the hydrophobic region of the bilayer. We then formulated a theory of gas solubility and tested it using solubility data of noble gases in several hydrophobic solvents and in the DMPC membrane. In this theory we considered the solubility as a process in which solute molecules adsorb into holes which the thermal movement of the solvent molecules has created in the medium. Adsorption of a solute molecule into a hole is accompanied by the creation of a new hole inside the solvent medium. The formation of holes and the accommodation of the solute molecules into the holes are treated as two independent steps. In the first step work is done against the cohesive forces which hold the solvent molecules together. Energy is gained in the second step because of solute adhesion to the solvent. The theory leads to an expression
+
AHo = NA(e11 - eSl) - R T
(2)
which describes satisfactorily the enthalpies of solution of the noble gases in hydrophobic solvents in general and in the phospholipid membrane of DMPC in particular. NA,R , and T have their usual meaning as Avogadro’s number, the universal gas constant, and the absolute temperature, respectively. e l l is the microscopic cohesive energy parameter, which measures the work which is done during the formation of a hole in the liquid. It depends only on the nature of the solvent and is solute independent. esl is the microscopic adhesive energy parameter, which measures the attraction between a solute molecule and the surrounding solvent molecules. Since it characterizes the intermolecular interactions between solute and solvent it depends on the nature of both. The characteristic energy parameters of the noble gas solubility in the hydrophobic region of the DMPC membrane are given in Table I. We have found that when we employ the noble gases as solutes and choose hydrophobic organic liquids or the hydrophobic region of the membrane as solvents then there is a correlation between the adhesive and cohesive parameters. The combining rule CSl
= 2(~1,41’2
(3)
by which the energy parameters are correlated means that any given adhesive parameter can be expressed in terms of the characteristic cohesive parameters of the pure entities. We have shown30that the energy parameters of these simple solutes in the solvent systems which are mentioned above depend on the intermolecular energy between close neighbors and on the number of neighbors surrounding each molecule as follows €11
=
2x11
(4)
192
The Journal of Physical Chemistry, Vol. 90, No. 1, 1986 €,I
= 2ZXSI
(5)
where z is the number of contacts between solvent pairs which are abolished when a hole is made and 22 is the number of interactions which are created when a solute is adsorbed to the hole. xll and xslare the energies of interaction between a pair of close neighbors. The interaction energy of a nonpolar molecule can be described approximately in terms of a Lennard-Jones (6-12) pairwise additive potential. We have shown that to calculate the enthalpy we have to consider only the attractive part of this potential. The repulsive part of the potential has to be considered only when calculating entropies. When calculating entropies we treated the solute molecules as hard spheres confined to rigid spherical holes. The cohesive energy of the solvent has an effect on the magnitude of the entropy because the attraction between the close neighbors determines the dimensions of the hole inside which the solute moves.
Results Before evaluation of the molecular details of the microscopic cohesive energy parameter of the membrane let us consider the general components of the partition function of the membrane. This is needed as a general structure in accordance with which the molecular variables of noble gas solubility will be established and related to each other. Our approach is also useful to the discussion of several events such as phase transitions and the transfer of matter through membranes. The partition function of the hydrophobic regions of membranes has been defined in terms of three energy contribution^.^,^' It is based on experiments on phase transitions in The energy contributions are (1) attractive energies between close chains; (2) steric or hard-core repulsive interactions between chains; (3) energies which are due to the conformational states of the hydrocarbon chains of the bilayer. In what follows we accept these contributions as the only factors which may have an influence on the magnitude of the cohesive energy parameter. To evaluate their relative importance to the formation of holes in the solvent medium, we start our calculations by looking at the parameters which characterize the accommodation of the noble gases into the holes, Le., the adhesive parameters. We then examine the nature of the combining rule of the geometric mean (eq 3) and use it to introduce the outcome of our calculations into the cohesive energy parameter of the membrane. This procedure furnished results which simplify the analysis of the cohesive energy parameter and the relative importance of the different factors which determine its properties. In the case of noble gas solute, the adhesive parameter depends only on the intermolecular attractions between solute and solvent. These features appear because solute noble gases and holes that already exist in the solvent medium have no internal degrees of freedom which may change when solution take place. Further examination shows that the combining rule of the geometric mean is also a relation between intermolecular attraction^.^^,^^ Since both the adhesive parameter and the combining rule deal only with intermolecular attractions it follows that any combination of them is also a function of the attractive forces and the attractive forces alone. Thus all we have to do to obtain the attractive component of the cohesive energy parameter catt, is to measure the adhesive energy parameters characterizing noble gas solubilities and transforming them by using the combining rule. The total cohesive interaction will then be given by the sum €11
=
t
+ Cattr
=
€
+ tsl/4E,,
(6)
where t stands for the energy contributions to the cohesive parameter which are not due to intermolecular attractions. (45) Chapman, D.;Williams, R. M.; Ladbrooke,B.D. Chem. Phys. Lipids 1967, 1, 445.
(46) Engelman, D. M.; J . Mol. Biol. 1970, 47, 115. (47) Wilkins, M. H. F.; Blaurock, A. E.; Engelman, D. M. Nature 1971, 230, 1 2 . (48) London, F. Discuss. Faraday SOC.1937, 8
Katz It can be easily seen from the above discussion that the energy of cohesion, Le., the work of hole formation in the hydrophobic region of the membrane, comes mostly, if not solely, from intermolecular attractions between close hydrocarbon chains. The effect of the conformational changes and the repulsive interactions shows up to be insignificant and negligible. This can be easily seen when eq 6 is compared with the experimental finding (eq 3). While this conclusion is well established for isotropic organic solvents1 it is quite surprising to find that it holds also for membranes. It is worth looking further into this surprising result since it is of biological significance and since it is relevant to the analysis of many experimental results, such as experiments on phase transitions in membranes. We therefore combine eq 2 and 6 to calculate the cohesive energy parameter from the experimental standard enthalpies of solution, using the parameter e as a free parameter to which arbitrary numerical values are assigned. Negative values represent contributions that act to repel the chains from each other. Positive values mean that extra energy, besides the attractive intermolecular energy, has to be invested in order to overcome existing energy barriers. The outcomes of these calculations using the standard enthalpies of solution of the noble gases argon and krypton in the DMPC membrane are presented in Figure 1. Inspection of this figure reveals that the calculation is sensitive to the existence of extra factors. These factors may have a significant influence on the magnitude of the cohesive energy parameter but the actual effect is rather small and the results are those expected to be found when the major contributions to membrane cohesion come from intermolecular attractions. Another interesting conclusion which can be obtained by inspecting Figure 1 is that the cohesive energy parameter assumes a numerical value which is different from zero, meaning that work has to be done in order to create a vacancy in the membrane. This fact reveals the dynamic nature of the membrane and indicates that vacancies in the membrane are caused by thermal fluctuations which carry neighbor molecules away from each other. If the holes were rigid and the thermal fluctuations were frozen in the membrane then the cohesive energy would appear to be zero while the adhsive parameters would have finite values which differ from zero. This conclusion is not surprising if we keep in mind the fact that the membrane is known to be fluid. Nevertheless we consider the result to be interesting because it shows that our conclusions are consistent with conclusions coming from other sources, because it shows the kind of information which solubility measurements in membranes can provide and because special attention is paid to holes in the analysis of membrane structure and a ~ t i v i t y . 4 ' , ~ * , ~ ~ , ' ~ The nature of the interacting constituents implies that the attractive intermolecular forces inside the hydrophobic region of the DMPC membrane are dispersive London forces. These forces arise because of electron oscillations inside the interacting atoms. The oscillations transform the neutral atoms into temporary dipoles because they shift the center of gravity of the charged electrons from that of the positively charged nucleus. These temporary dipoles produce electric fields and induce the formation of dipoles in nearby atoms by separating electrons from protons. The orienting dipoles are always in phase with the induced ones giving the dispersive energy between two molecules s and 1 as (7)
where ct is the molecular polarizability, I is the first ionization potential, and r is the radius of the atom. It follows from this equation that the interaction energy between two similar molecules is
which leads directly to the combining rule of the geometric mean.3' ~
~~~
~~
(49) Bocian, D. F.; Chan, S. I. Annu. Rev. Phys. Chem. 1978, 29, 307. (50) Lieb, W. R.; Stein, W D. Nature 1969, 224, 240.
The Journal of Physical Chemistry, Vol. 90, No. 1, 1986 193
Physical Characteristics of Membranes cal -
mo 1 12000
10000
8000
6000
4000
2000
0 -2100
0
-700
-1400
700
1400
2100
mo 1
Figure 1. Calculated cohesive parameters for the DMPC membrane. It is assumed that other factors besides the dispersive attractions between chains affect the magnitude of the cohesive parameter of the membrane. These contributions are negative when they operate to repel the chains. They are zero when only the dispersive attractions exist. They are positive when extra work has to be invested to overcome the effect of existing energy barriers. The calculation is made from experimentally determined values of solution enthalpies of the noble gases argon and krypton in the DMPC membrane (Katz, 1981). The combination of eq 2, 3, and 6 gives a quadratic equation which is easily solved. Introduction of possible contribution to the energy as free parameters to the quadratic equation affects the magnitude of the calculated cohesive energy parameter. The figure shows the value of the cohesive energy parameter as a function of the magnitude of the free parameter of the quadratic equation. The experimentally found results, given by the vertical line in the drawing, agree best with the assumption that the dispersive interactions are the interactions which determine the cohesive energy parameter.
It is easily seen that the typical feature of the dispersive energy is its additivity, which means that one obtains the total energy of a molecular entity by superimposing the energies of interaction of its parts. This feature makes it possible to express the cohesive energy parameter e l l of the membrane in terms of interaction energies between CH, units and to use eq 4 for the description of these interactions. The interaction between a pair of CH2units has been calculated to be51
x
= -5.61 X
kJ/mol
(9)
where r is the distance in nanometers between the centers of the interacting CH, units. The value of the cohesive energy parameter, obtained from eq 4 and 9, is therefore e l l = -5.61 X 10-3z/fi kJ/mol
(10)
Inspection of this equation shows that the ratio z/# is a scaling factor which determines the cohesive energy of the bilayer. According to this equation the difference in the cohesive energies in the different membranes is only a difference of scale, the scaling factor being Z I P . Let us find now the number z of interactions between close neighbors which contribute to the cohesive energy parameter characterizing the DMPC membrane. For this purpose we insert the cohesive energy parameter which is given in Table I into eq 10 and obtain z/r6 = 31.2/5.61 x 10-3 = 5.51 x 103 nm-6 (51) Salem, L. J . Chem. Phys. 1962, 37, 2100.
(11)
We then estimate the distance r between neighboring groups by dividing the molar volume of the hydrophobic region by the number of C H I units found per mole. This gives an average nm3 for each group of CH, of the volume of r3 = 23.45 X hydrophobic region (in this calculation we assume that the density of the membrane in the hydrophobic region is close to 1 g/cm3 52). Insertion of this value into eq 11 gives a value of z = 3.1 for the number of nearest-neighbor contacts between CH, units involved in the creation of holes and the determination of the cohesive energy parameter of the DMPC bilayer. When similar considerations are applied to bulk hydrocarbon solvents they give values of z = 2.83, 2.86, and 2.96 for the solvents n-hexane, n-dodecane, and cyclohexane, respectively. None of these values differ by more than 10%from the value z = 3.1 found for the hydrophobic region of the DMPC membrane. Our findings mean that the variability of the cohesive energy parameter from one hydrocarbon solvent to another is dominated mostly by the intermolecular distances between chains and that the mode of packing of the interacting units has only a mild effect on the energy. The findings verify previous theoretical calculations which claim that the interaction between CH2groups is extremely sensitive to the distance between them” and relatively insensitive to their mode of packing in the liquid state.” These findings provide the only direct evidence that the special arrangement of the chains in the bilayer has no, or only a minor, effect on the magnitude of the intermolecular attractions and their behavior in this respect is similar to that of the isotropic organic solvents. (52) Lecuyer, H.; Dervichian, D. G. J . Mol. Biol. 1969, 45, 39.
194 The Journal of Physical Chemistry, Vol. 90, No. 1, 1986 A closer look into the calculated results furnishes useful information about the dimensions of the holes which are expected to be found in the hydrophobic region of the bilayer. The importance of these data to the analysis of membranes comes from the fact that holes are created by the thermal motions of the solvent molecules. This implies that the size of the holes in a membrane is a measure of the ability of its chains to fluctuate, and its determination is therefore valuable to the analysis of the molecular factors which determine the behavior of the bilayer. To estimate the dimensions of the holes from the enthalpy measurements we note that the contribution of the neighbor solvent molecules to the adhesion of the solute which accommodates the hole can be calculated in two ways. In one method we assume that the number of the interacting solvent units is 22. We find the number z using the calculations made above. We have found that for DMPC membrane z = 3.1. The number of interactions of the solvent with a solute which occupies a hole is twice the number of cohesive interactions which have to be abolished in order to create a hole and hence we conclude that the number of adhesive interactions in DMPC is equal to 22 = 6.2. In the second method of calculation we assume that the interacting solvent molecules are located in a spherical shell of thickness h around the hole which has the radius R. We also assume that the concentration of the solvent in this spherical shell is the same as its average concentration in the medium giving p = N / V l where V I is the molar volume of the solvent. We have already found that p = 1/(23.45 X molecules/nm3. Obviously the two methods of calculation must give the same results; hence
Katz TABLE 11: Cohesive Energy Parameters, Close Neighbors, and Hole Volumes in Some Hydrophobic Solvents
solvent
Z
Ne1 I
n-hexane n-dodecane cyclohexane DMPC
2.83 2.86 2.96 3.10
16.57 16.87 18.62 31.21
NL)~ 14.4 64.7 66.2 25.4“ 20.4’
Molar hole volume calculated from entropy measurement^.^" Molar hole volume calculated in this article from enthalpy measurements. Units for the cohesive energy parameter are kJ/mol. Hole volumes are expressed in cm3/mol.
gatiod3 leads to the interesting result that, to a fairly good approximation, the holes in a given medium may be assumed to have the same size. This means that it is sufficient to consider the dimensions of the holes when the ability of different solvents to undergo structural fluctuations is compared. The results presented here can be used to evaluate the dielectric constant in the hydrophobic region of the membrane. This seems to be of interest to permeability studies in bilayers,54 in studies of the electron transport chain,55and to the studies of microviscosities in membranes by fluorescent probe^.^^,^' To this goal we consider the classical Clausius-Mossotti relation, which expresses the dielectric constant of a medium as a function of the molecular polarizability a and the number density of the molecules in the medium.58 This relation n - i
2z = 4rR2hp
(12)
We further notice that once a hole is formed the sphere of radius R + h / 2 becomes excluded for the solvent, meaning that a distance of R + h / 2 is the closest distance from the center of the hole that a center of the solvent molecule is allowed to approach. Locating the molecules of the first neighbor at the distance R + h / 2 from the center of the hole gives a first shell of close neighbor solvent molecules whose thickness is h. We have calculated h from the volume of a C H I group given above and found that h = 2.86 X 10-I nm. Inserting the available data and solving for the hole radius R as the unknown we obtain R = 2.01 X 10-I nm. Using this value we calculate a molar hole volume NAuh = 20.43 cm3/ mol. Hole volumes in bulk solvents and in the hydrophobic region of the DMPC membrane were also calculated from the entropies of solution of the noble gases in these solvents.30 Hole volumes based on entropy measurements of argon, krypton, and xenon in DMPC membrane gave a molar hole volume of 25.44 f 1.39 cm3/mol. The latter results deviate by about 20% from the results which are obtained by using enthalpy measurements. Considering the independence of the two methods of estimation and the difference in the calculating procedures this is really a remarkable agreement. Inspection shows that in calculating hole volumes from entropies we considered only the “hard” repulsive part of the intermolecular potential. We ignored in this calculation the interaction which a molecule undergoes with molecules other than the close neighbor^^^*^' and we have also ignored the distribution of the molecules around the hole, which is considered by some authors to present one of the most crucial problems in the theory of dense fluids.33 That so much can be done by using such drastic approximations is really remarkable. Inspection of Table I1 reveals that the dimensions of the holes which characterize the hydrophobic region of the DMPC membrane are significantly smaller than the hole dimensions which are evaluated for the bulk hydrocarbon solvents.30 The dimensions of the holes measure the disturbance caused by the irregular thermal movement of the molecules in the solvent medium. The fact that holes in the membrane are smaller than holes in bulk solvents means therefore that the resistance to structural changes is larger in the membrane than it is in the bulk solvents. Similar kinetic energies of the moving CH2units cause much less change in the bilayer. This resistance to changes is due at least partially to the high cohesive energy of the membrane. Further investi-
(Dis the dielectric constant) is strictly correct only for gases of nonpolar molecules. However, the deviations of the theoretical predictions from the experimental findings for the condensed phases of the nonpolar molecules have also teen found to be rather Hence we apply eq 13 both to the hydrophobic region of the DMPC membrane and to n-dodecane. Using the ratio
we obtain the value of 2.6 for the dielectric constant of the hydrophobic region when taking the value D = 2.014 as the dielectric constant of n-dodecane. In making-the above calculations we ignored the fact that both a and D are tensor q ~ a n t i t i e s .This ~~ is not a serious problem when isotropic solvents are considered but it may be a factor which affects the considerations in unisotropic solvents, such as the hydrophobic region of the membrane. We accepted the view that the tensorial character of a, and hence D,can be ignored when dealing with interactions between neighbor molecules.51 Discussion The results which were derived in the previous section are useful in two respects. First, they can be used to solve some open questions about the characteristics of the bilayer, and second, they teach us about limitations and ambiguities in deductions on membrane behavior which are based on solubility measurements. Here we relate to these two aspects. We make some deductions about membrane properties which are needed to the analysis of (53) Furth, R. Proc. Cambridge Philos. SOC. 1941, 37, 253. (54) Dilger, J. P.; McLaughlin, S. G.A.; McIntosh, T. J.; Simon, S . A. Science 1979, 206, 2106. ( 5 5 ) Bellemare, F.; Fragata, M. J . Colloid Interface Sci. 1980, 77, 243. (56) Shinitzky, M.; Dianox, A. C.; Gitler, C.; Weber, G.Biochemistry 1971, 10, 2106. (57) Shinitzky, M. Isr. J . Chem. 1974, 12, 879. (58) Debye, P. J. W. ‘Polar Molecules”; Dover: New York, 1960 (59) Frohlich, H. ‘Theory of Dielectrics”; Oxford, London, 1950.
The Journal of Physical Chemistry, Vol. 90, No. 1. 1986 195
Physical Characteristics of Membranes
TABLE 111: Relation between the Density of the Solvent and Its Cohesive Energy Density"
solvent
P
n-hexane n-dodecane cyclohexane
0.660 0.749
DMPC
0.779 1.020
,,,312
0.537 0.648 0.688 1.030
NP2
P3i2
P4
Cf
0.436 0.561 0.607 1.040
0.354 0.485
0.190 0.314 0.368 1.082
0.525 0.535 0.590
0.536 1.051
1.ooo
CllP'3~2 0.98 0.83 0.86
CflD'2
C'f
lP'2
1.21
1.32
0.95
1.03
0.97
1.02
u p is the density in g/cm3, C' is the cohesive energy parameter relative to the cohesive energy parameter of DMPS, pf is the density relative to that of DMPC, and cff is the relative cohesive energy parameter corrected for the difference in the number of close neighbors in the different solvents.
phase transitions in membranes, discussing the relation between the density of the membrane and its cohesive energy parameter, determining the attractive part of the energy of melting of the bilayer, and establishing a connection between the partition functions of solubility and that of phase transition. We then show that these deductions cannot be made from the solubilities of more complex solutes in the membrane and we indicate the importance of discriminating between the adhesive and cohesive contributions to the interaction of solutes with the bilayer. First we examine our conclusions about the dependence of the intermolecular attractive energy on the intermolecular distances. This appears to be of special importance in the discussion of biological systems where the behavior may depend on minor energy differences. Reference to the l i t e r a t ~ r e ' , ~reveals ~ , ~ ~how , ~ ~much attention is paid to the determination of the attractive energy as a function of the intermolecular distances and of the distribution of the interacting units. In contrast, when we inspect our treatment we notice a rather simplistic way of deriving the relation between intermolecular distances and the distributions and the energy which holds them together. We assume that only interactions between close neighbors contribute to the cohesive energy, and that these neighbors are equally spaced from the central reference molecule, and we also assume that the formation of a hole is equivalent to an infinite separation between close neighbors. Hence we consider the cohesive energy in terms of the energies of two well-defined states. The intermolecular energy which we use is therefore inversely proportional to the sixth power of the intermolecular separation or to the density of the solvent squared. To examine the validity of these assertions we compare the cohesive energy parameters of several solvents and their densities squared, using the DMPC bilayer as reference solvent. According to our calculations the ratio between the cohesive energy parameters of two solvents, corrected to take account of the small differences in their number of nearest-neighbors, is equal to the ratio between their densities squared. The comparison is presented in Table 111. Inspection of the table shows good agreement, to within a few percent, between the expectations and the results. This means that we can use our calculations to evaluate membrane behavior and to compare between the energies of different membranes, a step which is needed in a comparative study of membranes. The dependence of the attractive cohesive energy on the density is of importance in various investigations on membrane behavior.41 Several relations of the form attractive energy = up", where p is the density and a and n are pure numbers, have been discussed. The relations can be examined by comparison of the cohesive energy parameters. The examination, which is presented in Table 111, shows that the best description of the attractive energy is obtained when n = 2 is used. Reasonable results are obtained also when n = 3/2 is used. Other values of n furnish inferior results. The relation which we found between the cohesive energy and the density of the hydrophobic region is used to determine the
changes in the attractive energy which accompany phase transitions in membranes. This has been estimated indirectly from freezing energies of paraffins and but findings to support these estimations do not exist. Using the cohesive energy parameter of the membrane, presented in Table I, in conjunction with available data on the volume changes which accompany phase t r a n ~ i t i o ngives ~ ~ , ~an~intermolecular attraction of tattr= 34.6 kJ/mol holes. The difference between the attractive energies in the two phases is therefore 3.4 kJ/mol holes. To compare this result and the estimations which appear in the literature we must find the interaction per CHI unit. Our calculation is based on a derivation of the ratio between the hole volume and the volume of a CH2 unit in DMPC. There are two calculations of the hole volume: a value of 25.44 cm3/mol which is based on entropy measurements and value of 20.43 cm3/mol given in this article which is based on enthalpy measurements. These lead to values of 1.8 and 1.5 CH2units per hole, respectively, and provide the values 1.87 and 2.33 kJ/mol CH2 for the change in the attractive energy per CH2 unit when melting occurs. The value which has been estimated from the freezing energies of the bulk solvents is 0.9 kJ/m01,~lwhich is about half of our results. We now proceed to show that our deductions were made possible only because we exploited the simplicity in structure and behavior of the noble gases. This is an important point which has to be kept in mind when trying to answer questions about the physical state of the membrane from solubility measurements. Inspection shows that the enthalpy of solution is determined by several statistical mechanical function^.^' These functions describe the rotational degrees of freedom of the solute and the solvent molecules in the pure state and when in solution (four functions) and also the intermolecular functions in the pure components and in solution (three functions). Such a large number of variables can be determined only if a sufficient number of relations exist between them or if the number of variables is reduced by choosing appropriate solutes. Our findings have shown that these demands can be met by using noble gases since these solutes have no rotational degrees of freedom and since their intermolecular interactions in the pure state are vanishingly small. Inferences which are based on solubilities of more complex molecules will suffer from ambiguities because the increase in the number of unknowns is not accompanied by the introduction of new relations. Finally, it is important to notice that treating solubility in terms of two independent processes contributed significantly to our analysis. We obtained results which could not be derived from the considerations of the enthalpies of solutions alone. Also the interpretation is simpler and less ambiguous because the effect of the solvent is separated from that of the solute. (60) Marcelja, S. J . Chem. Phys. 1974, 60, 3599. (61) Billmeyer, F. W. J . Appl. Phys. 1957, 28, 1114 (62) Engelman, D. M. J . Mol. Biol. 1971, 58, 153.
