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CHINESE JOURNAL OF PHYSICS
WINTER, 1799
VOL. 17, No. 4
Theoretical Study of the Effect of Cholesterol
Bilayer Phase Transition*
W. H. C
HENG
on
the Lipid
($Q&$)
Teleconununicalion Laboratories P. 0. Box 71, Chung-Li,
Taiwan, Republic of China
(Received November 19, 1979)
Based on the structural properties of cholesterol and lipid molecules and on
some experimental studies concerning the effect of cholesterol on the lipid bilayer
phase transition, a model for lipid-choIestero1 mixtures is proposed. The scaled
particle theory for mixtures of hard disks is used to describle the hard core
interactions among the lipid, and lipid-cholesterol molecules. Together with a
semiempirical potential to describe the long range interactions, we find that the
model provides accurate description of the manner in which cholesterol disrupt
lipid phase transition. This indicates that the excluded volume effect is a major
underlying mechanism for lipid-cholesterol mixtures.
I. INTRODUCTION
H E physical properties of the lipid bilayers found in biological membranes have been the object
One important result of this research is the discovery that
most lipid bilayers undergo thermotropic transitions in the neighborhood of 20”-60”‘L). It is now
clear that the phase transition in lipid bilayers involves a cooperative structural change from a state
in which the lipids are closely packed and chains fully extended, to a state in which a large fraction
of molecules exhibit as many gauche rotations per molecuIet1~2).
Cholesterol is recognized as one of the major lipid constituents of many plasma membranes@).
The effect of cholesterol on the lipid bilayer phase transition has also been extensively studied”-“?
Early calorimetric studies of cholesterol-dipalmitoylphosphatidylcholine (DPPC) mixtures(j) showed a
decrease in the transition enthalpy, eventually vanishing at 50 mole percent cholesterol. Recent
studies on DPPC and dimyristoylphosphatidylcholine (DMPC) dispersions indicate that the transition
enthalpy decreases linearly with the percentage of cholesterol and vanishes at about 33 mole percent
cholesteroltd).
The study of pure lipid bilayer phase transition is only the first step toward understanding the
structural properties of biological membranes.
The effects of molecular heterogeneity in naturally
“JIP of a great deal of research work.
* Work supported in part by the National Science Council of the Republic of China.
( 1 ) For a review see, J. F. Nagle and H. L. Scott, Physics Today, Feb. 38 (1978).
(2) J. E. Rothman, J. Theoret. Biol. 37, 1 (1973).
( 3 ) G. J. Nelson, Blood Lipids and Lipoproteins: Quantitation, Composition and Metabolism (Wiley-Interscience
N. Y., 1972).
(4) B. P. Gaber and W. L. Peticolas, Biochim. Biophys. Acta, 465, 260 (1977).
(5 ) B. D. Ladbrooke, R. M. Williams and D. Chapman, Biochem. Biophys. Acta, 150, 333 (1968).
(6) H. J. Hinz and J. Sturtevant, J. Mol. Biol. 247, 3697 (1972).
(7) E. J. Shimshick and H. M. McConnell, Bichem. Biophys. Res. Commun. 53, 446 (1973).
( 8 ) W. Kleemann and H. M. McConnell, Biochem. Biophys. Acta, 419, 206 (1976).
(9) W. I. Calhoun and G. G. Shipley, Bichemistry, 18, 1717 (1979).
(10) T. N. Estep, D. B. Mountcastle, Y. Barenholz, R. L. Biltonen and T. E. Thompson, Biochemistry, 18,
2112 (1979).
(11) D.A. Pink and C. E. Carroll, Phys. Lett. 66A, 157 (1978).
204
1
-.
W.H. CHENG
205
occurring membranes experimentally can be studied in model systems such as lipid-lipid, lipid-protein,
and lipid-cholesterol mixtures which can provide a closer understanding of biological membranes than
the pure lipid systems. In a previous paper (12, To be referred to as Paper l), we have introduced
a theoretical model for lipid mixtures phase transitions and phase diagrams. This model has succeeded
in describing the pure lipid phase transitions, the phase diagrams for binary lipid mixtures, and lipidprotein interaction in lipids. The purpose of this paper is to present the first theoretical study of
the effect of cholesterol on the lipid bilaycr phase transition based on Paper 1. The lipid we used
in our study is dipalmitoylphosphatidylcholine (DPPC). The major advantage of this study is that
it is readily generalized from a pure lipid system to lipid-cholesterol mixtures by consideration excluded volume interactions between lipid and cholesterol.
II. METHOD
A potentially useful method for finding an approximate Hamiltonian to describe a complex system
such a lipid bilayer is to replace the exact intermolecular interactions by semiempirical potentials
which treat the individual interactions separately. In view of the results of experimental studies of
phase transitions from an order structure to a disordered structure in membranes”) and the theoretical
consideration of the type of forces present in lipid bilayers and biomembranes(r3), it seems reasonable
to propose a theoretical model for which we can write an effective Hamiltonian for a monolayer or
half a bilayer in the form
H=H,,tfH,ot+H~c
(1)
These are basically the same energy contributions to monolayers as used by Andersen et Cal.,
MarEelja(lS), Nagle(16J and Scott(171. The first term, Hart, is the sum of attractions holding monolayers
together. The second term H,,r, is the internal energy of the chain configuration. According to the
well-known rotational isomeric mode1(r8), each carbon-carbon bond can take up either a trans
conformation with energy E=O or one of two gauche conformations with energy E=500cal/mol.
The last term, Hh_-, is the excluded volume interactions among the molecules. The approximate
Hamiltonian which simplifies the intermolecular interactions of lipid bilayers has been obtained in
Paper 1 by using the results of quantum chemical calculations of long range forces, the rotational
isomeric theory, and the scaled particle theory of fluid hard-particle mixtures.
There are sixteen bonds per chain for DPPC. We consider only fourteen of these units as rotatable, since experiments show the CH, chains are rigid and tightly packed near the polar group(1B~20J.
Applying the kink and jog isomeric models, discussed by Seelig and Pechhold(21@J, we suggest seven
possible configurations for DPPC as indicated schematically in Fig. 1. Using the scaled particle
theory‘ the semiempirical attractive potential, and the rotational isomeric theory as described in Paper
1, the effective Hamiltonian for a monolayer or half a bilayer of DPPC, according to equation
F=E,,,+E,,,-TS is given by
- l- $JCKiln(~i/Wi)+ln(A- &&;A!)-
i
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
i=l
i=l
II. L. Scott and W. H. Cheng, Biophys. J. 28, 117 (1979). To be referred to as Paper 1.
L. Salem, J. Chem. Phys. 37, 2100 (1962).
R. E. Jacobs, B.S. Hudson and H. C. Andersen, Proc. Nat. Acad. Sci. (USA), 72, 3993 (1975).
S. MarEelja, Biochem. Biophys, Acta, 357, 165 (1974).
J. F. Nagle, J. Chem. Phys. 58, 252 (1973).
H. J. Scott, J. Chem. Phys. 62, 1347 (1975).
P. J. Flory, Sfafi~fical Mechanics of Chain Molecules (\Viley-Interscience N. Y. 1963).
\V. L. Hubbell and H.M. McConnell, J. Am. Chem. 93, 314 (1971).
B.G. McFarland and H.M. McConnell, Proc. Nat. Acad. Sci. (USA), 68, 1274 (1971).
A. Seelig and J. Seelig, Biochemistry, 13, 4839 (1975).
W. Pechhold, Kolloid 2. Z. Polym. 228, 1 (1968).
(2)
I‘
206
THEORETICAL STUDY OF THE EFFECT OF CHOLESTEROL
,STATE
I
2
3
4
5.
6.
7
Fig. 1. Kink and jog models for a lipid bilayer (DPPC), Here we show lipids only with one
chain. The circles indicate the head groups and zig-zag lines indicate the CH2 chains.
All the summations in this equation are over all various states of DPPC. Where N is the t.%tl
number of lipid molecules, A is the total area per molecule, ai is the fraction of molecules in s:.{I~
i, and C(16) is the attractive energy constant. The various chain state i, statistical weights Wi, I\W
age number of gauche rotations in each state gi, and the area associated with each state J&, de&k\1
in Paper 1, are listed in Table I. The numerical analysis involves finding the values of the ai \\hlch
minimize the free energy at fixed temperature and area. In keeping with the semiempirical natum \\f
the model, the parameter C(16) in the equation (2) is adjusted to equal 132 kcal/mole so that \W
calculated isotherms agree as closely as possible with the experiment. Then we use the isothcrm.~l
properties to consider the melting transition in bilayers.
Table I. Chain states i, statistical weights I+‘;, average number of gauche rotations
gi, and the area associated with each state &of DPPC
I
i
Wi
Si
Ai
1
24
280
1,328
328
256
24
0
2
3.79
5.90
4.78
4.84
4.50
40.8
43.5
46.6
50.2
54.4
59.3
65.2
___.
1
2
3
4
5
6
7
I
Having found the free energy to describe the pure lipid bilayer phase transitions, we must attt\l\’
the mixtures of lipid and cholesterol. It is known that the incorporation of cholesterol into multlllt
layers decreases the fluidity of hydrocarbon chains above phase transitions(4-1r). The disappc:tt;tttc’(’
of transition stems to be due to the rigid nature of cholesterol which prevents the cooperative 1st)
merization of the long hydrocarbon chains of the lipids. It is also possible that some hytlt~~%(“~
bonding occurs between lhe hydrosyl groups of cholesterol and carbonyloxygen groups of an ad,i;l~‘utll
which may act to reduce the number of gauche conformations in lipids. solllo /I’(
phospholipid(“s’,
pects of this behavior can be simulated by studying the stefic portion of lipid-cholesterol alone.
___.__-._._ -._ ~~. ~~ _ ____~. .~_..
(23) C. H. Iluang, Lipids, 12, 381 (1977) and Nature, 259, 242 (1976).
>
W. H. CHENG
207
To describe a mixture of DPPC and cholesterol, we use the seven-state DPPC model mentioned
earlier and add an eighth state that corresponds to another hard cylinder for the cholesterol molecule.
Then the total free energy of DPPC-cholesterol mixtures is given by
500
8
F/lvKT=m (,F,‘j ai)-
CC161
RT[A-(38X,~+A,x,h)]‘. ”
( i ai
/X)”
- I- i$, niIn(Nj/~Vi)+ln(A- 2 ~iAj)--~‘-g
i-1
{
( A - glaiAi)---I.
(3)
The summation from one to seven in this equation corresponds to DPPC states and state number
eight corresponds to cholesterol. Because the cholesterol molecule is smaller in cross section than a
double-chain lipid molecules, we choose the cholesterol parameters g8=0, A,=35A2, and W,=l. The
value A,=35Az for the hard-core area of cholesterol molecule is taken from the model-building conclusions drawn by Forslind and Kjellander(Z4j. The 38X,6+AoXch is the mean close-packed area for
lipid-cholesterol mixtures. Here X,, is the fixed composition of lipid, Xek=l -X1, is the fixed composition of cholesterol, and A,, the average soft-core area for the cholesterol molecule, is to be
evaluated semiempirically.
Let us consider the ratio of the excluded volume energy term to the attractive energy term in
equation (3) and take the limit as the packing of molecular area approaches to the close-packed
value of cholesterol molecules. Using X,.,=0 and Xch=lr this limit r can be written approximately as
As part of our semiempirical construction. We expect that the excluded volume interaction effect
should be stronger than the long range interaction in determining the bilayer phase transition as the
packing of molecular area approaches to the close-packed value of cholesterol molecules. It follows
that we have
A0<35A2
(5)
The precise location of the critical point, as we will show in Fig. 2, is fairly sensitive to the choice
of Ao. In order that our calculations will be agree with experimental data X,*=0.33, A0 is chosen
to be 30Ae.
The numerical calculation of the surface pressure is performed using the following relation
(6)
where rt is the surface pressure, K is the Boltzmann constant, T is the absolute temperature, F is the
Helmholtz free energy, and A is the total area per molecule. For ease of comparison to experimental
results, the values n and A are given in the conventional units of dynes/cm and AZ/molecule, respectively. From equation (6), we can obtain a set of pressure-area isotherms for various cholesterol
mole fractions.
III. RESULTS
The results of pressure-area isotherms for mixtures of DPPC-cholesterol at transition temperature
T=314”K are shown in Fig. 2. These monolayer isotherms are characterized by van der Waals loops
which, after Maxwell construction, lead to coexisting phases in’ a first order transition.
From this
Figure, wc find that the addition of cholesterol to lipid causes the phase transition of lipid to disappear
at cholesterol composition Xch=0.33. This result is in excellent agreement with the data of Hinz and
Sturtevant’O) although other worker claim the transition persists up to X,* =O.5(Jl. A plot of mole(24) E. Forslind and R. J. Kjellander, J. Theoret. Biol. 51, 97 (1975).
._..
THEORETICAL STUDY OF TCIE EFFECT OF CHOLESTEROL
208
cular area versus mole fraction of cholesterol phase diagram is shown in Fig. 3 based on Fig. 2.
The dashed line merely denotes the upper boundary of the region in which it is possible distinguish
fluid and solid phases.
cl”‘“,“”
111-11
45
50
6(
55
A(~‘/MOLECULE)
Fig. 2. Monolayer isotherms for DPPC-cholesterol mixtures, for various cholesterol
mole fraction at T=314”K. The transition vanishes as the mole fraction of
cholesterol approach 33%. This result is excellent agreement with the data of
Hinz and Sturtevantcle).
0.5)
A fd/MOLEC”LE)
Fig. 3.
A theoretical plot of molecular area vs. mole fraction of cholesterol phase
diagram. The dashed line denotes the upper boundary of the region in which
it is possible distinguish fluid and solid phases.
W. H. CHENG
209
As we described earlier, the rigid nature of cholesterol prevents the cooperative isomerization of
hydrocarbon chain of the lipids. From this fact, we suggest that consideration of only the hard-core
interactions between lipid and cholesterol molecules in our study gives US a good approximation to
This indescribe the experimental results of the effect of cholesterol on the lipid phase transition.
dicates that the excluded volume effect is a major underlying mechanism for lipid-cholesterol mixtures.
Together with our previous calculations in Paper 1, our basic conclusions is that the model describes
well the qualitative behavior of bilayer consisting of pure lipid, binary mixtures of lipid, lipid-protein
mixtures, and lipid-cholesterol mixtures. Clearly, there are also other contributions to the energy of
lipid bilayers such as the head group and water interactions in the interface regiono4). However, because of the complex nature of interactions between the head group and water, such an interaction is
difficult to explicitly include in our simple model.