a microscopic interaction model of maximum solubility of

A Microscopic Interaction Model of Maximum Solubility of Cholesterol inLipid Bilayers

Juyang Huang and Gerald W. FeigensonSection of Biochemistry, Molecular and Cell Biology, Cornell University, Ithaca, New York 14853 USA

ABSTRACT We recently reported the equilibrium maximum solubility of cholesterol in a lipid bilayer, x*chol, to be 0.66 in fourdifferent phosphatidylcholines, and 0.51 in a phosphatidylethanolamine (Huang, J., J. T. Buboltz, and G. W. Feigenson. 1999.Biochim. Biophys. Acta. in press). Here we present a model of cholesterol-phospholipid mixing that explains these observedvalues of x*chol. Monte Carlo simulations show that pairwise-additivity of nearest-neighbor interactions is inadequate todescribe all the x*chol values. Instead, if cholesterol multibody interactions are assigned highly unfavorable energy, then jumpsoccur in cholesterol chemical potential that lead to its precipitation from the bilayer. Cholesterol precipitation is most likelyto occur near three discrete values of cholesterol mole fraction, 0.50, 0.57, and 0.67, which correspond to cholesterol/phospholipid mole ratios of 1/1, 4/3, and 2/1, respectively. At these solubility limits, where cholesterol chemical potentialjumps, the cholesterol-phospholipid bilayer mixture forms highly regular lipid distributions in order to minimize cholesterol-cholesterol contacts. This treatment shows that dramatic structural and thermodynamic changes can occur at particularcholesterol mole fractions without any stoichiometric complex formation. The physical origin of the unfavorable cholesterolmultibody interaction is explained by an “umbrella model”: in a bilayer, nonpolar cholesterol relies on polar phospholipidheadgroup coverage to avoid the unfavorable free energy of cholesterol contact with water. Thus, at high cholesterol molefraction, this unfavorable free energy, not any favorable cholesterol-phospholipid interaction, dominates the mixing behavior.This physical origin also explains the “cholesterol condensing effect” and the increase in acyl chain order parameter incholesterol-phospholipid mixtures.

INTRODUCTION

We seek to understand general properties of biomembranesby exploring the molecular-level interactions among thecomponents. Since there are many chemically differentcomponents in a biomembrane, there are very many possi-bilities for molecular interactions just between nearest-neighbor molecules.

A reasonable starting point toward a description of themolecular-level interactions is to consider a chemicallywell-defined bilayer mixture of lipids. Cholesterol is thesingle most abundant molecule in the mammalian plasmamembrane, so there have been numerous studies of choles-terol/phospholipid bilayer mixtures (see, for example, Fine-gold, 1993). In a plasma membrane, cholesterol comprisesup to half of all the lipids (Bloch, 1991), and the molefraction is likely to be even higher in one of the bilayerleaflets.

When the cholesterol content is high enough to exceedthe maximum solubility of cholesterol in the lipid bilayer, amole fraction we refer to asx*chol, excess cholesterol mol-ecules leave the bilayer as a precipitate of cholesterol mo-nohydrate crystals. This maximum solubility represents theboundary of a first-order phase transition. Values ofx*chol

that describe this phase boundary in the high-cholesterolregime have been reported in numerous publications, butthere is a wide range of disagreement on the actual values(Horowitz et al., 1971; Gershfeld, 1978; Finean, 1990; Bachet al., 1998). Yet, we require a reliable value ofx*chol,because theoretical studies of cholesterol/phospholipid mix-tures offer the best possibility of developing our understand-ing of the intermolecular interactions only if we can pre-cisely compare the theoretical with the experimental results.

Recently, we discovered that the major source of thedisagreement lies in the sample preparation method. Con-ventional sample preparation methods that pass the mixturethrough a dry state can induce cholesterol/phospholipiddemixing in this dry state (Buboltz and Feigenson, 1999).The cholesterol crystals are carried over to the hydratedlipid dispersions, making the apparentx*chol value lowerthan the true equilibrium value. By using two new indepen-dent sample preparation methods that avoid lipid demixing,we were able to find the true equilibrium values ofx*chol

(Huang et al., 1999). These values show a very interestingpattern: for all of the phosphatidylcholines (PCs) studied,regardless of acyl chain type (12:0,12:0 or 16:0,16:0 or16:0,18:1 or 22:1,22:1) the value ofx*chol is 0.666 0.01 at24°C. In contrast, we found the distinctly different resultthat x*chol for (16:0,18:1) phosphatidylethanolamine (PE) is0.516 0.01. These new findings lead us to ask the follow-ing questions: why are the values ofx*chol insensitive to thetype of phospholipid acyl chains? Are some unique prop-erties of cholesterol-phospholipid organization revealed bythe observedx*chol values, which are very close to choles-terol/phospholipid mole ratios of 1/1 and 2/1? What kind of

Received for publication 2 September 1998 and in final form 19 January1999.

Address reprint requests to Dr. Gerald W. Feigenson, Section of Biochem-istry, Molecular and Cell Biology, Cornell University, 201 BiotechnologyBuilding, Ithaca, NY 14853-2703. Tel.: 607-255-4744; Fax: 607-255-2428; E-mail: [email protected].

© 1999 by the Biophysical Society

0006-3495/99/04/2142/16 $2.00

2142 Biophysical Journal Volume 76 April 1999 2142–2157

microscopic interactions in cholesterol/phospholipid mix-tures could give rise to the observedx*chol values of 0.66 and0.51?

In this study we present a microscopic interaction modelthat answers these questions. We used Monte Carlo simu-lation as a tool to translate our model into the correspondingchemical potentials and lipid lateral organizations. We findthat pairwise-additive interactionscan explain a value ofx*chol 5 0.51, if mixing of phospholipid with cholesterol issufficiently strongly favorable. However, this simple modelcannotpredictx*chol 5 0.66. We find that if a simple formof multibody interaction is used, in which cholesterol-cho-lesterol interaction energy becomes unfavorable moresteeply than additively for more cholesterols in contact, thenx*chol of either 0.51 or 0.66 can occur, depending upon thedetails of the nonlinear energy increase. Moreover, a valueof x*chol 5 0.57 is also shown to be possible. Our simula-tions show that at the solubility limits, highly regular lipiddistributions form in the bilayer, and the chemical potentialof cholesterol jumps steeply. The physical origin of thismultibody interaction can be explained by the hydrophobic-ity of cholesterol.

BASIC ASSUMPTIONS AND STRATEGY

Chemical potential of cholesterol

Cholesterol in a monohydrate crystal, as a pure substance,has a constant chemical potential,mchol

crystal. In contrast, thechemical potential of cholesterol in a lipid bilayer,mchol

bilayer

(xchol), is a function of the bilayer composition. Whenxchol

, x*chol, mcholbilayer (xchol) must be less than mchol

crystal for thebilayer to be the only stable phase. Whenxchol $ x*chol, thelipid bilayer phase and the cholesterol monohydrate crystalphase coexist. In this two-phase region, the chemical poten-tial of cholesterol in the bilayer must be equal to the chem-ical potential of cholesterol in the monohydrate crystal,mchol

bilayer (x*chol) 5 mcholcrystal.

A key experimental finding is the essentially identicalvalues ofx*chol in four PCs having different sets of acylchains. This finding leads us to deduce a general picture ofthe way in which cholesterol chemical potential changes asits mole fraction increases in a bilayer. The theoreticaltreatment will be designed to search for the microscopicenergies that reproduce this form of the chemical potentialchange.

We start with the chemical potential of cholesterol, at anygiven mole fraction, being different in phospholipids A, B,and C. A very simple form of this difference would be aconstant offset of the cholesterol chemical potential at everymole fraction in the three phospholipid environments. Fig. 1illustrates a general relationship between the shape of themchol

bilayer profile and the range of possiblex*chol values. Thisfigure also shows the chemical potential of cholesterol inthe pure monohydrate crystal,mchol

crystal, as a horizontal dashedline. In Fig. 1a, a hypotheticalmchol

bilayer curve in phospho-lipid A is drawn as a smoothly increasing function ofxchol

(solid line). The cholesterol mole fraction at whichmcholbilayer

intersects the chemical potential of crystalline cholesterolmonohydrate is thex*chol value in lipid A. Consider twoother phospholipids, B and C, with different sets of acylchains, and having cholesterol-phospholipid interactionsthat are slightly different from those in lipid A. Themchol

bilayer

profiles in phospholipids B and C would be slightly differ-ent from that in phospholipid A. We represent these profilesas the two dotted lines in Fig. 1a, obtained by adding orsubtracting a small constant offset to or from themchol

bilayer oflipid A. These two dotted lines intersectmchol

crystal at differentvalues ofxchol, thus resulting in different values ofx*chol.The range ofx*chol values for lipids A, B, and C is indicatedby the width of the shaded box. In Fig. 1b, a hypotheticalmchol

bilayer curve is drawn having a steep increase at somecholesterol mole fraction. With the same offset of chemicalpotential at each cholesterol mole fraction, the resultingrange ofx*chol is much narrower. Since thex*chol for four PCswith very different acyl chains are almost identical,what-ever is the nature of the microscopic interaction that in-duces the cholesterol precipitation, it must cause the chem-ical potential of cholesterol in the bilayer to increase sosharply atx*chol that other contributions can be neglected.

FIGURE 1 Relationship between the shape ofmcholbilayer profiles and the

range of possiblex*chol values. (a) A set of hypotheticalmcholbilayer profiles as

smoothly increasing functions ofxchol. The range of thex*chol values isindicated by the width of the shaded box. (b) A set of hypotheticalmchol

bilayer

profiles as steeply increasing functions ofxchol. The range of thex*chol

values is much narrower.

Huang and Feigenson Model of Cholesterol Solubility in Bilayers 2143

Single-state lattice model

The above analysis implies that we can safely ignore thevariation of mchol

bilayer in each different acyl chain environ-ment, since it does not show up in the measured value ofx*chol, as long as it is not of comparable energy. That is, theacyl chain contribution to the chemical potential doesnotneed to be assumed as zero, yetwe can completely ignoreacyl chain details in our model, and instead need only notewhether or not there is an acyl chain or a cholesterol at agiven lattice position.

The general mixing behavior of cholesterolsnear x*chol

Sincemcholbilayer (xchol) is a steeply increasing function ofxchol

near x*chol, the excesschemical potential of cholesterolshould be a steeply increasing function as well, because theideal part of the chemical potential (RTlnxchol) is a slowlyincreasing function aroundxchol 5 0.5 or 0.67. Thus, themicroscopic interaction that induces cholesterol precipita-tion strongly favors the mixing of cholesterol with phos-pholipid. Otherwise, if cholesterol molecules tended to clus-ter, the excess chemical potential of cholesterol woulddecrease as a function ofxchol, instead of increase. In termsof microscopic interaction energy,cholesterol-cholesterolcontacts are unfavorable or cholesterol-phospholipid con-tacts are favorable.

Type of lattice

The cross-sectional area of cholesterol is;37 Å2 (Lund-berg, 1982), that of a fluid phase phospholipid is on theorder of 65 Å2 (Nagle et al., 1996). Other researchers havemodeled the packing of cholesterol molecules and phospho-lipid acyl chains as a hexagonal lattice (Cruzeiro-Hansson etal., 1989; Scott, 1991). The justification for using a lattice ofacyl chainsand cholesterol, rather thanentire phospholipidsand cholesterol, has been discussed by Virtanen et al.(1995). We, too, will use the hexagonal packing model ofacyl chains and cholesterol in this study. Since the sizes andpacking details are surely not identical for cholesterol andphospholipid acyl chains, a distorted hexagonal latticeseems likely. Such a distorted hexagonal lattice will serveour purpose well, as long as each site has six nearest-neighbors. As we discussed above with reference to Fig. 1,we assume that all possible variations of the acyl chains(length, conformation, unsaturation) yield only one energystate, and all the interactions with neighbors as well as anyconformation changes on any parts of either the phospho-lipid or the cholesterol are included in the phenomenolog-ical interaction energy terms. In addition, although phos-pholipid headgroups and water molecules are not shownexplicitly in the lattice, their effects are implicit in thephenomenological acyl chain-cholesterol interaction.

Our goal is to formulate a simple and general form of themicroscopic cholesterol/phospholipid interaction Hamilto-

nian that could cause an abrupt increase of cholesterolchemical potential atxchol 5 0.66 or 0.51. We use MonteCarlo simulations as the tool to find the microscopic inter-actions: a hypothetical interaction Hamiltonian is fed intothe simulations; the simulations faithfully produce the cor-responding free energy, chemical potentials, and moleculardistributions of the simulated mixtures; the simulation re-sults are then compared to the experimental data, and theHamiltonian modified for a better fit.

THE MICROSCOPIC INTERACTION MODEL

The cholesterol/phospholipid bilayer is modeled as a two-dimensional hexagonal lattice. Each lattice site can be oc-cupied by either a phospholipid acyl chain or a cholesterolmolecule. We model the interactions between phospholipidsand cholesterols in the simplest possible way by limiting allthe interactions to nearest-neighbor only. The Hamiltonianhas two major components: one describing pairwise-addi-tive interactions,Hpair, and another for cholesterol multi-body interactions with its nearest-neighbors,Hmulti.

Htotal 5 Hpair 1 Hmulti (1)

The pairwise-additive part of the Hamiltonian includesthe interactions of acyl chain-acyl chain, cholesterol-cho-lesterol, and acyl chain-cholesterol contact pairs:

Hpair 51

2 Oi, j

EaaLaiLaj 11

2 Oi, j

EccLciLcj

11

2 Oi, j

Eac~LaiLcj 1 LciLaj)

(2)

whereEaa, Ecc, andEac are the interaction energies betweenacyl chains, between cholesterols, and between acyl chain-cholesterol, respectively;Lai and Lci are the occupationvariables (5 0 or 1) of acyl chains and cholesterols, respec-tively. The summationi is over all lattice sites, andj is overthe nearest-neighbor sites ofi only. The factor 1/2 is nec-essary to avoid counting each contact pair twice.

For this lattice system, the three interaction energies inEq. 2 can be further reduced to just one independent vari-able (Guggenheim, 1952). Eq. 2 is rewritten as:

Hpair 5Z

2 Oi

EaaLai 1Z

2 Oi

EccLci

11

2 Oi, j

DEm~LaiLcj 1 LciLaj),

(3)

whereZ is the number of nearest-neighbors to a lattice site,which is six for a hexagonal lattice.DEm is the pairwise-additive excess mixing energy of acyl chains and cholester-ols, defined as

DEm 5 Eac 2 ~Eaa1 Ecc!/2. (4)

2144 Biophysical Journal Volume 76 April 1999

In a canonical Monte Carlo simulation, the total numberof lattice sites (N), the number of acyl chains (Na 5 (i Lai),the number of cholesterols (Nc5 (i Lci), and the tempera-ture (T) are all fixed for each simulation. Therefore, the firsttwo terms in Eq. 3 are independent of lipid lateral distribu-tion, and the entire contribution of pairwise-additive inter-actions to the lipid mixing behavior is determined by thevalue ofDEm in the third term.

The cholesterol multibody interaction with its six nearest-neighbors is given by:

Hmulti 5 Oi

Os50

6

DEcCsLsiLci , (5)

where DEc is the strength of the cholesterol multibodyinteraction,Cs are the energy scaling factors, andLsi is theenvironment variable of a lattice site, which is defined as

Lsi 5 H 1, if site i hasscholesterol as it nearest neighbor0, otherwise.

In Eq. 5, the summations is over seven possible environ-ments for lattice sitei: a site can have zero to six choles-terols as nearest-neighbors. Thus, if a cholesterol is sur-rounded by s other cholesterols, then the multibodyinteraction energy for this cholesterol would beDEcCs. Noenergy difference is assumed for the different arrangementsof theses cholesterols among the nearest-neighbor sites.Seven energy scaling factors (C0, C1, . . . , C6) define therelative magnitude of the multibody interaction in the sevenpossible situations, andDEc determines the overall strengthof the cholesterol multibody interaction. The physical originof this multibody interaction term is described in theDiscussion.

SIMULATION METHOD

All the simulations were performed on a 1203 120 hex-agonal lattice with a standard periodical boundary condi-tion. Neighboring cholesterols and acyl chains can ex-change their position with a probability given by theMetropolis method (Metropolis et al., 1953). All simula-tions started from an ideal mixture of given composition.Equilibrium conditions were established after an initial25,000–70,000 Monte Carlo steps. The ensemble averageof the data was obtained in 10,000 Monte Carlo steps afterequilibrium. Data were averaged from three independentruns. As demonstrated in an earlier paper (Huang and Fei-genson, 1993), such a large-scale simulation makes thesimulation size effect negligible.

FREE ENERGY AND CHEMICAL POTENTIALS

In order to relate our microscopic interaction model toexperimentalx*chol data, it is crucial to be able to calculatethe chemical potential of cholesterol from computer simu-lations. We applied the Kirkwood coupling parameter

method (Haile, 1986; Chialvo, 1990; Kirkwood, 1935,1936) to calculate the excess Gibbs free energy of thecholesterol/phospholipid mixtures. Although this approachis computationally intensive, it provides complete informa-tion: mixing free energy, enthalpy, entropy, and chemicalpotentials of cholesterol and acyl chains.

Following similar steps described in our earlier paper (seeAppendix in Huang et al., 1993), the excess Gibbs freeenergy of a cholesterol/phospholipid mixture is given by

DGE~DEm, DEc! 5 N0 E0

DEm

K12 Oi, j

~LaiLcj 1 LciLaj!Llc50

NdlE

1 N0 E0

DEc

KOi

Os50

6

CsLsiLciLlE5DEm

NdlC

2 N0DEcC6

Nc

N(6)

whereN0 is Avogadro’s number,lE and lC are couplingparameters, and the angle brackets denote an ensembleaverage from Monte Carlo simulations. By performing nu-merical integration,DGE can be calculated. The excessenthalpyDHE and entropyDSE are given by:

DHE 5 N0K12 Oi, j

~LaiLcj 1 LciLaj!LDEm/N

1 N0KOi

Os50

6

CsLsiLciLDEc/N 2 N0DEcC6Nc/N

(7)

and

DSE 5 ~DHE 2 DGE!/T (8)

The excess chemical potentials of cholesterol and phos-pholipid (mchol

E andmlipidE ) can be obtained by differentiating

DGE,

DGE 5 mcholE Nc/N 1 mlipid

E Na/N (9)

For convenience, we choose the standard state ofmcholE as

that in which cholesterols are infinitely dilute in a phospho-lipid bilayer, and the standard state ofmlipid

E as that in a purephospholipid bilayer.

Since each phospholipid has two acyl chains (see Note 1at end of text) the mole fraction of cholesterol in a bilayeris given by

xchol 5 2Nc/~2Nc 1 Na!. (10)


COMPUTER SIMULATION RESULTS

Pairwise-additive interaction energies

First, we investigate the effects of a pairwise-additive in-teraction energy on the chemical potential and the moleculardistributions. As shown above, three interaction energies(Eaa, Ecc, andEac) can be specified for a mixture, but thesecan be reduced to a single parameterDEm (Eq. 4), whichdictates the mixing behavior: 1)DEm 5 0, ideal mixing.Excess Gibbs free energy5 0; 2) DEm . 0, mixtures tendto form clusters of like molecules, and even lateral phaseseparation is possible (Huang and Feigenson, 1993); 3)DEm , 0, cholesterol and acyl chains are mixed better thanrandomly. Like molecules tend to separate from each other.The excess chemical potential of cholesterol increases withincrease ofxchol, which is consistent with the general mix-ing behavior of cholesterol deduced from thex*chol data.This is the case we shall study.

Fig. 2a shows excess Gibbs free energies versusxchol forsome negative values ofDEm. These negative values ofDGE

indicate that the Gibbs free energies are lowered by mixingphospholipids with cholesterols. Fig. 2a has a distinct fea-ture thatDGE has a sudden change in slope at aboutxchol 50.5 whenDEm # 22 kT, indicating a new regime of lateralorganization of the molecules.

Fig. 3 shows some snapshots of acyl chain/cholesterollateral distributions simulated using pairwise-additive inter-actions. Each filled circle represents one cholesterol mole-cule, each open circle a phospholipid acyl chain. As shownin Fig. 3a, atxchol 5 0.5, with a low magnitude ofDEm (21kT), the distribution has no particular pattern: some choles-terol molecules have no cholesterol-cholesterol contact, oth-ers have up to four contacts. However, at the same compo-sition, with DEm 5 23 kT, cholesterol molecules form acrystal-like regular pattern (Fig. 3b): each cholesterol issurrounded by six acyl chains, and cholesterol-cholesterolcontacts are minimized. Although some minor defects arecommon, the order is very long-range, only limited by thesimulation size. Fig. 3,c andd are snapshots of the lateraldistributions simulated withDEm 5 23 kT at xchol 5 0.44and 0.56, respectively, slightly below or abovexchol 5 0.5.As xchol increases, the crystal-like regular pattern graduallygrows in size at lower concentration, encompasses thewhole lattice at the critical composition, then is graduallydestroyed whenxchol becomes higher.

The effect of this crystal-like regular distribution onchemical potential of cholesterol is shown in Fig. 4a. Whenthe value ofDEm is small,mchol

bilayer increases smoothly as afunction of xchol. With a large negative value ofDEm,mchol

bilayer shows a jump atxchol 5 0.5. The magnitude of thejump increases with the magnitude ofDEm.

The formation of the crystal-like regular distribution pat-tern and the sharp increase ofmchol

bilayer at xchol 5 0.5 can beeasily understood. A large negative value ofDEm makeslike-molecule contact very unfavorable (or equivalently,unlike-molecule contact very favorable). Energy is lowered

FIGURE 2 Excess Gibbs free energy,DGE, as a function of cholesterolmole fraction in a bilayer. Simulations use: (a) pairwise-additive interac-tions, which is equivalent to MIEP I; (b) MIEP II; (c) MIEP III; (d) MIEPIV.


by minimizing cholesterol-cholesterol contacts or maximiz-ing acyl chain-cholesterol contacts. Atxchol 5 0.5, the ratioof acyl chain to cholesterol is exactly 2/1. The crystal-likeregular pattern shown in Fig. 3b is theonlydistribution thatavoids any cholesterol-cholesterol contact at this composi-tion. The chemical potential of cholesterol in such a perfectpattern is still low. Now imagine that a tiny amount ofcholesterol is added to the mixture, replacing phospholipid,to raisexchol by an infinitesimal amount just above 0.5: eachadded cholesterol now occupies a site previously occupiedby an acyl chain, and would havethreecholesterol-choles-terol contacts in such a pattern (see Fig. 3b). The energycost for such an operation (losing three acyl chain-choles-terol contacts) is very high, with a large negative value ofDEm. Therefore, the chemical potential of cholesterol justabovexchol 5 0.5 rises sharply.

In order to induce a sharp jump in chemical potential ofcholesterol,DEm must be#22 kT (Fig. 4a; see Note 2).This DEm could arise from a22 kT favorable interactionenergy between an acyl chain-cholesterol pair or from a14kT unfavorable interaction energy between a cholesterol-cholesterol pair, or from some combination (Eq. 4).

Need for the multibody interaction

Fig. 4a shows that besidesxchol 5 0.5, the chemical po-tential of cholesterol hasno sharp increase at any othercomposition(see Note 3). Therefore, no matter how thethree interaction energies (Eaa, Ecc, andEac) are chosen toformulateDEm (using Eq. 4), it is impossible to produce asharp increase ofmchol

bilayer at xchol 5 0.67 by a pairwise-additive interaction Hamiltonian. This suggests that a newform of the interaction is needed. As we show in the nextsection, a relatively simple form of the cholesterol multi-body interaction can produce a sharp increase ofmchol

bilayer

either atxchol 5 0.5, or 0.57, or 0.67. The pairwise-additiveterm we studied in this section can be seen as a special caseof the multibody interaction.

The cholesterol multibody interaction energy

In the previous section we saw the limitation of the pair-wise-additive interaction Hamiltonian. The main questionstill remains: how to formulate a simple and general micro-scopic cholesterol/phospholipid interaction Hamiltonian

FIGURE 3 Snapshots of phospholipid and cholesterol lateral distributions simulated using pairwise-additive interactions. (a) xchol 5 0.50 andDEm 521 kT; (b) xchol 5 0.50 andDEm 5 23 kT. Cholesterols form a crystal-like regular distribution pattern: each cholesterol has no other cholesterol as itsnearest neighbor; (c) xchol 5 0.44 andDEm 5 23 kT; (d) xchol 5 0.56 andDEm 5 23 kT. (F) Cholesterol; (E) acyl chain.


that could cause an abrupt increase of cholesterol chemicalpotential atxchol 5 0.67 or 0.50. Since no pairwise-additiveterm will work, using a multibody interaction is a naturalsolution. However, unlike the pairwise-additive interactionshown in Eq. 2, there is no general form for the multibodyinteraction. Although there are many possibilities for thetype of multibody interactions (e.g. 3 body, 4 body, . . .), ina hexagonal lattice, a reasonable choice would be one in-volving all six nearest-neighbors. After a period of exten-sive simulation work, we discovered that a simple andreasonable form for the multibody interaction given in Eq.5 can solve the problem.

We start by noting that for a pairwise-additive interactionthe number of such pairs is counted, and the total interaction

energy increaseslinearly with the number of interactionpairs. For example, in Eq. 2Ecc is the interaction energy fora cholesterol-cholesterol pair. If there aren cholesterol-cholesterol contacts, then the total energy is simplynEcc.Thus, the total pairwise-additive energy is thesumof theenergies of each individual pair.

In contrast, the multibody interaction energy is a descrip-tion of the interactions of all nearest neighbors consideredas a group. The multibody interaction Hamiltonian in Eq. 5allows the total interaction energy to increasenonlinearlywith the number of cholesterol-cholesterol contacts. Forexample, if a cholesterol has two cholesterol-cholesterolcontacts, the interaction energy would bec2DEc instead of2DEc. Herec2 is a chosen constant. Thus, six parameters,

FIGURE 4 Excess chemical potential of cholesterol,mcholE , or of phospholipid,mlipid

E , as a function of cholesterol mole fraction. (a) mcholE simulated using

pairwise-additive interactions, which is equivalent to MIEP I; (b) mcholE , simulated using MIEP II; (c) mchol

E and (d) mlipidE , simulated using MIEP III; (e) mchol

E ,simulated using MIEP IV.


c1 . . . c6 (c0 is always 0) need to be specified for themultibody interaction in a hexagonal lattice.

Table 1 lists four sets of multibody interaction energyparameters (MIEP) used in this study. For example, inMIEP II, the energy cost of the first cholesterol-cholesterolcontact is 1 unit, for the second one 8 units, the third 3 units,and so on. Thus,c1 5 1, c2 5 1 1 8 5 9, c3 5 1 1 8 1 3 512, and so on. These sets describe, although not uniquely,how the energy increases as cholesterols make contact witheach other, for positive values ofDEc.

The first set, MIEP I, is constructed to create a significantbut equal energy cost for each cholesterol-cholesterol con-tact. Therefore, the total energy increaseslinearly with thenumber of cholesterol-cholesterol contacts (indicated by thelinear increase ofc1, . . . , c6). With MIEP I, the multibodyinteraction Hamiltonian is reduced to a pairwise-additiveHamiltonian withEaa 5 Eac 5 0, Ecc 5 2DEc, andDEm 52DEc (using Eq. 4). Therefore, we can consider that theresults from the previous section were obtained usingMIEP I.

The second set, MIEP II, is constructed to create a rela-tively small energy cost for each pair of cholesterols that arein contact but completely surrounded by phospholipids,with a much larger energy cost for three cholesterols incontact. The first cholesterol-cholesterol contact costs 1 unitof energy, whereas the next one costs 8 energy units, thusmuch higher than the first. Contacts that are even higher-order are not so critical, so all are assigned to 3 units. Fig.2 b shows the excess Gibbs free energy versusxchol simu-lated using MIEP II. The values ofDGE are negative,indicating that the Gibbs free energies are lowered by mix-ing phospholipids with cholesterols.DGE has a global min-imum as well as a sudden change of slope atxchol 5 0.57(cholesterol/phospholipid mole ratio of 4/3). Fig. 4b showsthe excess chemical potential of cholesterol versusxchol;mchol

E has a sharp jump at 0.57 forDEc $ 0.5 kT.Fig. 5 shows snapshots of cholesterol and acyl chain

lateral distributions simulated with MIEP II. With a lowmagnitude ofDEc (0.2 kT), as shown in Fig. 5a, at xchol 50.57 the distribution shows no particular pattern. However,at the same composition, withDEc 5 0.5 kT, cholesterolmolecules form an aligned dimer pattern (Fig. 5b): eachcholesterol has exactly one cholesterol-cholesterol contact.Figs. 5c and 7d are snapshots of the lateral distributionssimulated withDEc 5 0.5 kT at xchol 5 0.51 and 0.62,respectively. Atxchol 5 0.51, many cholesterol dimers haveformed, but no one preferred orientation is adopted (Fig.

5 c); at the critical mole fraction of 0.57, all dimers areperfectly aligned to one orientation (Fig. 5b); at xchol 50.62, the dimer pattern no longer exists (Fig. 5d).

By understanding this dimer pattern, one can understandhow other multibody interaction sets work. The formationof the dimer pattern and abrupt increase ofmchol

bilayeratxchol 50.57 are direct results of this particular choice of the multi-body interaction parameters. In Table 1, MIEP II is delib-erately formulated to make the second contact cost muchmore energy than the first. The energy cost for other,higher-order contacts are not critical to produce the dimerpattern, as long as they are positive, so we chose 3 for them.The mixture responds to MIEP II with a distribution thatminimizes the second cholesterol-cholesterol contact, asshown in Fig. 5b. At xchol 5 0.57, the cholesterol/acylchain ratio is 2/3. The dimer pattern is the only pattern atthis composition such that no cholesterol has the secondcholesterol-cholesterol contact, i.e., the lowest energydistribution.

Consider a tiny amount of cholesterol added to the mix-ture, replacing acyl chains and raisingxchol by an infinites-imal amount above 0.57. The added cholesterol would beforced into sites previously occupied by acyl chains witheither three or four cholesterol contacts (Fig. 5b). Thisnewly added cholesterol now touches the three or fournearest-neighbor cholesterols that already have a secondcholesterol-cholesterol contact. The energy cost for such anoperation would be very high. Therefore, the chemicalpotential of cholesterol sharply jumps atxchol 5 0.57.

In MIEP II, the second cholesterol-cholesterol contact isassigned to cost 8 energy units. In order to induce a sharpjump in chemical potential of cholesterol,DEc should be$0.5 kT (Fig. 4b). Thus, the actual energy cost for thesecond cholesterol-cholesterol contact should be 0.53 8 54 kT or higher.

Fig. 5e shows a metastable distribution atxchol 5 0.57simulated withDEc 5 0.6 kT. There are multiple domainsof aligned dimer patterns in this distribution, and eachdomain has its own orientation. In a hexagonal lattice, threesuch orientations are possible. At the domain boundaries,many cholesterol trimers are visible, which indicates thatthe energy of this multidomain distribution is higher thanthat of the single-domain distribution shown in Fig. 5b. Wefound that this metastable distribution at highDEc is verypersistent in simulations: it could not be relaxed to a single-domain distribution even with very long simulation times(up to 1,000,000 steps). However, it is possible to avoid the

TABLE 1 A list of multibody interaction energy parameter sets used in this study

Multibody InteractionEnergy Parameter Set Energy Cost for Each Additional Chol-Chol Contact

Eq. 5 Coefficients

c0 c1 c2 c3 c4 c5 c6

MIEP I 0 1 1 1 1 1 1 0 1 2 3 4 5 6MIEP II 0 1 8 3 3 3 3 0 1 9 12 15 18 21MIEP III 0 1 1 10 3 3 3 0 1 2 12 15 18 21MIEP IV 0 1 2 3 4 5 6 0 1 3 6 10 15 21


multidomain distribution by slowly cooling down the mix-ture from a high temperature.

In MIEP III, we continue the assignment of large, unfa-vorable interaction energy to yet higher-order cholesterol-cholesterol contacts. In MIEP III, the 4-cholesterol multi-body interaction is assigned the largest interaction energy.

Fig. 2c shows excess Gibbs free energy versusxchol simu-lated with MIEP III, and withDEc ranging from 0 to 1 kT.DGE has a sharp change in slope atxchol 5 0.67. The valueof 0.67 is also a global minimum ofDGE.

The excess chemical potential of the mixture componentsis shown in Fig. 4,c andd. With DEc $ 0.3 kT,mchol

bilayer has

FIGURE 5 Snapshots of phospholipid and cholesterol lateral distributions simulated using MIEP II. (a) xchol 5 0.57 andDEc 5 0.2 kT; (b) xchol 5 0.57andDEc 5 0.5 kT. Cholesterols form an aligned dimer pattern: each cholesterol has one other cholesterol as its nearest neighbor; (c) xchol 5 0.51 andDEc 50.5 kT; (d) xchol 5 0.62 andDEc 5 0.5 kT; (e) a metastable multidomain regular distribution atxchol 5 0.57 andDEc 5 0.6 kT. (F) Cholesterol; (E) acylchain.


an abrupt increase atxchol 5 0.67. The excess chemicalpotential of the acyl chains is plotted in Fig. 4d. It showsthe decrease atxchol 5 0.67 expected from the Gibbs-Duhem equation.

Fig. 6 shows snapshots of acyl chain-cholesterol lateraldistributions simulated with MIEP III. In Fig. 6a, atxchol 50.67, with a low magnitude ofDEc (0.1 kT), the distributionlooks almost random: a cholesterol molecule could havefrom zero to six cholesterol-cholesterol contacts. However,at the higherDEc of 0.6 kT shown in Fig. 6b, cholesterolmolecules form a “maze” pattern: the majority of choles-terols are surrounded by four acyl chains and two choles-terols. The majority of acyl chains are surrounded by fourcholesterols and two acyl chains. Figs. 6c and 10d aresnapshots of lateral distributions simulated withDEc 5 0.6kT atxchol 5 0.60 and 0.72, respectively. Asxchol increases,the maze pattern that gradually forms at lowerxchol, andencompasses the whole lattice atxchol 5 0.67, is graduallydestroyed whenxchol becomes higher. This indicates thatboth the formation and the disappearance of a regular dis-tribution pattern are continuous phase changes.

MIEP III is formulated so that thefirst two cholesterol-cholesterol contacts only cost 1 energy unit each, but thethird one costs 10 units, much higher than the first two. Theenergy costs for the fourth, fifth, and sixth contacts are notcritical to produce the maze pattern, so we chose 3 for them.The mixture responds to MIEP III with a distribution thatminimizes the third cholesterol-cholesterol contact, asshown in Fig. 6b. This regular distribution pattern is a littlemore complicated than the other two. Atxchol 5 0.67, thecholesterol/acyl chain ratio is 1/1. As shown in Fig. 7a, byforming alternating lines of cholesterols and acyl chains, allthe cholesterols can have exactly two cholesterols as nearestneighbors, and no additional cholesterol-cholesterol con-tacts occur. However, there are three possible orientationsof the pattern in Fig. 7a in a hexagonal lattice. In addition,the pattern shown in Fig. 7b has an energy identical to thatof the pattern in Fig. 7a, and it can also be arranged in threepossible ways. Unlike the multidomain aligned dimer patternshown in Fig. 5e, the patterns in Fig. 7 can switch from oneorientation to another with relatively low energy penalty. Themaze pattern is the result of this sixfold degeneracy.

FIGURE 6 Snapshots of phospholipid and cholesterol lateral distributions simulated using MIEP III. (a) xchol 5 0.67 andDEc 5 0.1 kT; (b) xchol 5 0.67andDEc 5 0.6 kT. Cholesterols form a maze pattern: each cholesterol has two other cholesterols as its nearest neighbors; (c) xchol 5 0.60 andDEc 5 0.6kT; (d) xchol 5 0.72 andDEc 5 0.6 kT. (F) Cholesterol; (E) acyl chain.


Consider a tiny amount of cholesterol added to the mix-ture to replace acyl chains and raisingxchol by an infinites-imal amount above 0.67. The added cholesterol is forcedinto sites previously occupied by acyl chains having fourcholesterol contacts. The newly added cholesterol nowtouches the four nearest-neighbor cholesterols already hav-ing three cholesterol-cholesterol contacts. The energy costfor such an operation would be very high. Therefore, thechemical potential of cholesterol sharply jumps atxchol 50.67.

In MIEP III, the third cholesterol-cholesterol contact isassigned to cost 10 energy units. In order to induce a sharpjump in chemical potential of cholesterol,DEc needs to be$0.5 kT (Fig. 4c). Thus, the actual energy cost for the thirdcholesterol-cholesterol contact should be 0.53 10 5 5 kTor higher.

MIEP IV in Table 1 is an interesting parameter set. Theenergy cost for each additional cholesterol-cholesterol con-tact was chosen to be 1 unit higher than the preceding one.Thus,ci increases nonlinearly from 0 to 21. Fig. 2d showsthe excess Gibbs free energy versusxchol simulated usingMIEP IV. Compared with MIEP I, II, and III, the slopechange inDGE at critical mole fractions is more subtle. Fig.4 e shows the excess chemical potential of cholesterol ver-susxchol. This curve has three step increases forDEc $ 3kT: a steep jump atxchol 5 0.5, and more gradual rises at0.57 and 0.67.

Fig. 8 shows a plot ofDGE and DHE, together with2TDSE for DEc 5 0.6 kT, simulated with MIEP II.DGE andDHE are both at global minima atxchol 5 0.57. However,2TDSE has a peak atxchol 5 0.57. This indicates that byforming the aligned dimer pattern atxchol 5 0.57, as in Fig.5 b, the excess entropyDSE is drastically reduced. Theentropy part of the Gibbs free energy2TDSE actuallyincreases to a peak atxchol 5 0.57. Whenever a regulardistribution is formed, regardless of the interaction type, the

excess entropy,DSE, reaches a minimum. Fig. 9 showsDSE

as a function of cholesterol mole fraction for a series ofDEm

values, simulated using pairwise-additive interactions, orequivalently, using MIEP I. AsDEm becomes more nega-tive, the crystal-like regular pattern atxchol 5 0.50 graduallyforms (Fig. 3b); by adopting fewer and fewer configura-tions in phase space, the entropy reaches a minimum atxchol 5 0.50. Once the pattern is completely formed atDEm # 22 kT, DSE stops changing atxchol 5 0.50.

2TDSE is always at a peak whenever a regular distribu-tion occurs. In contrast,DHE is always at a local (sometimesglobal) minimum; however,DGE is not always at a mini-mum, but usually has a sudden change of slope at a regulardistribution composition.

DISCUSSION

We have developed a model for cholesterol-phospholipidinteractions in a bilayer having high cholesterol content.Our model focuses on the key finding that cholesterolprecipitates from bilayer mixtures with PC at a cholesterol

FIGURE 7 Two possible regular distribution patterns atxchol 5 0.67.

FIGURE 8 DGE andDHE, and2TDSE as functions of cholesterol molefraction, forDEc 5 0.6 kT, simulated with MIEP II.

FIGURE 9 Excess entropy,DSE, as a function of cholesterol molefraction, at various values ofDEm, simulated using pairwise-additiveinteractions.


mole fraction of 0.66, and from bilayer mixtures of PE at0.51. Our confidence in the precise numerical value of thesemole fractions is based upon exhaustive experimentationdescribed in recent papers (Buboltz and Feigenson, 1999;Huang et al., 1999).

We find that the most simple model for the microscopicinteraction energies, in which cholesterol-cholesterol, cho-lesterol-phospholipid, and phospholipid-phospholipid ener-gies are all pairwise additive,canexplain the observedx*chol

value of 0.51 for 16:0,18:1-PE. This value of 0.51 appearsnaturally when the energy of unlike molecular contacts issufficiently more favorable than the energies of like con-tacts (see Eq. 4). This is a common situation in mixtures, aform of nonideal mixing in which the mixing is better thanrandom. However, if this pairwise additive description wereadequate, then we would actually learn little about thephysical origin of the interactions, since so many types ofinteractions fall into this category.

A key result from the thermodynamic analysis of oursimulations is that the pairwise additive treatmentcannotexplain the observedx*chol value of 0.66 for the four differ-ent PCs for which we have data. No choice of parameters ina pairwise additive model can yield a steep increase inmchol

bilayer (xchol) at xchol 5 0.66, as is shown in Fig. 4a.We could treat PE and the PCs separately, modeling the

PC/cholesterol mixtures, but not the PE/cholesterol mix-tures, with amultibodyinteraction energy treatment. How-ever, we find that we gain insight by considering PE and PCto be described by the same general type of interactionparameter sets. In this general multibody interaction energyparameter (MIEP) model, we notice the number of choles-terol nearest-neighbors for each cholesterol in the lattice.We then assign the same energy to each cholesterol-choles-terol contact as in MIEP I, or we assign a gradual increasein energy as the number of multibody contacts increases asin MIEP IV, or else we assign a large jump in energy toparticular contacts, e.g., to three-body contacts using MIEPII, or to four-body contacts using MIEP III.

Depending upon the choice of MIEP coefficients, that is,whether and how we assign energy jumps to particularcholesterol-cholesterol multibody contacts, computer simu-lations reveal different patterns for cholesterol-phospholipidlateral distributions:

1. There are three possible critical cholesterol mole frac-tions at which highly regular distributions of cholesterolcan occur (see Note 3). They arexchol 5 0.50, 0.57, and0.67, corresponding to cholesterol/phospholipid mole ra-tios of 1/1, 4/3, and 2/1, respectively. In these regulardistributions, a cholesterol has exactly 0, or 1, or 2 othercholesterols as nearest-neighbors, if there are no defectsin the distribution patterns.

2. When a regular distribution occurs,mcholbilayeralwaysjumps

steeply at this critical mole fraction. Ifmcholbilayer jumps high

enough, it intersectsmcholcrystal. Thus, the MIEP model

predicts that the measuredx*chol values are most likely tobe in the vicinities of three possible critical cholesterol

mole fractions. The enthalpy is always at a local (orglobal) minimum, indicating that a regular distribution isa minimum energy state. The entropy is also at a mini-mum, because the mixture adopts only a few particularconfigurations in phase space.

3. For the regular distribution to occur atxchol 5 0.5, theenergy cost for every cholesterol-cholesterol contactmust be higher than;4 kT; for a regular distribution tooccur atxchol 5 0.57, the energy cost for the additional,second cholesterol-cholesterol contact must be;4 kThigher than that of the first one; and for the regulardistribution to occur atxchol 5 0.67, the energy cost forthe third cholesterol-cholesterol contact must be;5 kThigher than that of the first two.

The umbrella model

In our cholesterol multibody interaction model, in order tohave a steep jump in chemical potential at critical molefractions, the energy cost for some higher-order cholesterol-cholesterol contacts must become much higher than that forlower-order contacts. Here, we propose a simple physicalorigin, which we term the “umbrella model,” for the suddenincrease in energy cost as cholesterols come into contactwith each other in a bilayer.

Cholesterol is largely nonpolar. The lone hydroxyl rep-resents,5% of the mass, and perhaps 1/4 of the surface thatis exposed at the bilayer interface. Exposure to water of thenonpolar part of cholesterol would yield a very unfavorablecontribution to the free energy (Privalov and Gill, 1988;Levy and Gallicchio, 1998). When cholesterols are incor-porated into a phospholipid bilayer, phospholipid head-groups provide “cover” to shield the nonpolar part of cho-lesterol from exposure to water. This is illustratedschematically in Fig. 10a. Phospholipid headgroups act likeumbrellas. The space under the headgroups is shared by acylchains and cholesterols.

As the cholesterol content in a bilayer increases, polarphospholipid headgroups reorient in order to provide morecoverage per headgroup for the increasing fraction of cho-lesterol molecules, as drawn in Fig. 10b. The headgroupumbrellas are “stretched” to provide more coverage area.Under the umbrella acyl chains and cholesterol moleculesbecome tightly packed. No cholesterol is exposed to waterat this point. Cholesterol hydroxyl groups also interact at theaqueous interface to provide partial coverage, but thesehydroxyls cannot completely cover the nonpolar part ofcholesterol without help from phospholipid headgroups.

For a given cholesterol molecule, an additional choles-terol nearest-neighbor comes at the cost of losing an acylchain contact. If neighboring phospholipid headgroups arerequired to shield a given cholesterol from water, then weexpect the energetic cost of each additional cholesterol-cholesterol contact to increase (see Note 4). In particular, iflosing one more phospholipid contact would expose thecholesterol molecule to the aqueous phase, then the energycost would jump sharply. As illustrated in Fig. 10c, if the


phospholipid headgroups are stretched to their limits, theycan no longer provide shielding for additional cholesterols.Exposure of cholesterol to water is very unfavorable. Thismay well be the big increase in energy cost for acquiring anadditional cholesterol-cholesterol contact (or equivalently,for losing an acyl chain contact) needed to produce theregular distribution of cholesterol and the steep jump ofmchol

bilayer at critical mole fractions. To lower the overall freeenergy, instead of allowing the hydrophobic regions ofbilayers to be exposed to water, excess cholesterol mole-cules precipitate, forming cholesterol monohydrate crystals,as shown in Fig. 10d. Therefore, in our umbrella model,x*chol has a physical meaning:x*chol is a cholesterol molefraction at which the capability of phospholipid headgroupsto cover cholesterol molecules from water has reached itsmaximum. Any additional cholesterol in the bilayer wouldbe exposed to water.

The difference inx*chol values of PE and PC may originatefrom the size of their headgroups (Wilkinson and Nagle,1981). Here, we mean the “effective area” of the headgroup,including bound water. The smaller headgroup of PE wouldbe less effective than that of PC in providing shielding, soPE/cholesterol mixtures would have a high energy cost evenfor the first cholesterol-cholesterol contact. According toour simulations, relatively high energy cost for the firstcholesterol-cholesterol contact results inmchol

bilayer risingsharply at 0.50, and thusx*chol 5 0.50. In contrast, the largerheadgroup of PC might well accommodate the firsttwocholesterol-cholesterol contacts, but be overwhelmed by thethird one. The energy cost profile could then be modeled byMIEP III, i.e., very high energy cost for the third choles-terol-cholesterol contact. This would result inx*chol 5 0.67.So if the size of the phospholipid headgroup is the domi-nating factor, then the lack of acyl chain dependence ofx*chol for PCs is explained.

Would the energy cost associated with aqueous exposureof cholesterol, together with any phospholipid deforma-tions, be in the range of 4–5 kT, as estimated from ourcomputer simulations? There are no data available for cho-lesterol-containing mixtures, but studies of the hydrophobicinteraction and lipid deformation in other cases suggest that4–5 kT is within the expected range. For example, King andMarsh measured the critical micelle concentration of spin-labeled PCs by ESR spectroscopy. They estimated the freeenergy of transferring a PC monomer from an aqueousphase to a micelle to be21.1 kT per CH2 group (King andMarsh, 1987). Fattal and Ben-Shaul calculated the lipiddeformation free energy caused by lipid-protein hydropho-bic mismatch in a bilayer, using a molecular model. At amismatch length of 3.5 Å, the deformation free energy canbe as high as 2 kT per lipid (Fattal and Ben-Shaul, 1993).

We have found only one other case in the literature wheremultibody interactions were used for membrane simula-tions. Drouffe and co-workers attempted to simulate a self-assembled two-dimensional membrane from individual am-phiphilic “particles” using molecular dynamics simulations(Drouffe et al., 1991). They found that a membrane wouldnot be stable unless a multibody interaction, which assignsa large free energy cost for exposing the hydrocarbon chainsto water, is included in the simulation. This finding supportsthe importance of a hydrophobic effect as an essentialmembrane interaction.

The umbrella model and the “condensing effect”

The umbrella model implies the physical origin for thelong-known “cholesterol condensing effect” (the averagemolecular area in a cholesterol-phospholipid monolayer isless than the sum; or, the range of motion of phospholipidacyl chains in a bilayer is reduced by cholesterol (Leathes,1925; Demel et al., 1967; Stockton and Smith, 1976; Vistand Davis, 1990). These effects are also related to thedecrease of bilayer permeability with increasing cholesterol

FIGURE 10 The physical origin of the maximum solubility of choles-terol in a bilayer, illustrated by the “umbrella model.” Cholesterol mole-cules are represented as ovals. (a) In a lipid bilayer, the polar phospholipidheadgroups act like umbrellas, shielding the nonpolar part of cholesterolmolecules from water; (b) as the concentration of cholesterol increases,phospholipid headgroups are stretched to provide coverage for the increas-ing number of cholesterols; (c) the phospholipid headgroups are stretchedto their limit and can no longer provide shielding for all cholesterolmolecules. Some cholesterol molecules (shaded ovals) are exposed towater, causing a steep jump in chemical potential of cholesterol. (d) Tolower the free energy, excess cholesterol molecules precipitate and form asecond phase: cholesterol monohydrate crystals. The bilayer phase nowcontains the maximum number of cholesterols that the phospholipid head-groups can cover.


content (Kinsky et al., 1967). In the umbrella model, acylchains and cholesterols become more tightly packed ascholesterol content increases, because they share the limitedspace under phospholipid headgroups. The hydrophobicnature of cholesterol thus forces cholesterol and acyl chainstogether. The model also implies that while acyl chains arecompressed, phospholipid headgroups expanded.

Thus we propose that the hydrophobic interaction couldbe the common driving force for cholesterol precipitationfrom the bilayer, as well as for the condensing effect/acylchain ordering.

Regular distribution and multibody interaction

The regular distribution pattern shown in Fig. 3b is iden-tical to one of the so-called “superlattice” distribution pat-terns of pyrene-PC or cholesterol in phospholipid bilayers.The proposed superlattice patterns were initially based onobservation of a series of “kinks” or “dips” in the ratio ofexcimer-to-monomer fluorescence at particular mole frac-tions of pyrene-PC in bilayers (Somerharju et al., 1985;Tang and Chong, 1992; Chong et al., 1994). The bulkypyrene moieties were thought to form hexagonal superlat-tices in order to maximize separation from each other. Later,fluorescence data on cholesterol/phospholipid mixtures in-dicated that cholesterol molecules might also form super-lattices in lipid bilayers (Chong, 1994; Virtanen et al., 1995;Liu et al., 1997). Superlattice patterns atxchol 5 0.118,0.154, 0.20, 0.25, 0.33, 0.40, and 0.5 were suggested basedon geometrical symmetry arguments. A regular distributionat xchol 5 0.67 (Fig. 7a) has been proposed, but does not fitthe general form of the superlattice (Lundberg, 1982; Vir-tanen et al., 1995). Sugar et al. explored the origin ofsuperlattice patterns by using Monte Carlo simulations,introducing a long-range pairwise-additive repulsive inter-action (Sugar et al., 1994). They found that long-rangepairwise-additive repulsion can generate a superlattice pat-tern atxchol 5 0.5, like the pattern in Fig. 3b, but cannotproduce any other large-scale superlattice patterns. How-ever, the general form of microscopic interaction that couldproduce the superlattices has remained unclear. In contrast,we have shown that the superlattice pattern in Fig. 3b canalso be generated with a nearest-neighbor pairwise-additiveinteraction,without a long-range force.

Pairwise-additive interactions are commonly used tomodel microscopic interactions between phospholipid mol-ecules (Jorgensen and Mouritsen, 1995; Sugar et al., 1994;Huang et al., 1993), phospholipid and sterols (Cruzeiro-Hansson et al., 1989), as well as phospholipid and proteins(Zhang et al., 1993). In contrast, multibody interactions arerarely used for modeling lipid bilayers. This study clearlydemonstrates that pairwise-additive interactions are inade-quate to model the observed maximum solubility of choles-terol in bilayers; a multibody interaction is indispensable.The multibody interaction introduced in Eq. 5 can generatea series of regular distributions atxchol $ 0.5. This new

class of regular distributions is based on molecular contact,not on maximum molecular separation, as proposed for thebilayer superlattices. It is interesting to note that the focus ofthe studies of Sugar et al. was to obtain a regular distributionof bulky molecules, whereas ours was to obtain a steep jumpin cholesterol chemical potential. By calculating the freeenergy, we show that these two seemingly unrelated phe-nomena are actually the two inseparable consequences of aclass of multibody interactions.

The usefulness of the multibody interaction treatment isnot limited to description of the cholesterol interaction inbilayers of very high cholesterol content. It is readily ex-tended to the low cholesterol content regime by introducinga different multibody interaction, associated with the localenvironment of each acyl chain. We have successfully sim-ulated many cholesterol superlattice patterns atxchol , 0.5(Huang and Feigenson, manuscript in preparation). We cannow conclude that regular distributions are generally cre-ated by multibody interactions.

Acyl chain effects

The chemical potential of cholesterol in bilayers must havesome contribution from the microscopic interaction of cho-lesterol with its acyl chain neighbors. Cruzeiro-Hansson etal. (1989) calculated a cholesterol/phospholipid phase dia-gram for the lowxchol regime, in which different interactionenergies were assigned between cholesterol and acyl chainsfor different acyl chain conformations. We looked for sucheffects by measuringx*chol in long- and short-chain PCs. The(pure PC) di22:1-PC bilayer is nearly 40% thicker than thedi12:0-PC bilayer (Caffrey and Feigenson, 1981), so wemight expect some difference in the interactions with cho-lesterol. Instead, we find thatx*chol are essentially identicalin these two PCs, as well as in 16:0,18:1-PC and di16:0-PC,so the cholesterol-acyl chain contribution tomchol

bilayer (xchol)must be considerably smaller than that from cholesterol-cholesterol nearx*chol.

Our Hamiltonian is built to explore the cholesterol-cho-lesterol interactions that dominate in causing cholesterolprecipitation nearx*chol. There are many other types ofinteractions that are included in our Hamiltonian, whichcould become dominant under different conditions. Forexample, at lowerxchol, where cholesterol-cholesterol con-tacts are almost absent, cholesterol-acyl chain interactionscould have an important role. However, the lowxchol regimeis not the focus of this study.

Absence of cholesterol-phospholipid complexes

As seen above, there are indeed “special mole ratios” ofcholesterol/phospholipid, e.g., at 1/1 or 4/3 or 2/1. Distinctstructural and thermodynamic events take place, such as theappearance of a regular distribution and a chemical potentialjump. However, these special mole ratios that we identify


haveno connection whatsoeverto any chemical equilibriumbinding reaction, such as 2Chol1 PL º PL(Chol)2 or4Chol1 3PLº PL3(Chol)4. To clarify this point, considerthe aligned dimer pattern atxchol 5 0.57 shown in Fig. 5b.Superficially, it might appear to be a cholesterol-to-acylchain stoichiometric 2/3 complex. However, at the lowercholesterol composition shown in Fig. 5c, many cholesteroldimers are already formed, but no particular cholesterol/phospholipid ratio is associated with each dimer! The dimerpattern atxchol 5 0.57 is simply a minimum energy pattern,in which all cholesterols avoid the costly second cholester-ol-cholesterol contact:there is no chemical association be-tween phospholipid and cholesterol.In fact, there is nochemical association energy term in our Hamiltonian, andyet dramatic events can happen at “special mole fractions.”Therefore, it is incorrect to generalize all distinct distribu-tions that occur at special mole fractions to be complexformation.

Testable predictions

Our MIEP model predicts thatx*chol is most likely to occurat certain discrete values, in the vicinities of eitherxchol 50.500, or 0.571, or 0.667. So far, we found that four PCshave x*chol at 0.66, and one PE at 0.51. Of course, moreequilibrium x*chol data for other phospholipids would pro-vide a valuable test of this model.

x*chol 5 0.57 is a new critical mole fraction that naturallyarises from our model. There is no known report ofx*chol at0.57 so far, nor have its regular distribution patterns (Fig.5b) been described previously. Yet, the cholesterol-choles-terol multibody interactions that would give rise to thesepatterns and tox*chol 5 0.57 would seem to be accessible,given the great variety of phospholipid headgroups. In termsof the microscopic energy, the requirement is that the three-body interaction of cholesterols should have a jump inenergy, compared with the two-body cholesterol-cholesterolinteraction, i.e., MIEP II should apply. It would be interest-ing to find some phospholipids havingx*chol 5 0.57. Rea-sonable candidates would be phospholipids with headgroupsize intermediate between that of PE and PC, such as monoor dimethyl-PE. Mixtures of PE and PC might showx*chol 50.57, for example if PE and PC in a bilayer mixture distrib-ute randomly around cholesterol-cholesterol dimers.

If cholesterols do distribute regularly in bilayers atx*chol,as pictured in Figs. 3b and 6b, such organization of cho-lesterol and acyl chains might be detectable, for example byneutron diffraction.

In this study we have partially characterized a particularinteraction that dominates all others in the special case ofbinary phospholipid-cholesterol mixtures at highxchol. Al-though the study is restricted in this way, the finding ofhighly unfavorable cholesterol-cholesterol interaction seemsapplicable to real plasma membranes.

NOTES

1. In this simulation, two acyl chains belonging to the same phospho-lipid are not physically linked. A new simulation procedure to implementsuch a link has been recently reported (Sugar et al., 1998). We will try toincorporate it in our future simulations. We anticipate that this newprocedure might have some effect on the entropy contribution to the Gibbsfree energy.

2. In Fig. 4 and all other figures,xchol represents the cholesterol molefraction in the bilayer. It can be different from the overall cholesterolcontent. For example, if the chemical potential jumps high enough atxchol 5 0.5 and reaches the value ofmchol

crystal, any additional cholesterolwould precipitate to form cholesterol monohydrate crystals;mchol

bilayer wouldstop at the value ofmchol

crystal, and the cholesterol mole fraction in the bilayerwould remain constant, even as the overall cholesterol content exceeds 0.5.

3. There are other theoretical critical mole fractions, higher than 0.67,at which regular distributions can occur (each cholesterol having three ormore cholesterol contacts). Since we have not yet found an experimentalx*chol value higher than 0.67, we do not discuss these here.

4. The hydrophobic interaction has an overall unfavorable contributionto the free energy (Privalov and Gill, 1988; Levy and Gallicchio, 1998). Atroom temperature it is energetically favorable, but entropically very unfa-vorable. Since our lattice model does not explicitly contain water mole-cules, this unfavorable free energy contribution is phenomenologicallyexpressed as an unfavorable interaction energy.

The authors thank Jeffrey Buboltz for many helpful discussions.

This work was supported by National Science Foundation Grant MCB-9722818.

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a microscopic interaction model of maximum solubility of

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