Calculating Partition Coefficients of Chain Anchors in Liquid-Ordered and Liquid-Disordered Phases

Transcription

1 Biophysical Joural Volume 98 May Calculatig Partitio Coefficiets of Chai chors i Liquid-Ordered ad Liquid-Disordered Phases Mark J. Ulie, Gabriel S. Logo, M. Schick, ad Igal Szleifer * Departmet of Biomedical Egieerig, Northwester Uiversity, Evasto, Illiois; The bdus Salam Iteratioal Cetre for Theoretical Physics, Trieste, Italy; ad Departmet of Physics, Uiversity of Washigto, Seattle, Washigto BSTRCT We calculate partitio coefficiets of various chai achors i liquid-ordered ad liquid-disordered phases utilizig a theoretical model of a bilayer membrae cotaiig cholesterol, dipalmitoyl phosphatidylcholie, ad dioleoylphosphatidylcholie. The partitio coefficiets are calculated as a fuctio of chai legth, degree of saturatio, ad temperature. Partitioig depeds o the differece betwee the lipid eviromets of the coexistig phases i which the achors are embedded. Cosequetly, the partitio coefficiet depeds o the ature of the achor, ad o the relative compositios of the coexistig phases. We fid that saturated achors prefer the deser liquid-ordered phase, ad that the fractio of achors i the liquid-ordered phase icreases with icreasig degree of saturatio of the achors. The partitio coefficiet also depeds upo the locatio of the double bods. chors with double bods closer to the middle of the chai have a greater effect o partitioig tha those ear the ed. Doublig the umber of saturated chais icreases the partitioig ito the liquid-ordered phase for tails that are early as log or loger tha those comprisig the bilayer. Partitioig of such chais icreases with decreasig temperature, idicatig that eergy cosideratios domiate etropic oes. I cotrast, partitioig of shorter chais icreases with icreasig temperature, idicatig that etropic cosideratios domiate. INTRODUCTION Oe of the most iterestig, ad cotroversial, models of biological membraes posits that they are ot compositioally homogeeous, but that aggregates of saturated lipids ad cholesterol float, like rafts, i a sea of usaturated lipids (1 6). further hypothesis is that the rafts ad the sea are i fact domais of two coexistig phases deoted liquidordered (lo) ad liquid-disordered (ld), respectively (7). Coexistece of two liquid phases is, ideed, commoly observed i model membraes that cosist of a terary mixture of saturated ad usaturated lipids ad cholesterol (8). Oe reaso for the great iterest i this picture is that a ihomogeeous membrae would provide a substrate withi which differet proteis would be eriched, either i the rafts or i the sea, thereby icreasig their efficacy. This erichmet follows from the differet hydrocarbo chais that achor may membrae proteis: extracellular achors attached to the membrae via a glycophosphatidyliositol achor that bury two such chais ito the ocytoplasmic moolayer, itracellular achors attached to the cytoplasmic moolayer by meas of a sigle chai ofte derived from saturated myristic or palmitic acid, or from a bulky ad usaturated preyl group. Presumably saturated achors prefer the eviromet of the raft (i.e., liquidordered phase), rich as it is i saturated lipids, whereas usaturated ad/or otherwise bulky achors prefer the eviromet of the sea of usaturated lipids (i.e., liquid-disordered phase). To test this hypothesis, systems of lipids ad lipidated peptides have bee studied to determie the effect Submitted October 22, 2009, ad accepted for publicatio Jauary 19, *Correspodece: igalsz@orthwester.edu Editor: Thomas J. McItosh. of lipid chai legth, degree of usaturatio, etc., o the peptides partitio coefficiet, i.e., the ratio of their cocetratio i the lo phase to that i the coexistig ld phase (9 15). These studies have foud that the partitioig ito the lo phase is greatest for saturated multichai achors. The partitioig icreases with icreasig degree of saturatio ad/or umber of saturated chais. The results also idicate that temperature has a greater effect o the partitioig of loger carbo chai achors (16). s itriguig as the phase-separatio hypothesis is for uderstadig the heterogeeity of biological membraes, there are several competig theories. (Two excellet reviews o this topic ca be foud i Hacock (17) ad Semrau ad Schmidt (18) ad their cited refereces.) Oe suggestio is that rafts be idetified with the fluctuatios that arise i a terary fluid mixture that cotais a iterfacially active aget, as i a microemulsio (17,19,20). other arises from a recet experimet (21), which suggests that membraes could be close to a miscibility critical poit. This would imply the existece of strog compositio fluctuatios withi the system: fluctuatios that could be idetified as rafts. The existece of a critical poit suggests aother possibility. Suppose that the system were i a oe-phase regio, say the ld phase, but ot far from ld-lo coexistece. If a array of proteis that favored the lo phase were itroduced ito the system, the it is likely that they would iduce a wettig trasitio at coexistece. I fact, before the miscibility critical poit is reached alog coexistece, such a trasitio has to occur accordig to Cah s well-kow argumet (22,23). Off of coexistece, there would be a regio of lo-like fluid surroudig ay such protei. This is a regio i which the radius would be fiite, because the system is off of Ó 2010 by the Biophysical Society /10/05/1883/10 $2.00 doi: /j.bpj

2 1884 Ulie et al. coexistece (so that the lo phase is ot thermodyamically stable), ad because the proteis are ot i a array (24) (i.e., they occur as moomers or i small clusters). There are also dyamic models of raft formatio (25) that, biologically, take recyclig ad diffusio ito accout. Give this cotroversy o the ature ad existece of rafts, it would be very useful to calculate the partitio coefficiet withi the phase separatio hypothesis ad a model system, ad to determie its variatio with the architecture of the achor. To do so, of course, requires ot oly a reasoable descriptio of the achors, but also a model that produces coexistig liquid-ordered ad liquid-disordered phases. Such a model would be comprised of a miimum of three compoets: saturated lipids, usaturated lipids, ad cholesterol. We have carried out such a program. The partitio coefficiets are calculated as a fuctio of chai legth, degree of saturatio, ad temperature. Perhaps our most importat observatio is model-idepedet: that the partitioig depeds strogly o the lipid eviromet i which the achors come ito cotact, ad therefore the partitio coefficiet depeds ot oly o the ature of the achor, but, for a give achor, must also deped o the overall compositio of the system. This compositio determies the differece i cocetratios of the compoets i the coexistig lo ad ld phases. Such differeces ca be ifiitesimal, such as at a critical poit (21,26). I this case, the partitio coefficiet must be uity, or the differeces betwee lo ad ld phases ca be large (i which case, the partitio coefficiet may differ substatially from uity). From our theoretical model-membrae system, we report calculated partitio coefficiets for sigle chais of various legths ad saturatio ad for a few cases of double-chaied molecules. The results reproduce the observatios that partitio coefficiets icrease with chai legth ad degree of saturatio. They also predict that variatio of the legth of the achors, which are short compared with those makig up the bilayer, has little effect o the partitio coefficiet, ad that double bods located ear the eds of chais have less effect tha those ear the middle. MODEL TERNRY SYSTEM We cosider a fluid, plaar, symmetric lipid bilayer of thickess L. This is composed of a terary system of cholesterol, a lipid with two saturated hydrocarbo chais of 16 carbos, C16:0, such as dipalmitoylphosphatidylcholie (DPPC), ad a lipid with two moousaturated hydrocarbo chais of 18 carbos, C18:1, such as dioleoylphosphatidylcholie (DOPC). The chais are described by Flory s Rotatioal Isomeric States Model (27,28) i which each CH 2 group is i oe of three cofiguratios: the lowest eergy tras, or the gauche-plus or the gauche-mius (both of which are of a eergy, 500 cal/mol, greater tha that of the tras cofiguratio). These states are thermally populated. There are N molecules of which a fractio, x c, is cholesterol; x s, saturated lipid; ad x u, usaturated lipid. Whe ecessary, we idicate molecules i the ier leaflet by a subscript (i) ad those i the outer leaflet by (o). We treat separately the cotributios to the free eergy arisig from the lipid hydrocarbo chais ad from the polar headgroups. From our mea-field theory, we write the total free eergy per molecule, f h F/N, i the form of a sum of the two itralayer free eergies ad a couplig betwee them, bf ðt; a; x s ; x u ; x c Þ¼ X bf d þ bf coupl ; (1) d ¼ i;o where b ¼ 1/k B T. The itralayer free eergies arise from the iteractio of the chais with the polar headgroups ad exteral water; from the etropy of mixig of the three compoets; from the free eergy arisig simply from the may differet cofiguratios of the sigle chais; ad from a iteractio betwee chais i the same leaf. These four cotributios are writte bf d hbg 0 a þ X x i 2 l x i l 2 i =a þ bf cofig;d þ bf oriet;d ; ð2þ i ¼ s;u;c where g 0 is the water-hydrocarbo iterfacial free eergy, ad a is the molar area of the midplae of the bilayer, the total volume of the bilayer divided by its thickess L: aðx s ; x u ; x c ;LÞ ¼ 2x s s ð s þ 1Þ þ2x u ðð u 1Þ s þ 2 CH Þþx c c c : L Here s, CH, ad c are the volumes of a CH 2 uit, of half a CH ¼ CH uit, ad of each carbo uit i cholesterol. The thermal wavelegth of the i th compoet is deoted l i. The cofiguratioal free eergy of the chais is bf cofig;d h X x i X 2 tails i P i;d ða i;d Þðl P i;d ða i;d Þþbeða i;d ÞÞ; i a i;d (3) where tails i is the umber of tails of species i; i.e., tails s ¼ tails u ¼ 2; tails c ¼ 1: Further P(a j, d ) is the probability of fidig the chai of molecular species j i leaf d i a particular coformatio specified by the idex a j, d, oe with a total iteral eergy e(a j, d ). The iteral eergy arises from the gauche bods whose eergies exceed that of the tras cofiguratio. The orietatioal iteractio betwee chais i ad j i the same leaf is writte bf oriet;d h X i X j x i x j bj i;j 2 2 tails i tails j s Z xi;dðzþ x j;dðzþ dz; where hx i, d (z)i is the esemble average of the local desity of bods weighted by their relative orietatio to the bilayer ormal. (4)

3 Partitio Coefficiets of Lipid chors 1885 The explicit couplig of the chais from the two differet leaves is due to the iteractio betwee chais that overlap ear the midplae. It is of the same form as the iteractio betwee chais withi the same leaf, bf coupl ¼ X i X j x i x j bj i;j 2 2 tails i a s tails j Z xi;oðzþ x j;iðzþ dz: I sum, the free eergy of the system is give by Eqs The two leaves are ot oly coupled explicitly by this iteractio but also implicitly by the fact that these overlappig chais each cotribute to the volume of the hydrophobic regio, a regio whose desity we costrai to be costat (i.e., that it be icompressible). The costat desity implies adz ¼ X x i 2 tails i h i;d ðzþidz (6) i;d for 0 < z < L, where h i, d (z)i is the esemble-averaged cotributio to the volume of the hydrophobic core at z from the molecules of species i i the leaf d. The costrait is imposed by meas of a local Lagrage multiplier, p(z), which appears i the probability fuctios P(a i ). They have the same form for all of the compoets, Pða i Þ¼ 1 Z exp beða i Þ bpðzþ i ða i ; zþdz q i Z (5) bb i ðzþx i ða i ; zþdz ; (7) where q i is the coformatioal part of the partitio fuctio of a sigle chai of compoet i q i ¼ X Z exp beða i Þ bpðzþ i ða i ; zþdz a i Z bb i ðzþx i ða i ; zþdz : (8) The mea-fields bb i (z) act upo the orietatio of a molecule of species i at the coordiate z ad result from the average orietatioal iteractios arisig from the other molecules. They are determied by the equatios bb l ðzþdz ¼ bj llx s xs;o ðzþ þ x a s;i ðzþ dz s bj llx u a s xu;o ðzþ þ x u;i ðzþ dz bj lcx c s xc;o ðzþ þ x c;i ðzþ dz: The fields actig o saturated or usaturated lipids are idetical, bb s ðzþdz ¼ bb u ðzþdzhbb l ðzþdz; whereas the field actig o the cholesterol orietatio is distict ad is give by (9) bb c ðzþdz ¼ bj lcx s xs;o ðzþ þ x s;i ðzþ dz a s bj lcx u xu;o ðzþ þ x a u;i ðzþ dz s bj ccx c s xc;o ðzþ þ x c;i ðzþ dz: (10) Equatios 6, 9, ad 10 comprise the set of self-cosistet equatios whose solutio yields the equilibrium values of the fields p(z), b l (z), ad b c (z) from which all desities, ad the free eergy follows. Further details of the model ad its solutio ca be foud i ppedix. The model produces a phase diagram that exhibits three phases: oe rich i cholesterol ad saturated lipids, which we idetify with the liquid-ordered (lo) phase; oe rich i usaturated lipids, which we idetify with the liquiddisordered (ld) phase; ad oe cosistig almost etirely of saturated lipids, the gel phase. The phase diagram at 290 Kelvi (K) is show i Fig. 1. This figure, alog with Fig. 2 of Elliott et al. (29), is i good qualitative agreemet with the experimetal phase diagrams for the DPPC, DOPC, ad cholesterol mixture preseted i Davis et al. (16), ad with the PSM, DOPC, ad cholesterol mixture determied by de lmeida et al. (30). Marsh (6) presets several other experimetal, three-compoet phase diagrams of the geeral form that we see i this work. Examples of such systems preseted i Marsh (6) are PSM, POPC, ad U ld C lo FIGURE 1 The phase diagram of the DPPC (S), DOPC (U), ad cholesterol (C), lipid bilayer at 290 K as obtaied from our model. The regio of the gel phase is deoted G, that of the liquid-ordered phase is deoted lo, ad the liquid-disordered phase, ld. Bold lies represet the phase boudaries. Tie lies coect poits at compositios i coexistece with oe aother. The stars represet the locatio where the calculatios for the partitio coefficiet took place for T ¼ 290 K. The regio of three-phase coexistece is deoted 3. Parameters are J ll ¼ k B T*(T* ¼ 315 K), J lc ¼ 0.85 J ll, ad J cc ¼ 0.80 J ll. 3 G S

4 1886 Ulie et al. lo ld x lo ld x : :1 c9 0.4 :2 c9,12 :3 c9,12, :3 c6,9, :4 c5,8,11, B cholesterol alog with DPPC, POPC, ad cholesterol, ad DSPC, SOPC, ad cholesterol. THE PRTITION COEFFICIENT We ow add to the terary mixture a ifiitesimal amout of a fourth compoet, a achorig molecule whose coefficiet of partitio betwee coexistig lo ad ld phases we wish to compute. The partitio coefficiet of this molecule betwee liquid-ordered ad liquid-disordered phases is, by defiitio, k :0 :1 c9 :2 c9,12 C :3 c9,12,15 C:3 c6,9, C:4 c5,8,11, FIGURE 2 Partitio coefficiet for various sigle achor chais embedded i a membrae cosistig of model DPPC, DOPC, ad cholesterol. The mole fractios of the chai achors i the liquid-order ad liquid-disorder phases are x lo ad x ld, respectively. The umber of carbos i the chai is. () The temperature is 290 K. The cocetratios of the lo phase are x c ¼ 0.48, x s ¼ 0.30, ad x u ¼ 0.22, whereas those of the ld phase are x c ¼ 0.27, x s ¼ 0.21, ad x u ¼ (B) The temperature is 300 K. The cocetratios of the lo phase are x c ¼ 0.52, x s ¼ 0.26, ad x u ¼ 0.22, whereas those of the ld phase are x c ¼ 0.32, x s ¼ 0.22, ad x u ¼ ¼ xlo : (11) x ld The chemical potetial of this molecule must be the same i the coexistig phases, ad from this coditio, we obtai the partitio coefficiet. We start by decomposig the chemical potetial of the added molecule as bm ¼ l x þ b~m ; where b~m is that part of the chemical potetial that does ot deped upo the cocetratio of the added molecule. From the equality of the chemical potetial i the coexistig phases, we obtai k ¼ xlo x ld ¼ exp b~m ld b~mlo : (12) I ppedix B the chemical potetial of a achorig molecule, with a umber of chai achors, tails, is derived followig the potetial distributio theorem (31). It is give exactly by l 2 bm ¼ l x þ l l X hexpðbu Þi N1 ; (13) a where h.i N 1 represets the esemble average of the Boltzma factor of the achor molecule. The average is over all the cofiguratios of the three-compoet system i the absece of the achor. The sum over a is a sum over all of the coformatios of the iserted achor molecule, u is the total eergy of iteractio experieced by the iserted chai achor molecule, ad l is its de Broglie wavelegth. s the esemble average caot be readily evaluated, we approximate it from our mea-field calculatio. The expressio for this chemical potetial i our model is derived i ppedix B ad is foud to be bm ¼ l x þ b~m ; (14) l 2 ¼ l x þ l l Ytails k ¼ 1 q k ; (15) where q k (T, a, x s, x u, x c ) is the cofiguratioal part of the partitio fuctio of the chai achors that excludes their traslatioal degrees of freedom. Because the cocetratio of the achor is ifiitesimal, the partitio fuctio is idepedet of its cocetratio. From the equality of the achor chemical potetial i the coexistig phases, Eq. 12, we immediately obtai k ¼ xlo x ld ¼ Qtails lo a q k T; a lo ; xs lo; xlo u ; xlo c k ¼ 1 Qtails ald q T; a ld ; xs ld; xld u ; ; (16) xld c ¼ 1 where a lo ad a ld are the area per molecule i the liquidordered ad -disordered phases, respectively. The ecessary partitio fuctios ca be calculated i a maer similar to

5 Partitio Coefficiets of Lipid chors 1887 those of the other three compoets. The cofiguratioal part of the partitio fuctio for a chai achor embedded i the outer leaf of a membrae i the ld phase is q T; a ld ; x ld s ; xld u ; X xld c ¼ exp beða ;o Þ a ;o Z p b ld ðzþ ða ;o; zþ þb ld l ðzþx ;o ða ;o ; zþ (17) dz ; where the fields bp ld ðzþ ad bb ld 1 ðzþ are kow from the terary system ad do ot have to be recalculated. similar expressio obtais for a chai embedded i the coexistig lo phase. Because the partitio fuctios deped upo the cocetratios i the coexistig phases, it is clear from Eq. 16 that the partitio coefficiet must also deped upo these cocetratios. I fact, there are some limitig behaviors that oe ca determie without further calculatio. The most simple behavior obtais whe the system approaches a critical poit. Because the differece i the lipid compositios of the coexistig phases vaishes as the critical poit is approached, the partitio coefficiet must approach uity there. s a secod example, suppose that the terary system phase separates ito lo ad ld phases with saturated lipid cocetratios xs lo, ad xld s. If oe were to add a idetical two-tailed saturated lipid to this system, its partitio coefficiet would obviously be give by k ¼ xs lo=xld s. If we cosider a achor with these same two tails, the to the extet that its partitio coefficiet is domiated by the hydrocarbo tails, we expect its partitio coefficiet to be the same. Further, i a homologous series of saturated, twotailed achors (di C m :0) with the two-tailed saturated lipids of the membrae beig characterized by the iteger m mem, we ot oly expect the partitio coefficiet k (m) to take for m ¼ m mem but also to approach this value smoothly as m approaches m mem. Because the coexistig cocetratios of lipids deped upo the overall cocetratio of the system, the partitio coefficiet of this achor must also deped upo the overall compositio of the system, ad ot just o the compoets of which it is comprised. further result, valid withi mea-field theory, ca be derived cocerig the effectiveess of multiple achors compared to a sigle achor; that is, the relatio betwee the partitio coefficiet of a achor with multiple chais as compared to a achor with a sigle chai. From Eq. 16, oe fids easily that the value xs lo=xld s k chais ¼ a ld a lo ð chais 1Þ ½k ð1þš chais ; (18) which shows how much more effective a multichai achor ca be i effectig a large partitio of protei. Lastly, withi the same mea-field approach, oe also fids that the partitio coefficiet of a achor with two differet chais, say a saturated oe ad a usaturated oe, k su, is related to the partitio coefficiets of achors with two of the saturated chais, k ss ad two of the usaturated chais, k uu, accordig to k su ¼½k ss k uu Š 1=2 : (19) We treat the achors o the same level of approximatio as the lipids i the lipid bilayer. The exact chemical architecture of the likers i the achor is ot take ito accout explicitly, just as the glycerol carbo backboe is ot take ito accout explicitly withi our models of DPPC ad DOPC. The lipid bilayer ad the achor chais are ultimately take to be idepedet, ad the field variables are solved for i a mea-field maer so that everythig is treated withi the same level of approximatio. It is also importat to ote that we assume that the partitioig of molecules that have these lipid achors is domiated by the partitioig of the lipid achors themselves. We oly treat the chais of the lipids explicitly, ad we treat the absorbig molecules at the exact same level of approximatio. We could cosider the etire adsorbig molecule alog with the headgroups of the lipids explicitly i this theory; however, that is curretly beyod the scope of this work. s ca be see from our derivatio of the partitio coefficiet i ppedix B, we perform these calculatios i the ifiite-dilutio limit, ad therefore the chais of the achor have o effect o the structure of the equilibrium lipid bilayer. Were oe iterested i the effects of a large cocetratio of achors, oe would have to cosider a mixture of four compoets by meas of a straightforward, but tedious, extesio of the calculatio preseted here. Give the agreemet foud with experimets, we believe that ifiite dilutio is a good approximatio. RESULTS ND DISCUSSION We have calculated the partitio coefficiets of various achors betwee lo ad ld phases that are i coexistece with oe aother at a temperature of T ¼ 290 K ad of T ¼ 300 K. We first focus o the results for T ¼ 290 K. We choose a coexistece of lo phase characterized by a area per molecule of Å 2 ad a ld phase characterized by Å 2. The cocetratios of the lo phase are x c ¼ 0.48, x s ¼ 0.30, ad x u ¼ 0.22, whereas those of the ld phase are x c ¼ 0.27, x s ¼ 0.21, ad x u ¼ The partitio coefficiets for several sigle-chai achors are show i Fig. 2 as a fuctio of the chai legth. The otatio C m : kcp 1,., p k idicates a C m chai that cotais k cis double bods. The positios of the double bods i the C m chai are give by p 1,.p k, where p j idicates that there is a double bod betwee C p ad C pþ1. Thus C m :2c9, 12 is a double-usaturated chai with cis double bods betwee C9 ad C10 ad betwee C12 ad C13. Examples of the molecules with chais described i Fig. 2 are (from top to bottom, startig o the black curve) myristic acid, C 14 :0,

6 1888 Ulie et al. ad stearic acid C 18 :0; (o the red curve) oleic acid C 18 :1 c9 ad palmitoleic acid C 16 :1 c9; (o the gree curve) lioleic acid C 18 :2 c9, 12; (o the blue curve) a-lioleic acid C 18 :3 c9, 12, ad 15; (o the mageta curve) g-lioleic acid C 18 :3 c6, 9, ad 12; ad (o the brow curve) arachidoic acid C 18 :4 c5, 8, 11, ad 14. We ote that several treds observed i experimet (9,32,33) are reproduced here. I particular, the partitio coefficiets for the saturated achors icrease with the legth of the achor chai provided oly that the chais are loger tha 11 carbos. The rate of icrease is ot uiform. Ideed, we observe that saturated chais of legth 18 ad 20 partitio equally, just as observed i experimet (9). We fid that the partitio coefficiet icreases with icreasig degree of saturatio, agai i agreemet with experimet (9). Oe also otes from a compariso of the results for C m :3 c9, 12, ad 15 ad C m :3 c6, 9, ad 12 that the locatio of the double bods affects the partitio coefficiet. I particular, it is larger if the double bods are earer the eds of the chai. Presumably, this is because the order parameter, eve of the saturated chais comprisig much of the bilayer, decreases toward the ed of the chais. Thus, double bods of the achor disturb the lo phase less whe they are close to the chai eds. We have also calculated partitio coefficiets for three classes of double-tailed achors: those with two saturated chais, di-c m :0; those with oe saturated ad oe moousaturated chai, C m :0 C m :1 c9; ad those with two moousaturated chais, di-c m :1 c9. The results are show i Fig. 3. s expected, the partitio coefficiet of the diusaturated chais is least; that of the mixed chais is larger, ad that of the disaturated chais is largest. We ote, however, that the differeces are small. The value of the partitio coefficiet of di-c 16 :0 is 1.43 ad is show by a star i the figure. This value correspods to the partitio coefficiet of DPPC i the equilibrium bilayer which, from the cocetratios give above, is xs lo=xld s ¼ 0:30:0:21 ¼ 1:43. The agreemet is to be expected, as stated earlier ad show explicitly i ppedix B. The value of the partitio coefficiet of di-c 1 8:1 c9 is 0.42, also show by a star i the figure, ad correspods to the partitio coefficiet of DOPC i the equilibrium bilayer, xu lo=xld u lo ld x lo ld x di-saturated (di- :0) sat.-usat. ( :0-1:9) 0.5 di-usaturated (di- :1:9) B di-saturated (di- :0) sat.-usat. ( :0- :1c9) di-usaturated (di- :1c9) FIGURE 3 Partitio coefficiet for various double-chai achors embedded i a membrae cosistig of model DPPC, DOPC, ad cholesterol. The mole fractios of the chai achors i the liquid-order ad liquid-disorder phases are x lo ad x ld, respectively. The umber of carbos i the chai is. The stars represet the values of the partitio coefficiet i the bulk lipid bilayer. () The temperature is 290 K. The cocetratios of the lo phase are x c ¼ 0.48, x s ¼ 0.30, ad x u ¼ 0.22, whereas those of the ld phase are x c ¼ 0.27, x s ¼ 0.21, ad x u ¼ (B) The temperature is 300 K. The cocetratios of the lo phase are x c ¼ 0.52, x s ¼ 0.26, ad x u ¼ 0.22, whereas those of the ld phase are x c ¼ 0.32, x s ¼ 0.22, ad x u ¼ ¼ 0:22:0:52 ¼ 0:42. To study the effect of temperature o the partitio coefficiet, we repeat our calculatios for a temperature of T ¼ 300 K. We choose a coexistece at which the lo phase has a area per molecule of 47.0 Å 2, ad cocetratios of x c ¼ 0.52, x s ¼ 0.26, ad x u ¼ 0.22, whereas the ld phase has the larger area per molecule of Å 2, ad cocetratios x c ¼ 0.32, x s ¼ 0.22, ad x u ¼ These cocetratios are close, but ot idetical, to those at the lower temperature. The results for the sigle chai achors are preseted i Fig. 2 B ad those for the double tails are preseted i Fig. 3 B. The value of the partitio coefficiet of DPPC at this temperature is The fact that the partitio coefficiet of DPPC decreases with icreasig temperature for saturated chais >12 carbos is to be expected from our observatio that the coefficiets are directly related to the cocetratios of the compoets of the membrae, ad that the differece i cocetratios betwee coexistig phases decreases with icreasig temperature for log saturated chais. Ideed, our result is cosistet with the experimetal results of Davis et al. (16), as show i their Fig. 11. The effect o the partitio coefficiet both of temperature ad of multiple achors o the partitio coefficiet is show i Fig. 4. The effect of temperature o the partitio coefficiet is depedet o the legth of the saturated chais relative to those that are a major compoet of the bilayer itself. ccordig to Fig. 4, the partitioig ito the lo phase of saturated chais loger tha 11 carbos icreases

7 Partitio Coefficiets of Lipid chors 1889 lo ld x di-sat. (di- :0) T = 290K di-sat. (di- :0) T = 300K sat. ( :0) T = 290K sat. ( :0) T = 300K FIGURE 4 Partitio coefficiet for various saturated chai achors embedded i a membrae cosistig of model DPPC, DOPC, ad cholesterol. The temperatures are 290 K ad 300 K. with decreasig temperature, idicatig that eergetic cosideratios domiate etropic oes. However, for saturated chais shorter tha 10 carbos, this partitioig icreases with icreasig temperature, idicatig that etropic cosideratios are domiat. This is i accord with the work of degees (34) o the chai-legth depedece of the adsorptio of oe polymer ito a layer of aother. Temperature has very little effect o the partitioig of usaturated chais as ca be see from a compariso of Fig. 2, ad B, for sigle chai achors ad Fig. 3, ad B, for double tail achors. other commoly studied double-chai achor motif that we have cosidered is C 18:0 C 20 :4c 5, 8, 11, ad 14, which correspods to a GPI-achored protei. t a temperature of T ¼ 300, we obtai a partitio coefficiet of The experimetal results for this achor are that ~20 30% associate with the liquid-order phase (12). To compare our partitio coefficiet value with this umber, we have to take ito accout that the partitio coefficiet gives the mol fractio of the chai achors i the lo phase relative to the mol fractio of the chai achors i the ld phase, ot the fractio of the total amout of the chai achors i oe phase. However, a simple calculatio shows that the fractio of achor i the lo phase, f, is related to the partitio coefficiet by N lo f ¼ ¼ k lo = ld ða ld =a lo Þ N lo þ N ld 1 þ k lo = ld ða ld =a lo Þ ; (20) where lo = ld is the ratio of the extesive areas of the lo ad ld phases. This ratio depeds upo the average compositio of the system ad the phase diagram accordig to the usual lever rule. If we take the extesive area of the lo phase to be equal to that of the ld phase, we fid that 23% of this chai achor molecule associates with the lo phase, which is well withi the experimetal rage. The 20 30% value obtaied i Kahya et al. (12) is for the huma placetal alkalie phosphatase (PLP) GPI-achored protei. PLP has bee extesively reported as a membrae protei with a very high affiity for the lo phase (>95%, accordig to (17)). This coclusio was maily draw from deterget-resistat membrae compoets ad the assumptio that lo domais ad their associated proteis existig i the model membraes at 37 C preserve their compositios uder colddeterget extractio (17). However, it has bee show via both experimet ad theory that this assumptio is ot valid (35). I cotrast, the experimet of Kahya et al. (12) uses florescece microscopy to study the itact membrae of domai-formig giat uilamellar vesicles ad this provides a much more reliable measure of the partitioig of PLP. With the use of this same techique, aother GPI-achored protei, Thy-1, was show to partitio ito the lo phase by at most 40% (36). This result idicates that our calculatios of the partitio coefficiet are i the experimetal rage for a variety of differet proteis that are attached to the membrae by the same achorig motif. This suggests that exploratio of the partitioig of lipid achors themselves is a good model for the partitioig of proteis that use these lipids to attach to the membrae. CONCLUSIONS We have preseted the first microscopic model calculatio of the extet to which various commo achors partitio themselves betwee liquid-ordered ad liquid-disordered phases. The partitio coefficiets are calculated as a fuctio of chai legth, degree of saturatio, ad temperature. We fid excellet qualitative agreemet betwee theory ad experimet: i.e., that the partitioig ito the lo phase is the least for highly usaturated chais ad icreases as the degree of saturatio icreases; that the partitioig of saturated chais icreases with their legth provided their legth is comparable with the chais of the bilayer; ad that the partitioig of saturated chais ito the liquid-ordered phase icreases with decreasig temperature, but that the temperature has little effect o the absolute values of the partitio coefficiet for the shorter saturated chais. We foud that the partitioig of short chais icreases with icreasig temperature, idicatig that their partitioig, i cotrast to that of loger chais, is domiated by etropic effects. We also foud that for usaturated achors, the partitio coefficiet depeds o the locatio of the double bods. Usaturated achors with double bods ear the chai eds cause a smaller decrease i the fractio of achors i the liquidordered phase tha achors with double bods closer to the middle of the achor. We are also able to predict how the partitio coefficiet varies as the umber of saturated or usaturated chais i the achor icreases. The effect of doublig the umber of saturated chais i a achor is to icrease the partitioig ito the liquid-ordered phase whe the tails are early as

8 1890 Ulie et al. log or loger tha those comprisig the bilayer, but the effect of doublig the umber of saturated tails is miimal whe they are relatively short. Doublig the umber of chais ad reducig the temperature has essetially o effect o the partitioig of usaturated chais. Lastly, we fid that the partitioig depeds o the differece betwee the lipid eviromets of the coexistig phases i which the achors are embedded. Cosequetly, the partitio coefficiet depeds ot oly upo the lipid compoets ad the ature of the achor, but also upo the relative compositios of the coexistig phases. PPENDIX : THE MODEL Because the model has bee preseted ad explored previously (29), our descriptio here ca be brief. The volume of the moomer uit CH 2 is take to be s ¼ 27 Å 3, ad that of a CH 3 group is 54 Å 3. Each half of the CH¼CH segmet of the cis-usaturated bod is assiged a volume of CH ¼ 0.8 s. The umber of carbos i each saturated chai is s ¼ 16, whereas the umber of carbos i each usaturated chai is u ¼ 18. The local orietatio of the chai is coveietly specified by the uit ormal to the plae determied by the k th CH 2 group, u s;k ¼ dr k =jdr k j; dr k hr k1 r k þ 1 ; with k ¼ 1,.. s 1. Just as for the lipids, the cofiguratio of the cholesterol is completely specified by the locatio of all of its carbo atoms ad of the agles betwee them. The orietatio of the small acyl chai of the cholesterol is specified i the same maer as for the lipid chais. The orietatio of its rigid rigs are specified by the uit vector, u c, from the 3rd to the 17th carbo i the molecule, usig the covetioal labelig, ad a ormal vector to the plae of those rigs. The umber of carbos i cholesterol is c ¼ 27, ad the volume of the carbos is c ¼ 21.0 Å 3. Because the orietatioal iteractio betwee bods depeds upo the local orietatio of these bods with respect to the bilayer ormal, c, it is coveiet to itroduce a fuctio, x s (a, z), that measures the local desity of bods weighted by their orietatio with respect to the bilayer ormal (37), x i ða; zþdz ¼ X i1 i;k ða; zþgðu k $cþdz; (21) k ¼ 1 where i, k (a, z)dz is the volume that segmet k i molecule i has i coformatio a at positio z. The fuctio g is chose to be gðu$cþ ¼ðm þ 1=2Þðu$cÞ 2m : For large m, g z m exp( mq 2 ), where q is the agle betwee the two uit vectors. Matchig lipid parameters, we have take m ¼ 18. We take the stregths of the local iteractios betwee bods i lipid tails to be the same irrespective of whether the bods are i a saturated or usaturated chai: J ss ¼ J uu ¼ J su h J ll. Thus, the differece i free eergy betwee saturated ad usaturated lipids arises solely from the fact that the cofiguratios of these lipids differ due to the presece of the double bod i the usaturated lipids, ot to ay explicit differece i iteractio eergies. To evaluate the sigle-molecule partitio fuctios, q i (Eq. 8), we geerate ~10 8 cofiguratios of each molecule. We the search the compositio space at a give temperature for phase coexistece. Phase equilibria is determied by stadard thermodyamic equalities (38): that the chemical potetials of the three compoets be equal i each phase; that the surface tesios be equal i each phase; ad that the commo value of the surface tesio be zero. The stregth of the iteractio betwee chai segmets, J ll, is set such that the calculated mai-chai trasitio temperature of a lipid with two saturated tails of 16 carbos, C16:0, is equal to that of dipalmitoylphosphatidylcholie (DPPC), T ¼ 315 K. This determies that J ll ¼ 3: k B T ðt ¼ 315 kþ: To obtai the best qualitative fit of our phase diagram to those of experimet, we have chose J lc ¼ 0:85 J ll ad J cc ¼ 0:80 J ll : For a temperature T ¼ 290 K we obtai the phase diagram show i Fig. 1. Whe we choose T ¼ 300, we obtai a phase diagram very similar to that show i Fig. 2 of Elliott et al. (29) for that temperature. Our parameters differ from those of Elliott et al. (29) because we have icluded the traslatioal etropy i the free eergy, ad because we employ a order-of-magitude more cofiguratios for each molecule. Fig. 2 of Elliott et al. (29) also icludes a plot of the order parameters of the saturated tails i the three phases, lo, ld, ad gel, which coexist at the triple poit. PPENDIX B: CHEMICL POTENTIL The chemical potetials of the three compoets that comprise the bilayer ca be calculated by takig the appropriate derivative of the free eergy with respect to the umber of molecules, vfmft m i ¼ ; (22) vn i T;;N½iŠ where N[i] meas that all of the species except species i are held fixed ad is the area of the bilayer. Because the two leaves are idetical, there is o differece betwee the chemical potetials of species i i either leaf irrespective of whether the species ca iterchage betwee the two leaves. The expressio for the free eergy withi mea-field theory, F mft ðt; N s ; N u ; N c ; Þ; is obtaied by substitutig the probability distributio fuctios (Eq. 7) ito the free eergy (Eq. 1) ad icludig the icompressibility costrait. We fid that all chemical potetials ca be writte i the form! bm j ¼ l x jl 2 j tails j l q j ; j ¼ s; u; c: (23) I particular, the chemical potetial of the two-tailed, saturated lipid compoet of the membrae is xs l 2 s bm s ¼ l : (24) q 2 s We ow tur to the determiatio of the chemical potetial of the achors. There are several ways that it ca be determied, ad we choose to follow the method of Widom s (31) potetial distributio theorem. We begi with the expressio of the chemical potetial of achor as QN bm ¼l ; (25) Q N1 where Q N is the partitio fuctio of the four-compoet system of N molecules with molecules 1 to N s correspodig to the saturated lipids, N s þ 1to N s þ N u to the usaturated lipid, N s þ N u þ 1toN s þ N u þ N c ¼ N 1tothe cholesterol molecules, ad the fial N th molecule is the chai achor

9 Partitio Coefficiets of Lipid chors 1891 molecule (molecule ). Thus, the chai achor molecule is i the limit of ifiite dilutio. The partitio fuctio Q N 1 is that of the three-compoet system without the chai achor ad thus cotais N 1 molecules. We separate the partitio fuctios ito traslatioal ad cofiguratioal parts so that the chemical potetial of achor is the icompressibility field, bp(z), ad the third is the cotributio of the aligmet field, bb l (z). We ca ow write the chemical potetial of a achor as bm x l 2 " R R # / expð bun ðr 1 ;.; r N NÞÞdr 1.dr N N ¼ l l R / R ; (26) exp bu N1 r 1 ;.; r ðn1þ ðn1þ dr 1.dr ðn1þ ðn1þ where i is the total umber of degrees of freedom for the molecule (umber of chai uits ad the degrees of freedom of the headgroup of molecule i), ad the cofiguratios of the molecules are specified by r 1 ;.; r ðn1þ ðn1þ : x l bm ¼ l 2 Xtails l X ( exp beða Þ k ¼ 1 a Z ) b ½pðzÞ ða ; zþ þb l ðzþx ða ; zþšdz (30) The potetial eergy of the total system ca be writte as U N ðr 1 ;.; r N NÞ¼U N1 r 1 ;.; r ðn1þ ðn1þ þ u N r 1 ;.; r ðn1þ ðn1þ; r ðn;headþ ; a ; where u N r 1 ;.; r ðn1þ ðn1þ; r ðn;headþ ; a (27) is the total eergy of iteractio experieced by the N th molecule with its headgroup at positio, (r (N, head) ), ad the chais of the N th molecule are i coformatio, a. The total eergy of the three-compoet system without the chai achor is U N 1. We ca ow decompose the cofiguratioal part of Eq. 26 to write where tails x l 2 ¼ l tails l q ; (31) is the umber of chais of the achorig molecule, ad we have used Eq. 8. This is precisely the same form as the chemical potetial of the compoets of the bilayer, Eq. 23. Because of this, it is easily see that the partitio coefficiet of a achor which is idetical to either the saturated or the usaturated compoet of the bilayer must be the same as the partitio coefficiet of that bilayer compoet. This stems from the fact that the partitio coefficiet is determied from the equality of the chemical potetial i coexistig phases ad from the form of the relatio betwee chemical potetial ad cocetratio. 2R R P 3 x l bm ¼ l 2 expðbun1 Þdr 1.dr ðn1þðn1þ expðbu N Þdr ðn;headþ l4 a R 5: (28) expðbu N1 Þdr 1.dr ðn1þ ðn1þ Because the system is homogeeous, u N is take to be idepedet of r (N, head) so that we ca itegrate over r (N, head) to give the area of the bilayer leaflet. We the get the followig exact relatio for the chemical potetial of the achor, This work was supported by the Natioal Sciece Foudatio uder grat No. DMR (to M.S.), grat No. CBET (to I.S.), ad the Natioal Istitutes of Health grat No. NIH GM (to I.S.). x l bm ¼ l 2 l X a hexpðbu N Þi N1 ; (29) where h.i N 1 is a caoical average i the (N 1)-molecule system, i which the cofiguratios of the N 1 molecules are ot iflueced by the presece of the N th test molecule. t this poit, we ca use the mea field approximatio to calculate the cofiguratioal part of the chemical potetial. If there are multiple chai achors o the absorbig molecules, we assume that the chais are idepedet of each other. There are three cotributios to u N : oe is the iteral eergy of the chai, the secod is the cotributio of REFERENCES 1. Simos, K., ad D. Toomre Lipid rafts ad sigal trasductio. Nat. Rev. Mol. Cell Biol. 1: Edidi, M The state of lipid rafts: from model membraes to cells. u. Rev. Biophys. Biomol. Struct. 32: Muro, S Lipid rafts: elusive or illusive? Cell. 115: Simos, K., ad W. L. Vaz Model systems, lipid rafts, ad cell membraes. u. Rev. Biophys. Biomol. Struct. 33: McMulle, T., R. N. Lewis, ad R. McElhaey Cholesterolphospholipid iteractios, the liquid-ordered phase ad lipid rafts i model ad biological membraes. Curr. Opi. Coll. It. Sci. 8:

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