Monte Carlo dynamics study of motions in &s-unsaturated hydrocarbon chains

Transcription

1 Monte Carlo dynamis study of motions in &s-unsaturated hydroarbon hains Y. K. Levine Department of Moleular Biophysis, Buys Ballot Laboratory, The Netherlands University of Utreht, 3508 TA Utreht, A. Kolinski Department of Chemistry, University of Warsaw, Pasteura I, Warsaw, Poland J. Skolnik Department of Moleular Biology, Researh Institute of the Sripps Clini, La Jolla, California (Reeived 25 Marh 1991; aepted 21 May 1991) A Monte Carlo dynamis study of the motions of hydroarbon hains ontaining is double bonds is presented. The simulations utilize the high-oordination {2 1 0) lattie for the simultaneous representation of the tetrahedrally bonded arbon atoms and the planar unsaturated segment. Results on single hains undergoing free motion in spae and tethered to an impenetrable planar interfae are reported. The introdution of a is double bond into a hydroarbon hain indues a slowdown in the dynamis. The simulations show this to be a universal result independent of the representation of the hain on the lattie. In ontrast, polyunsaturated hains are found to be more mobile than saturated ones. 1. INTRODUCTION The haraterization of the orientational and motional properties of hydroarbon hains in lipid bilayer systems has been the subjet of many experimental and theoretial investigations. I-3 Despite these efforts, the full understanding of physial experiments in terms of the behavior of the hydroarbon hains on a mirosopi sale has been impeded by a variety of fators. These inlude the multipliity of motions aessible to the lipid moleules and the impliit model dependene of the interpretation of the measurements. Consequently, questions pertaining to the nature of the motions, the importane of olletive motions, and the extent and time sale of rotational diffusion have yet to be answered. One of the most intriguing aspets of the work is the eluidation of the hanges in the behavior of the alkane hains brought about by the introdution of is double bonds. It is known that the position of the is bond in a hydroarbon hain influenes the thermodynami properties of the lipid bilayer. For example, the transition temperature for a bilayer onsisting ofphosphatidylholine moleules with hains 18 arbon long dereases by as muh as 60 K as the unsaturated bond is moved away from the head group towards the 9-10 position in the middle of the hain.4 The transition temperature, however, inreases by the same amount for hains with a is bond near the free terminal methyl group. Interestingly, naturally ourring lipids often possess a single is-double bond in the hains at the 9-10 position (oleoyl hains).i-3 The most ommon polyunsaturated 18 arbon hains ontain unsaturated is bonds in the lower half of the hain, in positions 9-10/12-13 (linoleoyl hains) or 9-10/12-13/15-16 (linolenoyl hains).le3 Inreasing unsaturation in the lipid hains has been shown to ause a general derease in the orientational order as monitored by *H nulear magneti resonane (NMR),5 eletron-spin resonane (ESR) spetrosopy,6 and fluoresene depolarization.7*8 However, the latter two tehniques indiate that the redution in order is aompanied by a slowdown in the rates of motion of the hains. This latter finding ontrasts with the effet of inreasing temperature whih indues a redution in the moleular order, but an inrease in the rotational dynamis. A greater insight into the dynamis of hydroarbon hains and in partiular, the effets of unsaturation on their motional properties, is only possible through the interpretation of the experimental data with theoretial simulations of hain motion. A number of moleular dynamis (MD) simulations have been performed previously on lipids ontaining alkane hains, 9-1 but these have been limited to time sales of 100 ps or less. Consequently, little has been learned onerning isomerizations, overall rotation, and the importane of olletive motions in determining hain dynamis. In ontrast, the tehnique of Brownian dynamis (BD) i2-16 overs the ns time sale and lends itself to the study of loal onformational motions. Indeed, the BD approah, rather than MD simulations, has proved useful in the analysis of 13C relaxation time measurements in lipid systems.r6 The time sale of the simulations of hain behavior an be extended even further by the use of Monte Carlo dynam- is (MCD) tehniques. * * This method has been used with muh suess in the study of dynami proesses in polymer hains in melts and isotropi solvents. 7-9 It has also been applied reently to studies of the dynamis in monolayers of saturated lipid moleules2 The advantage of MCD over MD and BD simulations is that its fundamental time sale is that required for loal onformational modifiations. The effiieny of MCD is ahieved by performing the simulations on a lattie whih permits the use of fast integer arithmeti operations. An added advantage is the simpliity and rigor of the algorithms for avoiding bond utting during the onformational hanges. The tehnique is partiularly attrative for studies of global relaxation phenomena, where the 3828 J. Chem. Phys. 95 (5), 1 September / Amerian Institute of Physis

2 Levine, Kolinski, and Skolnik: Motions in hydroarbon hains 3827 detailed motions of the hains within loal onformational wells beome unimportant and in this respet omplements MD and BD studies. The drawbak of the MCD tehnique, however, is the use of a lattie representation of the hydroarbon hain. The diamond (tetrahedral) lattie is often used in the simulations as it reprodues the tetrahedral oordination of the arbon atoms (Fig. 1). Nevertheless, it has been found in pratie that the omputational advantages outweigh the simplifiations arising from onfining the hains to a lattie. Moreover, it has been shown for various polymer systems that this lattie representation works well and moreover, the simulations are in good agreement with the results of MD simulations in the intermediate to long time regime. -19 While the diamond lattie provides a faithful representation of the rotational isomeri states of an alkane hain, it annot reprodue the planar onfiguration of a is double bond (Fig. 1). This is merely a refletion of the different oordinations of the sp3 and sp2 valene states of the arbon atoms. * In order to implement the MCD algorithms for unsaturated hains, we have utilized the high oordination {2 1 0) lattie.2 *22 This lattie has proved partiularly suessful in studies of the folding of polypeptide hains* and the range of dynami moves it affords has been disussed in detail previously.22 Here we present an initial study of the dynamis of alkene hains using the {2 1 0) lattie representation. Results I: I=. FIG. 1. A 12 atom alkane hain in the extended all-rruns onformation (top) and a 12 atom alkene hain with a is double at the entral 6-7 position (bottom). The saturated segments of the alkene hain are in the alltmns onformation. on single hains undergoing free motion in spae and tethered to an impenetrable planar interfae are reported. We shall show that the introdution of a is double bond into a hydroarbon hain indues a slowdown in the hain dynamis, in qualitative agreement with experimental results on lipid bilayer systems.6-8 Our findings indiate strongly that this is a universal result independent of the representation of the hain on the lattie. Interestingly, the simulations predit an enhanement of the rates of motion in the polyunsaturated linolenoyl hains. The results of the simulations an be tested diretly by arrying out NMR relaxation experiments on seletively labeled alkene and alkane hains. This work is in progress.26 In addition, experiments on lipid bilayer systems ontaining linolenoyl hains are being undertaken using ESR and fluoresene depolarization tehniques. II. METHOD OF SIMULATION A. Geometrial representation of hain moleules on a lattie The model hydroarbon hain onsists of 18 beads onneted by 17 bonds and is onfined to a ubi lattie. On this lattie, the individual beads are separated by a distane of ~ 5, and the bonds are obtained by a yli permutation of the 24 vetors of the type ( + 2, f 1,O). One lattie unit is thus equivalent to a distane of nm. The vetors are hosen from this basis set under the ondition that the distane R 13 between beads i and i + 2 lies in the interval dlo<r 13<dl8. Note that this does not faithfully reprodue the tetrahedral valene angle ( ) ourring in real alkane hains.27 However, the hoie was ditated by the need to obtain a uniform sampling of the allowed onfiguration spae for isolated hains. We have found that simulations arried out with a fixed angle between pairs of adjaent vetors either led to kineti trapping or to a biased sampling of the allowed hain onfigurations. With this representation of the hain onfiguration, we have admitted the existene of multiple dihedral onformation states about eah interior bond. It is important to note here that the onformational states ourred in pairs, haraterized by a rotation of the same magnitude, but of opposite senses. This property is analogous to that of the gauhe states of a real polymethylene hain2 and ensures that on average the model hain is ahiral. The ourrene of multiple hoies for the dihedral angles about the interior bonds leads to more disordered (oiled) onformations, with a resulting shorter end-to-end distane than would be expeted for real alkane hains.27 We have ounterated this artifiial shortening of the hain by penalizing the probability of ourrene of the effetive gauhe states. The penalty was introdued by imposing restritions on the distane R 14 between the ith and (i + 3) th beads in the hain, thus effetively aounting for the soalled pentane effet in real hains.27 This effet preludes the ourrene of the sequene gauhe minus/gauhe plus isomeri states about adjaent bonds as they lead to strongly repulsive steri interations between nonbonded methylene groups. In our algorithms we rejeted onfigurations in whih R 14<1/5 and aepted onfigurations with J. Chem. Phys., ol. 95, No. 5,1 September 1991

3 3828 Levine, Kolinski, and Skolnik: Motions in hydroarbon hains d/s <R 14(2/30 with a weighting fator of20%. The value of the weighting fator was hosen suh that it yielded endto-end distanes for the model saturated hain orresponding to those found for real alkane hains at K.* In order to implement exluded volume effets, in partiular to exlude hain utting, every bead was surrounded by 12 oupied lattie sites, forming a fae-entered-ubi (FCC) envelope about its position. These sites were obtained from the yli permutation of the vetors { f 1, & 1, 0). This onstrution ensured that the hain was unable to ross itself on undergoing onformational hanges. The representation of the &double bond on our lattie was ditated by the planar onfiguration of the segment C- = -. * Forty-eight distint triplet vetor ombinations taken from the set { f 2, f 1, 0) were found to produe a planar hain segment, though with valene angles differing from the expeted value of 120. We have found that in fat the geometrial representation of the unsaturated segments is of little onsequene for the hain dynamis. The simulations show that the predominant effets on hain dynamis are due to the rigidity of this segment, i.e., the lak of rotations about the entral C = C bond. We believe this to be a universal feature independent of the lattie representation of the hains. 6. Monte Carlo algorithm for hain motions The time evolution of the model hain onformations was assumed to be desribed by stohasti kinetis. Thus, we employed a dynami Monte Carlo tehnique to solve a stohasti kinetis master equation. 7*18 The use of this method to provide insight into the dynamis of maromoleular systems has a long history in polymer physis, where it has been shown to be in aord with the results of moleular dynamis simulations in the intermediate to long time regime. -19 The onformational dynamis of the model saturated hain is assumed to arise from loal rearrangements involving the transfer of a single bead, hosen at random, to a different lattie site, subjet to the preservation of the length of the bonds to its two neighboring atoms. In this way, new onformations an be introdued loally. The first (or last) bond is allowed to undertake random orientations. These latter moves introdue new loal orientations whih diffuse into the interior parts of the hain. This set of loal modifiations is found to span all possible onformations of the hain and allows the study of its dynami properties. The rigidity of the segment ontaining a is double bond neessitated the introdution of larger sale moves involving two or three beads. A four-bond motion was required if the first (or last) bead of the -C-C = C-C- segment (Fig. 1) is piked. This move involves the double bond itself and either the two preeding or two following bonds. Three beads are thus involved in a move whih keeps all the other beads of the hain in fixed positions. A three-bond move is allowed if either of the two beads onneted by the double bond are piked. This move involves a simple interhange of the two adjaent single bonds, thus onserving the onfiguration of the unsaturated segment. In addition, the penalty imposed on the distanes R 14 was modified in the neighborhood of the is double bond. This is neessary in order to aount for the important fat that there is little steri hindrane to rotation about the single bonds immediately adjaent to the is bond.23*24j7 This was implemented simply by allowing all values for the distane R 14 whih inludes the is double bond. Effetively, this means that the is double bond deouples the effets of steri hindrane between the two hain segmentsjoined to it. The hosen moves are subjeted to two aeptane tests. In the first plae, the new R 14 distanes between the beads involved in the move and their neighbors are heked and the move allowed only if it satisfies the riteria set out above. Seond, the move is only aepted if the final lattie positions of the beads are unoupied. This test is applied to both a bead and its assoiated FCC envelope. The fundamental time step of the algorithm is taken to be suh that every bead in the hain has an equal hane of undergoing a motion. 7 8 This is ahieved simply by attempting 18 moves, with eah bead having an equal probability of being piked. It must be emphasized that this time step is not defined on an absolute sale. The orrelation funtions for motion are obtained on invoking the ergodi hypothesis and replaing ensemble averages over a alulated trajetory in onfiguration spae by time averages. The deay of the time orrelation funtion G(t) of the end-to-end vetor R(t), G(t) = (R(O)*R(r)) was used to determine the length of the omputed trajetory. The total time of the simulation was hosen suh that this funtion deayed to less than 10% of its initial value. We have typially omputed 2X 10 onfigurations at intervals of 40 elementary time steps for the saturated hains and 100 time steps for hains ontaining is double bond segments. The orrelation funtions were omputed out to times orresponding to not more than 1000 intervals. With this hoie, the statistial flutuations in the alulated average hain parameters obtained from runs starting with dif- ferent initial onformations varied by less than 10%. The time orrelation funtions exhibited deviations between runs mostly after deaying to less than 30% of their initial values. The alulation of a trajetory for the saturated hain required about 90 minutes entral proessing unit (CPU) time (about 4 h for an unsaturated one) on a Spare 1 + workstation. HI. RESULTS AND DISCUSSION A. Free hain 1. Saturated hains The hain onfigurations obtained from a simulation run onsisted on average of the following R 13 distanes: 4% with R 13 equal to /lo and 1/12,25% with R 13 = d/14, ll%withr 13 = l6,and56%withr 13 = dl8.thedistribution of the square end-to-end distane R * of the saturated hains was found to be fairly narrow with a standard deviation of 10% around the average value of (R ') ~500. This was also refleted in the deay of time orrelation funtion of the end-to-end vetor whih was found to be domi- J. Chem. Phys., ol. 95, No. 5,l September 1991

4 Levine, Kolinski, and Skolnik: Motions in hydroarbon hains nated by orientational hanges. The probability of a truns onformation about any bond in the hain was found to be , exept for the two terminal bonds where a somewhat lower probability 0.18 was observed. These findings are onsistent with the results of the rotational isomeri state theory for polymethylene hains.* Every bond vetor in the hain was found to be essentially uniformly distributed over the 24 basis vetors. The small deviations from the uniform distribution were statistial in harater and varied from run to run. In order to gain insight into the dynamis of the hain, we divided the hain into four equal segments and onsidered the vetors onneting beads four bonds apart: l-5,5-9, 10-14, and This division is symmetri about the enter of symmetry of the hain at the bond onneting beads 9 and 10. We thus expet the vetors R l-5 and R 14-l 8 as well as R 5-9 and R to exhibit the same dynami behavior. This in fat forms a stringent test of the simulations. In addition, we also onsidered the vetor R 7-12 whih appears to reflet losely the average behavior of the hain. This oarse sampling of the hain behavior was hosen as we expet it to be less sensitive to the details of motion onferred by the underlying lattie. The time orrelation funtion G, (t) = (osp(0) l os p ( t) ) was alulated for eah of the five hain segments under onsideration. Here p is the angle between the given vetor and an arbitrary fixed vetor in the lattie. The deay of the orrelation funtion was found to be virtually independent of the hoie of referene vetor, the deviations being of the same order as the statistial flutuations observed between the results of different runs. This indiated that the orientational motions of the hain segments were isotropi. Consequently, we hose the Z axis of the lattie as our referene diretion for ase of omputations. The temporal behavior of the orrelation funtions for the motion of the vetors R l-5, R 5-9, and R 7-12 are shown in Fig. 2 for a saturated hain. The value of the orrelation funtion at time I = 0 was found to be 0.33 * 0.01, lose to the value of expeted for a uniform sampling of the angles fl. We have therefore normalized the orrelation funtions relative to time t = 0. The rates of motions are haraterized by the generalized orrelation time 7, defined as the area under the normalized orrelation funtion. The orrelation time was obtained numerially from a leastsquares fit of G, (t) to a triexponential deay funtion. The deay of G, (t) for the R IO-14and R vetors is indistinguishible from that shown in Fig. 2 for the R 5-9 and R l-5 vetors, respetively, and the orresponding orrelation times are given in Table I. The deay of the outer segmental vetors R l-5 and R is signifiantly faster than that for the inner vetors R 5-9 and R However, the orrelation funtion for the vetor R 7-12 whih spans the enter of mass of the hain, deays more slowly than those for the other vetors. This orrelation funtion is more sensitive to the reorientational modes of the hain as a whole; its deay is affeted to a lesser degree by the internal modes of motion. It turns out that the rate of deay of the orrelation funtions is sensitive to the effets of steri hindrane, or in 5 1.oo SATURATED CHAIN 0.80 * 1-5 ECTOR Y 5-9 ECTOR ECTOR FIG. 2. The time-orrelation funtions G, (t) (see the text) for the vetors in a saturated 18 bead model alkane hain with free terminal motion. The orrelation funtions for the pairs of vetors (R l-5, R 14-18) and (R 5-9 and R 10-14) are virtually indistinguishible and reflet the symmetry of the hain about the 9-10 bond. our ase, the restritions imposed on the R 14 distanes in the hain. A muh faster deay of G, ( t) is observed for the athermal model hain, i.e., in the absene of any restritions on R 14. Now the differenes in the deays of the inner and outer vetors is onsiderably smaller. The observed differenes are due solely to the diffusion of new orientations from the terminal bonds into the enter of the hain. Nevertheless, G, (t) for the vetor R 7-12 exhibits an appreiably slower deay refleting the overall orientational motion of the hain. The effets of steri hindrane on the dynamis of the model hain an be rationalized as follows. The restritions on the distanes R 14 introdue a ooperativity into the onformational motions by preluding ertain onformational sequenes about neighboring bonds. The onsequenes of ooperativity on the hain dynamis are marked for the motions of the entral segments of the hain as they are oupled to the outer segments. However, the terminal segments whih are bonded only at one end an undergo rapid and unrestrited end relaxation. TABLE I. The generalized orrelation times haraterizing the rates of motion of free hains. The values are given in units of MCD time steps. R l-5 RS-9 R IO-14 R R7-12 Chain Saturated is is is wM oooo 9-10/l%13 is /12-13/15-16 is Downloaded 07 Apr 2004 to Redistribution J. Chem. Phys., subjet ol. 95, No. to AIP 5,1 liense September or opyright, 1991 see

5 3830 Levine, Kolinski, and Skolnik: Motions in hydroarbon hains 2. Chains ontaining is segments Simulations of hains with the is double bond at the 3-4,6-7,9-10,12-13, and hains werearried out using the motions of the beads desribed above. Idential results 0.80 were obtained for the 3-4 and hains as well as for the 6-7 and ones refleting the symmetry properties of the 18 bead hains. Consequently, we shall here onsider only 0.60 the three hains 3-4, 6-7, and 9-10 whih exhibit distint properties. The distributions of the square end-to-end distane R 5 of the hains were found to be as narrow as those shown by the saturated hains. However, the enter of the distribution (R ) -450 for the 3-4 unsaturated hain and (R ) ~405 for the 6-7 and 9-10 hains was signifiantly lower than found for the saturated hains (R ) =: 500. The probability of a tram onformation about any bond in the enter of the hain was found to be , exept for the two bonds adjaent to the is double bond. Here a lower probability of was found, somewhat higher than the value of expeted in our model for a hain with no steri hindrane for bond rotations. The is double bond vetor in every hain was found to be essentially uniformly distributed over the 24 possible basis vetors. The small deviations from the uniform distribution were statistial in harater and varied from run to run. The orrelation funtions G, (t) for the end-to-end vetors exhibited substantially slower deays than that obtained from simulations of the saturated hain. Moreover, the deay slowed down markedly on moving the is double bond from the 3-4 position to the enter of the hain at the 9-10 position. Again, the deay was predominantly due to the orientational motions of the end-to-end vetor. This slowdown neessitated the omputation of muh longer trajetories than was the ase for the saturated ones. Our hoie of an inrease in the simulation time by a fator of 2.5 represents a useful ompromise between the length of the run and statistial signifiane. The slowdown in the dynamis of the orientational modes of motion was also refleted in the orrelation funtions of the hain segments ontaining the is double bond At the time origin, the orrelation funtions for the four hain vetors were found to have a onstant value G, (0) = 0.33 f 0.01 as expeted for a random orientational distribution in spae. We have therefore normalized the or relation funtions so that G, (0) = 1. The orrelation funtions were omputed relative to the Z axis of the lattie for onveniene as the same behavior was obtained if an arbi- 5 trary lattie vetor was hosen. Figures 3-5 show the orientational orrelation fun- tionsg, (t) forthemotionsofthevetorsr 1-5, R 5-9, R IO- 14, R 14-18, and R 7-12 for the hains with the is bonds at the 34,6-7, and 9klO positions. The orresponding orrelation times are given in Table I. Interestingly, the deays found for the symmetri hain ontaining the is double bond at the 9-10 position exhibit not only the expeted symmetry, but are also faster than those observed for the saturated hain. Only the vetor R 7-12 exhibits the slowdown in the orientational dynamis. All the orrelation funtions were found to exhibit long time FIG. 3. The time-orrelation funtions G, (t) (see the text) for the vetors of an 18 bead model alkene hain with a is double bond at the 34 position. Both terminal ends of the hain undergo unrestrited motion. tails aounting for up to 30% of the amplitude at t = 0. This ontribution depends strongly on the relative position of the vetor and the is double bond. The question now arises as to whether the slowdown in dynamis aused by the introdution of a isdouble bond is a real physial effet or an artifat of the rules for bond motions underlying the model. In this ontext, we need to note that a short hain ontaining six beads with a is bond at the 34 (symmetry) position exhibits a orrelation funtion for the 1-6 vetor virtually indistinguishible for that omputed - 1.oo J. Chem. Phys., ol. 95, No. 5,l September CHAIN IO- 14 ECTOR ECTOR 7-12 ECTOR il~ ~~~~ ~, ~ ~~~~,~ ~ ~~,11 ~111I, CHAIN lo-14 ECTOR ECTOR 7-12 ECTOR I ( ( ( 1 b 4 13 * ( , 1 f g * r 3 ( 13 I FIG. 4. The time-orrelation funtions G, (t) (see the text) for the vetors of an 18 bead model alkene hain with a is double bond at the 6-7 position. Both terminal ends of the hain undergo unrestrited motion.

6 Levine, Kolinski, and Skolnik: Motions in hydroarbon hains CHAIN * l-5 ECTOR 7-12 ECTOR,111(11111,IIIIIIIJI,III(I II(I hik STEPS (TIME) FIG. 5. The time-orrelation funtions G, (t) (see the text) for the vetors ofan 18 beadmodel alkenehain withaisdoublebondat the9-lo position. Both terminal ends of the hain undergo unrestrited motion. The funtions forthepairsofvetors (R l-5, R 14-18) and (R 5-9and R 10-14) arevirtually indistinguishible and reflet the symmetry of the hain about the 9-10 bond. for a six-bead saturated hain. This, taken with the observation of an isotropi distribution of orientations of the double bond vetor in spae, indiates that the intrinsi motions assigned to the rigid is segments do not by themselves aount for the slowdown in dynamis. The dynami effets arise simply from the restritions on the motions of the rigid is double bond segment due to its attahment on both sides to fairly immobile hain segments. This an indeed be seen from Figs. 3-5, whih show that the slow-down effets beome progressively more signifiant as the is double bond is moved away from the free terminal towards the middle of the hain. It appears that the rigid segment forms a bottlenek for the diffusion of new orientations along the hain. We note further that the rigid is double bond segment also deouples the ooperative motions within the hain due to steri hindrane, so that the saturated segments beome somewhat more mobile in the unsaturated hain than in a saturated one of the same overall length. Our onlusions that the insertion of a rigid strutural entity into an otherwise flexible hain introdues a bottlenek for the diffusion of new orientations along the hain are reinfored by simulations of hains ontaining a nonplanar onformation for the is double bond. There are 48 distint vetor triplets representing eah optial isomer of the segment. The slowdown in the dynamis is again observed if the is segment is restrited to the onfiguration of a single isomer. However, the dynamis revert to those observed for a saturated hain on allowing interonversions between the two isomers during the onformational motions. This is equivalent to admitting hindered rotations about the double bond. Further evidene supporting our ontentions is provided by simulations of polyunsaturated hains ontaining is double bonds in the 9-10/12-13 and 9-10/12-13/15-16 positions. The orrelation funtions G, (t) for these hains are shown in Figs. 6 and 7. The hain ontaining two is double bonds exhibits the slowdown in dynamis, partiularly for the vetors near the is double bonds (Table I). However, the hains ontaining three is double bonds exhibit rapid deays, faster even than those found in the saturated hains (Table I). We note that these hains onsist of a wholly unsaturated segment attahed to a saturated one. Only the short, seven bead long, saturated segment is subjet to steri hindrane effets. Consequently, this segment exhibits faster motions than a orresponding segment in an 18 bead hain. The mismath between the motions of the two segments is now only operative at the point of their attahment in the middle of the hain. Work is urrently under way using Brownian dynamis to examine the lattie independene of the results.** B. Tethered hains The hains were onstrained to move freely over a plane, e.g., the XY plane of the lattie, but the first bead was only permitted to move by one lattie unit along the normal to the plane, the 2 axis. Furthermore, the hain was restrited to the half-spae above the plane, so that it oupied lattie sites whose Z oordinate was positive. This was implemented simply in the algorithm by a permanent oupation of the lattie sites in the negative half-spae. The XY plane was hosen for onveniene, as idential results were obtained from simulation runs using other rystallographi planes of the ubi lattie. It is important to note that now the orientational average (os p ), the order parameter (P, ),does not vanish identially if the vetor orientations are restrited to the interval A 1.oo UNSATURATED g-10/12-13 * l-5 ECTOR x 5-9 ECTOR 0 lo-14 ECTOR b ECTOR ECTOR CHAIN CIS FIG. 6. The time-orrelation funtions G, (1) (see the text) for the vetors of an 18 bead model alkene hain with two is double bonds at the 9-10 and positions. Both terminal ends of the hain undergo unrestrited mo- J. Chem. Phys., ol. 95, No. 5,1 September 1991

7 3832 Levine, Kolinski, and Skolnik: Motions in hydroarbon hains 1.oo /12-13/15-16 CHAIN again orresponding to the area under the normalized orrelation funtion. For omputational purposes, we have trunated the integral after MCD time steps as to a good approximation G I (40 000) - (os fl) 2. FIG. 7. The time-orrelation funtions G, (t) (see the text) for the vetors of an 18 bead model alkene hain with three is double bonds at the 9-10, 12-13, and positions. Both terminal ends of the hain undergo unrestrited motion. O<os P< 1. Consequently, the orrelation funtion G, (t) will deay to a onstant plateau at long times, suh that G, (t+ o ) = (COS fl) 2. At time t = 0, we have G, (0) = (COS P >. For a uniform distribution of orientations over the half-plane, we expet (os p ) = 4 and (COS P) = 1. In this ase, we define the orrelation time r as 7== u s 0 m [G~(t)-C,(oo)]dt/[C,(0)-GG,(r,)] SATURATED IO- 14 ECTOR ECTOR (1) CHAIN 1. Saturated hains Figure 8 shows the deay of G, (t) for a saturated hain omposed of 18 beads. The derease in the values of G, (0) as well as of the heights of the long time plateaus indiates that the motions of the vetors beome less restrited on moving from the tethered first bead towards the free terminal of the hain. Indeed, these values indiate that the vetors R and R an aess an almost uniform distribution of orientations ( - 1 (os /?< 1) relative to the Z axis, the normal to the plane. However, the orientations of the first two vetors R l-5 and R 5-9 are restrited by the presene of the impenetrablexyplane to the range O<os & 1. The R 7-12 vetor exhibits an intermediate behavior whih may be identified with the average motion of the hain. The effetive orrelation times for the motions of the four vetors R l-5, R 5-9, R 10-14, and R are given in Table II. The progressive derease of r on moving from the tethered head to the free terminal segment of the hain is onsistent with the observed inreased range of orientations of the vetors. Interestingly, a smaller value of r is found for the initial l-5 segment than for the 5-9 segment. Again the motion of the R 7-12 vetor reflets the average hain motion. 2. Cis-unsaturated hains The tethering of the hains to the plane has effetively removed the enter of symmetry of the hain, so that now a distint behavior is expeted for hains ontaining is double bonds at different plaes along the hain. This is illustrated in Fig. 9, where the orrelation funtions G, (t) for the 9-10 hain are shown. A omparison of Figs. 8 and 9 shows that the motions of R 5-9 and R 7-12 are more restrited in the unsaturated hain than in the saturated one. In marked ontrast, the introdution of the is double bond allows the vetors R IO-14 and R to aess a larger range of orientations. This is onsistent with our onlusions above about the influene of the steri hindrane fator on hain motion. The effetive orrelation times (Table II) yield a different piture of hain dynamis. The values for all these parameters, with the exeption of that for R 5-9, are found to be signifiantly higher than those of the saturated hain. TABLE II. The generalized orrelation times haraterizing the rates of motions of tethered hains. The values are given in units of MCD time steps I,, I,,,,,,,,,,,,, I 1 ( ( ( ( ( (,,, ( ( FIG. 8. The time-orrelation funtions G, (t) (see the text) for the vetors in a saturated 18 bead model alkane hain tethered to the impenetrable XY plane. The first bead is allowed to move up and down the Z axis by one lattie unit. RI-5 R5-9 R9-10 R R7-12 Chain Saturated is 13OOG is lok is 6COO is C /12-13 is /12-13/15-16 is CGU J. Chem. Phys., ol. 95, No. 5,l September 1991

8 Levine, Kolinski, and Skolnik: Motions in hydroarbon hains h CHAIN * l-5 ECTOR 5-9 ECTOR : lo-14 ECTOR r ECTOR 0 7-l 2 ECTOR /12-13/15-16 CHAIN * l-5 ECTOR : 5-9 ECTOR i : lo-14 ECTOR A ECTOR 7-l 2 ECTOR FIG. 9. The time-orrelation funtions G, (t) (see the text) for the vetors in an 18 bead model alkene hain ontaining a is double bond at the 9-10 position. The hain is tethered to the impenetrable XY plane, but the first bead is allowed to move up and down the Z axis by one lattie unit. FIG. 11. The time-orrelation funtions G, (I) (see the text) for the vetors in an 18 bead model alkene hain ontaining three is double bonds at the 9-10, 12-13, and positions. The hain is tethered to the impenetrable XY plane, but the first bead is allowed to move up and down the Z axis by one lattie unit. Simulations of hains ontaining is double bonds at the 3-4, 67, 12-13, and positions show that in general the introdution of the double bond tends to lower the orientational averages of the vetors. The most signifiant effets being observed are for the 6-7 hain. However, the most striking feature of the orrelation funtions is the substantial inrease of the effetive orrelation times relative to those of the saturated hain (Table II). -7 G g-10/12-13 * l-5 ECTOR :, 5-9 ECTOR lo-14 ECTOR ECTOR ECTOR CHAIN FIG. 10. The time-orrelation funtions G, (t) (see the text) for the vetors in an 18 bead model alkene hain ontaining two is double bonds at the 9-10 and 12-I 3 positions. The hain is tethered to the impenetrable XY plane, but the first bead is allowed to move up and down the Z axis by one lattie The largest effets are found for the segments adjaent to or ontaining the is double bond. The only exeptions are the R l-5 and R 5-9 vetors in the and hains whih exhibit similar orrelation times to those of the saturated hains. The most remarkable effets of unsaturation on the dynamis of the hains are observed in polyunsaturated hains. The orrelation funtions G, (t) for hains ontaining is doublebondsatthe9-10/12-13and9-10/12-13/15-16positions are shown in Figs. 10 and 11, respetively. The introdution of a seond double bond at the position an be seen to ause a signifiant redution in the orientational parameters of the R 10-14, R 14-18, and R 7-12 vetors. The third is double bond at the position further redues these parameters. Thus the motions of the vetors in these hains are less restrited than those in the equivalent saturated ase. The orrelation times for the 9-10/12-l 3 hain (see Table II) are similar to those observed for the 9-10 hain with two lear exeptions. A large inrease is found for the R lo- 14 vetor (from 4400 to 7400 time steps), but a large derease (from 8900 to 6400 time steps) for the orrelation time of R The additional bond at position 15-16, however, auses a marked redution in all the orrelation times beyond R l-5 (Table II). These are in fat signifiantly lower than those found for the saturated hain with the sole exeption of that for the initial segment. We note that a similar effet was desribed above for the free hain. These results indiate that there is no simple relation between the restrition on the angular exursions of the vetors as refleted by the orientational averages and the orrelation times defining the rates of reorientational motions. This finding is in agreement with experimental observations of the behavior of lipid bilayer systems. 8 J. Chem. Phys., ol. 95, No. 5,1 September 1991

9 3834 Levine, Kolinski, and Skolnik: Motions in hydroarbon hains ACKNOWLEDGMENTS We thank the Sientifi Diretorate of NATO for the award of a Collaborative Researh Grant No. CRG to YKL and JS. This work was also supported in part by a grant from the Polymer Siene Programme of the National Siene Foundation to JS. D. M. Small, in ThePhysiuZChemistryofLipids, edited by D. J. Hanahan (Plenum, New York, 1986), ol. 4. B. L. Silver, The Physial Chemistry of Membrane (Solomon, New York, 1985). M. D. Houslay and K. K. Stanley, Dynamis of BiologiaI Membranes (Wiley, Chihester, 1982). P. G. Barton and F. D. Gunstone, J. Biol. Chem. 250,447O ( 1975). A. Seelig and J. Seelig, Biohemistry 16,45 (1977). 6L. J. Korstanje, E. E. van Faassen, and Y. K. Levine, Biohim. Biophys. Ata 982, 196 (1989). G. Deinum, H. van Langen, G. van Ginkel, and Y. K. Levine, Biohemistry 27,852 (1988). G. van Ginkel, H. van Langen, and Y. K. Levine, Biohimie 71, 23 (1989). 9P. van der Ploeg and H. J. C. Berendsen, J. Chem. Phys. 76,3217 ( 1982); Mol. Phys. 49,233 (1983). OS. Northrup, J. Phys. Chem (1983). I A. J. Kox, J. P. J. Mihels, and F. W. Wiegel, Nature 287, 317 (1980). I2 E. Helfland, Z. R. Wasserman, and T. A. Weber, Maromoleules 13,526 (1980); J. Skolnik and E. Helfland, J. Chem. Phys. 72, 5489 ( 1980). I3 R. M. Levy, P. G. Woylnes, and M. Karplus, J. Am. Chem. Sot. 103,5998 (1981). C. James and G. T. Evans, J. Chem. Phys. 76,268O ( 1982). I5 W. F. van Gunsteren, H. J. C. Berendsen, and J. A. C. Rullmann, Mol. Phys.44,69 (1981). I6 R. W. Pastor, R. M. enables, and M. Karplus, J. Chem. Phys. 89, 1112 (1988). Monte Carlo Methods in Statistial Physis, edited by K. Binder, (Springer, Berlin, 1986). J. Skolnik and A. Kolinski, Adv. Chem. Phys. 78, 223 ( 1990). A. Baumgartner, Annu. Rev. Phys. Chem. 35,419 (1984). M. Milik, A. Kolinski, and J. Skolnik, J. Chem. Phys. 93,444O ( 1990). * J. Skolnik and A. Kolinski, Siene 250, ( 1990). A. Kolinski, M. Milik, and J. Skolnik, J. Chem. Phys. 94,3978 ( 1991). x J. Skolnik and E. Helfland, J. Chem. Phys. 72, 5489 ( 1980). 24 J. Skolnik, Maromoleules 14,646 ( 198 1) 25A. Rey and J. Skonik (unpublished). 26R. R. old (personal ommuniation). P. J. Flory, StatistialMehanisof Chain Moleules (Intersiene, New York, 1969). J. Chem. Phys., ol. 95, No. 5,l September 1991