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# Review Sheet for PHYS 107 at UA

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Date Created: 02/06/15

RAPID COl IlIUNI PHYSICAL REVIEW E VOLUNTE 64 010901 R Composite membrane deformation on the mesoscopic length scale Michael F BrowngtXlt Robin L Thurmondf Steven W Dodd1 Dorte Otten2 and Klaus Beyer2 1Department of Chemistry University of Arizona Tucson Arizona 85 721 2Inslfilfulffiir Physikalisehe Bioehemie Universitiit Miinehen D8033 6 Miinehen Germany Received 14 October 199939 revised manuscript received 10 October 200039 published 26 June 2001 The physics of soft materials can be investigated using nuclear spinlattice relaxation which depends on the spectral densities of motion in the MHZ range For the rst time N1R relaxation has been used to study in uences of the acyl length polar head groups a cosurfactant and cholesterol on the viscoelastic properties of membrane lipids The results imply the concept of elastic deformation is relevant on lengths w the bilayer thickness and less involving a broad spectrum of collective modes which contribute to the forces between lipid bilayers DOI 101103PhysRevE64010901 The properties of exible amphiphilic layers composed of phospholipids are of fundamental interest to physicists with regard to their microstructures 1 5 and play a role in key biological functions 6 These and other fascinating soft ma terials can be studied using nuclear spinlattice relaxation for reviews see Refs 78 In this Rapid Communication we report NMR relaxation data that provide knowledge of the viscoelastic properties of lipid bilayers including their dependence on composition The results imply that the con cept of elastic deformation of membranes is applicable on the mesoscopic length scale approaching the molecular di mensions with quasicoherent modes on the order of the bi layer hydrocarbon thickness and less 9 We suggest the entire spectrum of uctuations is important for repulsions due to entropic con nement including long wavelength un dulations as well as higher frequency excitations Our development is based on a composite membrane de formation model 10 which explains the combined angular and frequency dependencies of the nuclear spin relaxation rates R12 of uid bilayer lipids A special feature is that quasicoherent order uctuations give a relaxation enhance ment visavis simple hydrocarbon uids We have postu lated 11 that the bilayer interior can be modeled in terms of a director eld nr Given the numerous degrees of free dom the system is treated as a continuous medium where collective motions are related to the elastic moduli Use of the equipartition theorem together with integration over the modes gives a characteristic frequency w dependence of the relaxation 12 The experimentally found of 2 dependence of R12 suggests that collective uctuations of the individual segments occur in the MHZ range 10 More generally one should consider the modes in relation to the interbilayer separation Figure 1 depicts the types of bilayer deformations for a free membrane The elastic shape uctuations are mod eled as unconstrained splay twist and bend excitations to gether with effective axial rotations of the lipids 10 In the case of splay deformations the divergence Vnr0 For gtAuthor to whom correspondence should be addressed Addi tional address Universitat Wurzburg Physikalisches Institut EP5 Am Hubland D 97074 Wurzburg Germany Present address R W Johnson Pharmaceutical Research Insti tute 3210 Merry eld Road San Diego CA 92121 1063651X200164101090142000 64 0109011 PACS numbers 87167b 76607k 8764Hd twist deformations the curl is along the local director such that nrVgtltnr 0 For bend the curl is perpendicular viz nrgtltVgtltnr 0 Such a continuum picture repre sents a signi cant departure from previous molecular theo ries for lipid bilayer relaxation The absence of molecular details is both the strength and weakness of the approach Now in 2H NMR spectroscopy of lipid bilayers reviewed in 78 the observables are the order parameters as a function of the acyl chain position i and the corresponding RE relaxation rate pro le These manifest a hierarchy of motions which contribute to the dynamical roughness of the bilayer Using the notation of Ref 10 the order parameters are S8932 123 cos2 3533371 where 3 is the angle be tween the principal axis system P of the electric eld gra dient EFG viz the ith C ZH bond and the macroscopic bilayer normal director frame D The order parameters rep resent the motional amplitudes whereas the NMR relaxation also includes the motional rates As a rule the dynamics can involve i segmental motions of the exible phospholipids ii molecular motions and iii collective motions For di saturated phospholipids in the liquidcrystalline L a state the dependence of the NMR relaxation on temperature and frequency disfavors a noncollective model involving mo lecular motions or superimposed internal chain rotations One is left with collective motions as a source of the relax ation 1011 a planar 4 I b splay c twist d bend amp FIG 1 Excitations of a uid lipid bilayer within the continuum elastic approximation a Planar bilayer b splay c twist and d bend deformations together with axial rotations about the local director 2001 The American Physical Society BROWN THURMOND DODD OTTEN AND BEYER The spinlattice relaxation is due to orientational uctua tions of the individual C ZH bonds which induce transi tions between the various Zeeman levels According to BlochWangsnessRed eld theory R12 is given to second order by 8 3 RlZZ7TZX2QJlwD4J22wDa 1 where XQ is the static quadrupolar coupling constant Here Jmwmetexpiwtdt where Gmt are rank2 auto correlation functions of the perturbing Hamiltonian m 12 and 00D is the deuteron Lannor frequency Note the EFG uctuations can encompass rapid local motions of the static EFG tensor eg due to transgauche isomerizations of the acyl chains as described by a fast order parameter Sf Additional slower motions of the remaining residual EFG tensor can also occur as described by an order param eter SE The slow motions can arise from quasicoherent uctuations of a threedimensional 3D splay twist and bend or 2D splay nature together with acyl rotations These lead to SODS2 E 2S 2 where 5 is a geometric factor 10 The irreducible spectral densities 10 are then JmwgtJ1wJ 2 1wgtJ 139c 1wgt 2 in which the small contribution from segmental motions is neglected The rst term describes collective 3D membrane deformations and assumes a single elastic constant K for splay K 11 twist K22 and bend K33 uctuations For a lipid bilayer smecticA this is clearly an heuristic approxi mation as one expects K11ltK22 K33 for longer wave lengths However it is premature to discuss individual NMR active modes in our view To linear order 12 the contribu tion from order uctuations is 11 5 mm gs DDwlt2d2gtDElmlt0mgt2 ID 2lt0DLgtI2L 3 where the highfrequency cutoff 1011 is neglected and m 12 Inclusion of higher order director uctuations 13 gives an additional correction for m 2 Here Dm0 is a Wigner rotation matrix element and the Euler angles QDL describe the transformation from the director D to the labo ratory L frame The above formula corresponds to the over damped regime such that each of the modes relaxes with a single exponential time constant For 3D director uctua tions d 3 yielding an w lZ frequency dispersion Follow ing Ref 12 the viscoelastic constant is D 3kBTH57WS 2COI2 where K is the effective elas tic constant for 3D excitations and 7 is the corresponding viscosity One can also consider 2D membrane deformations d 2 eg smectic undulation waves splay resulting in D gtD with an 0 1 dependence which is not observed ex perimentally at MHz frequencies 11 The J 1w term in Eq 2 corresponds to effective rotations of the exible lip RAPID COl Il IUNICATIONS PHYSICAL REVIEW E 64 010901R 60 a DLPCd46 b DMPcd54 50 40 g 44 30 Z i 39 24a a fquot 10 4e a g A 0 a l l l a 395 N Iv c DPPcd62 d DSPcd7O 5o z 40 30 20 5 ii If 10 C A30 C a y I50 0 10 a quot55 C j 980 C 0 i 0 002 004 o 002 004 006 i 2 ISCDI FIG 2 Squarelaw functional dependence of spinlattice relax ation rates and order parameters IS along the acyl chains index 139 for a homologous series of PCs in the La phase showing in uence of the acyl length bilayer thickness Data are for unori ented dispersions containing 50 wt H20 67 mM phosphate buffer pH 70 at 554 MHZ and various temperatures T Tm 6 0C a DLPCd46 b DMPCd54 c DPPCd62 and d DSPCd70 with acyl lengths of n 12 l4 l6 and 18 carbons respectively ids and nally J20139c 1w to a geometrical crossterm Ac cording to theory 10 the spectral densities scale in closed form with the square of the order parameter SOD as a char acteristic signature assuming SE2 and 5 do not depend appreciably on chain position In our experimental NMR relaxation studies we investi gated a series of phospholipids in the La state with the gen eral structure RCOOCH2RCOOCHCHZOX where R de notes the fatty acyl chains and X the polar head group The in uences of the acyl length bilayer thickness lipid polar head groups interfacial area per molecule addition of a cosurfactant and incorporation of cholesterol were studied in terms of the bilayer viscoelastic properties The effort to ob tain these results was substantial and entailed chemical syn thesis and 2Hisotopic labeling of phospholipids solidstate NMR studies at different magnetic eld strengths and test ing of theoretical models For phosphatidylcholine PC the polar head group is PO3 CH2CH2NCH33 Use of acyl chainperdeuterated phospholipids n 12 to 18 carbons al lowed us to observe simultaneously the entire hydrocarbon region of the bilayer As shown in Fig 2 a squarelaw mc tional dependence of the RE rates on along the chains describes the data for the homologous PCs for T26 C above the main phase transition temperature Tm The slopes of Figs 2a 2d suggest the bending rigidity K K11t depends only weakly on temperature in the La 0109012 COIVIPOSITE MEMBRANE DEFORMATION ON THE state increasing with the bilayer thickness t Our conclusions agree with earlier studies of temperature 14 and chain length 15 effects on the elastic properties of PC bilayers The fact that a broad range of data can be superimposed in terms of a nearly universal squarelaw functional dependence is striking For bilayers of DMPCd54 we obtain D 102 X 10 5 s and by treating higher order terms 10 we esti mate that K 12x 10 11 N and n071 P An heuristic cal culation assuming t4 nm gives KK11t05gtlt10 19 J ie 11kBT which implies a fairly soft bilayer in accord with micromechanical measurements K may be lengthscale de pendent 16 For the other PCs tincreases by 12 nm for C12 to C18 chains 15 giving a 30 calculated increase in K The angular amplitude of the uctuations of the local di rector N relative to the average director D assuming s 2gt 11732lt3 D06 is 812VD1Q30 Then from the relation 101213 that 812VDkBch772K the cutoff wavelength is Ac277qc8 A less than the bilayer thick ness t and close to the lateral lipid dimensions 1 Clearly an aspect of our work involves extension of the concept of membrane elasticity to relatively short distances Using con tinuum mechanics 9 one can also estimate the area elastic modulus K from the relation KKaf248 giving 150 mJ m 2 in the range for uid bilayers 16 The individual monolayers are treated as homogenous thin lms with zero 2D shear modulus 9 Clearly our NMR relaxation ap proach should be distinguished from previous NMR line shape studies 1718 which probe membrane elasticity at lower frequencies Modeling the lipid bilayer as an ordered uid thus yields our conjecture that qmodes of substantial amplitude in uence the relaxation within the MHz regime with wavelength components on the order of the bilayer thickness and smaller ie the mesoscopic length scale Some readers may think it surprising that our NMR re sults for soft membrane bilayers involving exible lipid molecules with many degrees of freedom can be interpreted using fairly simple concepts drawn from the physics of ma terials Within our framework the membrane lipids are ef fectively tethered to the aqueous interface via their polar head groups and the bilayer interior is essentially liquid hy drocarbon The reason why the R12 relaxation is governed by collective order uctuations is that the local segmental mo tions of the lipids are very fast 10 11s with spectral densities extending to very high frequencies Hence rela tively slow quasicoherent order uctuations provide a frequencydependent enhancement versus simple nparaf nic liquids eg nhexadecane 8 We also investigated bilayers containing phosphatidyle thanolamine PE with the polar head group PO3 CH2CH2NH3I Figure 3 shows the RE rates and the IngDI values are correlated reasonably well over the entire temperature range by a squarelaw functional dependence Yet a completely universal behavior is not observed for the entire series of phospholipids investigated Rather compared to DPPCa162 alone Fig 3a increasing the mole fraction of DPPEd62 yields a progressive reduction in R for a given order parameter Figs 3b and 3c Use of the vis RAPID COlIIlIIUNICATIONS PHYSICAL REVIEW E 64 010901R 50 39 I I I x I a DPPCd62 40 y 30 Wquot B V 20 V Pg 10 w 0 I I I I I I b DPPEdszzDPPC 11 A 40 DPPEzDPPCd 11 v 62 3 30 v3 EN 39 40 II 20 V g V 1 my 0 I I I I I I c DPPEd 4O 62 30 2o D W xv r T5 13 10 wi iv39 39v o FEE I I o 002 004 006 i 2 ISCDI FIG 3 Dependence ofRYZ on square of for phospholipid bilayers in the La phase showing effect of phosphoethanolamine head groups interfacial area per molecule Results are shown for unoriented dispersions containing 50 wt H20 20 mM MOPS buffer pH 71 at 461 MHZ and various temperatures a DPPC5162 42780 0C b DPPEd z DPPC 11 and DPPEDPPCd6211 65 OC and c DPPEd62 60780 0C coelastic constant for DMPCd54 as a reference 10 leads to KDPPEDDWcDDPPE23KDWC40gtlt10 11N about a threefold increase in K due to the presence of PE head groups Assuming t5 nm results in K43kBT and a cor respondingly larger value of the area expansion modulus K a To explain these observations we note that for PBS the re duction in the interfacial area per lipid gives an increase in acyl ordering We propose that a reduction in entanglement of the acyl chains con gurational entropy yields an in crease in bilayer stiffness and a possibly greater rate of ef fective axial chain rotations The result is a weaker repulsive force for PEcontaining bilayers due to quasicoherent elastic modes with a reduced hydration of the membrane dispersion and a concomitantly smaller interlamellar separation 19 This would give evidence for a connection between bilayer properties on the mesoscopic scale as studied by NMR relaxation and macroscopic properties of the bulk material The physical relevance of our ndings is further sug gested by the opposite in uences of a nonionic surfactant 0109013 BROWN THURMOND DODD OTTEN AND BEYER 50 1 a DMF39CdeD ZEa 21 40 40 b DMPCdSA 30 20 S 391 30 10 3 N A o v 0 001 002 003 I 20 4 1 I e 7 i I 1 10 4 I r r I T 1 c DMPCdmmhol 11 0 1 O 005 01 015 02 i 2 ISCDI FIG 4 Squarelaw dependence ongiZ on ISggIZ for bilayers of DIVIPCd54 summarizing in uences of a nonionic cosurfactant C12E8 in the La phase and cholesterol in the liquidordered phase Results are shown for bilayers aligned at 6900 containing excess HZO at 461 MHz a DMPCd54 C12E8 21 at 42 0C b DIVIPCd54 at 40 OC and c DIVIPCd54cholesterol 11 at 40 OC The inset shows an expansion of the data for DMPCd54C12E8 12 vs DIVIPCd54 CUES and cholesterol as shown in Fig 4 Previous studies of macroscopic bending deformations have revealed that a cosurfactant favors a decrease in K whereas cholesterol leads to an increase 2 The presence of CUE8 in the DMPCci54 bilayer 12 molar ratio Fig 4a yields a larger slope of the squarelaw plot versus DMPCd54 alone Fig 4b cf also inset which we interpret as due to softening of the bilayer We nd that KDMPcCIZE8075gtlt 10 11 N nearly a twofold decrease with K6kBT The increased entropic RAPID COI II IUNI PHYSICAL REVIEW E 64 010901R repulsion would result in greater swelling of the membrane dispersion as found for nonionic surfactants 20 By con trast a squarelaw plot for DMPCd54zcholesterol 1 1 is shown in Fig 4c evincing a reduction in slope compared to DMPCd54 alone Here we obtain KDWCchol93 X 10 11 N roughly an eightfold increase 21 Assuming t 5 nm gives K IOOkBT for the bending rigidity a dramatic stiffening Similar conclusions have been reached from NMR transverse relaxation studies 18 The fact that bilayer additives in uence the R12 data parallelng earlier results for larger distance scales 217 supports our hypothesis 10 that quasicoherent modes are already present on the order of the bilayer thickness and less and that all modes should be con sidered unless one can show they are decoupled from the physical properties of interest In closing this work provides striking evidence that mem brane defonnational uctuations occur over a wide range of length and timescales which depend on the bilayer lipid composition Clearly it is of interest to investigate the corre spondence to other techniques for studying membrane defor mation involving splay uctuations at larger wavelengths and to molecular dynamics simulations Moreover these ndings may be of relevance to the often debated issue of the nature and functional form of the repulsive forces between amphiphilic layers This work was supported by the RontgenProfessorship of Physics at the University of Wiirzburg MFB the US National Institutes of Health MFB the US NIH RLT the Deutsche Akademische Austauschdienst DO and the Deutsche Forschungsgemeinschaft KB 1 M Bloom E Evans and O G Mouritsen Q Rev Biophys 24 293 1991 2 E Sackmann in Handbook of Biological Physics edited by R Lipowsky and E Sackmann Elsevier Amsterdam 1995 Vol 1 pp 213304 3 W Pfeiffer S Konig J F Legrand T Bayerl D Richter and E Sackmann Europhys Lett 23 457 1993 4 W Helfrich Z Naturforsch A 33a 305 1978 5 J N Israelachvili and H Wennerstrom J Phys Chem 96 520 1992 6 M F Brown Chem Phys Lipids 73 159 1994 7 M Bloom C Morrison E Sternin and J L Thewalt in Pulsed Magnetic Resonance NMR ESR Optics edited by D M S Bagguley Clarendon Press Oxford 1992 pp 2747 316 8 M F Brown and S I Chan in Encyclopedia ofNuclear Mag netic Resonance edited by D M Grant and R K Harris Wiley New York 1996 Vol 2 pp 8717885 9 R Goetz G Gompper and R Lipowsky Phys Rev Lett 82 221 1999 10 A A Nevzorov T P Trouard and M F Brown Phys Rev E 58 2259 1998 11 A A Nevzorov and M F Brown J Chem Phys 107 10 288 1997 12 P G de Gennes and J Prost The Physics of Liquid Crystals Clarendon Press Oxford 1993 13 R L Vold R R Vold and M Warner J Chem Soc Fara day Trans 2 84 997 1988 14 P Meleard C Gerbeaud T Pott L FernandezPuente I Bi vas M D Mitov J Dufourcq and P Botherel Biophys J 72 2616 1997 15 L FernandezPuente I Bivas M D Mitrov and P Meleard Europhys Lett 28 181 1994 16 W Rawicz K C Olbrich T McIntosh D Needham and E Evans Biophys J 79 328 2000 17 F Auguste P Barois L Fredon B Clin E J Dufourc and A M Bellocq J Phys France 4 2197 1994 18 G Althoff N J Heaton G Grobner R S Prosser and G Kothe Colloids Surf A 115 31 1996 19 R P Rand and V A Parsegian Biochim Biophys Acta 988 351 1989 20 U Olsson and H Wennerstrom Adv Colloid Interface Sci 49 113 1994 21 T P Trouard A A Nevzorov T M Alam C Job J Zajicek and M F Brown J Chem Phys 110 8802 1999 0109014

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