# Intermonolayer Friction and Surface Shear Viscosity of Lipid Bilayer Membranes

By Otter, W K den Shkulipa, S A

ABSTRACT The flow behavior of lipid bilayer membranes is characterized by a surface viscosity for in-plane shear deformations, and an intermonolayer friction coefficient for slip between the two leaflets of the bilayer. Both properties have been studied for a variety of coarse-grained double-tailed model lipids, using equilibrium and nonequilibrium molecular dynamics simulations. For lipids with two identical tails, the surface shear viscosity rises rapidly with tail length, while the intermonolayer friction coefficient is less sensitive to the tail length. Interdigitation of lipid tails across the bilayer midsurface, as observed for lipids with two distinct tails, strongly enhances the intermonolayer friction coefficient, but hardly affects the surface shear viscosity. The simulation results are compared against the available experimental data.

INTRODUCTION

Membranes play a vital rule in living cells, where they act as semi permeable barriers, host numerous proteins, and provide mechanical strength while retaining a high degree of flexibility ( 1- 3). Membranes consist of lipids, i.e., biological amphiphiles, which in an aqueous environment cluster into locally planar hi layered structures under a combination of hydrophobic and hydrophilic interactions. In the absence of covalenl bonds between the lipids. a bilayer behaves as a twodimensional liquid whose resistance against shear deformations is characleri/ed by the surface shear viscositTJ,,. Any relative motion between the two leaflets of a bilayer is opposed by a friction force: the ratio between the force per unit area and the slip velocity is known as the intennonolayer friction coefficient h. Living cells actively control these two flow properties by varying the mixture of lipids and sterols present in their membranes. Here we are interested in the flow properties of homogeneous hi layers, and how these are related to the constituent lipids.

Current experiments to measure surface shear viscosities of membranes are all based on a theory by Saffman (4) relating T/S to the translalional diffusion coefficient D of a tracer particle confined to a membrane, i.e.. a geometry specitic analog of the familiar Stokes-Einstein relation. Most viscosity measurements employ labeled membrane-bound proteins as the diffusing tracer panicle (5.6). while recently latex spheres (7-9) and phase- separated lipid domains (10) have been used. Typical surface shear viscosities reported in the literature lie in the range of IO 7-IO ” surface poise (where 1 SP – 10″ Pa m s). The best-studied tracer particles, however, are the lipids constituting the hilayer (6,11- 15). hut their lateral diffusion coefficients are rarely converted into surface viscosities because these tracers are considered too small for Saffman’s continuum hydrodynamics model to hold true, and because lipids are strongly affected by microdomain formation within the membrane (6,9). Fluorcscence-alter-pholohleaching experiments (II) indicate thai the symmetric phosphatidyleholine lipids 1.2- dilauroylphosphatidylcholine ( DLPC. diC 12:01. 1.2-dimvristol- phosphatidylcholine (DMF3C’. diCI4:0), and 1.2-dipalmiloyl- phosphatidylcholine (DPK”, diC16:0) have surprisingly similar diffusion coefficients at 5O0C. despite the variation of their saturated tails from 12 to Ih carbon atoms, while the diffusion coefficient of the asymmetric l-palmitoyl-2-oleoyl- phosphatidyleholine (POPC. C 16:(VC IK: 11,,> is lower by approximately one-third. In nuclear magnetic resonance measurements (13). however, the lipids DMPC. POPC. and 1,2-dioleoyl- phosphatidylcholine (DOPC. diCIH:lt.i() are found to possess nearly identical diffusion coefficients, suggesting a remarkable insensitivity to tail length, symmetry, and saturation. A recent nuclear magnetic resonance study (15) reports a decrease in the lateral diffusion with increasing tail length for monoiinsalurated lipids with tails of up to 22 carbons. Addition of cholesterol, which is known to increase the ordering of the lipids and to induce domain formation, slows the lateral diffusion down, as does a reduction of the temperature (11.13). The surface shear viscosity, for which much fewer data are available, is expected to follow trends counter to those of the diffusion coefficient, i.e., TJS declines with increasing I) and vice versa.

The intennonolayer friction coefficient, whose existence first came to prominence a decade ago ( 16-20). has been measured by a number of techniques. In the experiments described by Evans and Ycung (IM). Raphael and Waugli (21 ). and by Chi/mad/hev et al. (22), a mechanical force was used to pull a bilayer through a region of extremely high local curvature, causing the leaflets to slip past one another. Merkel et al. (23) lixed the bottom monolayer of a membrane to a glass substrate and deduced a friction coefficient from the diffusion of tracer lipids in the top monolayer. Pfeiffer et al. (24) and Pott and Melcard (25) derived friction coefficients from lhc decay rales of the time-correlations of thermal undulations in a hi I aver slack and in vesicles, respectively, bin the interpretation of these measurements proved to be complicated. From these various experiments, a typical range of 10s-K)” Pa m ‘ s is obtained for the interimmolayer friction coefficient. The relation between lipid architecture and intermonolayer friction has hardly been explored. The friction coefficient of an OMPC bilaver lies ~3.V/ r below that of an SOPC hilayer, with the latter showing a remarkable insensitivity of h to temperature over the range from 15 to 350C (private communication with E. Yeung, University of Alberta, Canada). Cholesterol was found to increase the friction coefficient by roughly 50%, while the value for a mixed bovine brain sphingomyelincholesterol (CHOL) membrane is nearly live times thai of l-stearoyl-2-oleoyl-phosphatidylcholine (SOPC, ClSt(VU)CHOL and l- oleoyl-2-myristoyf-phosphatidylcholine (OMPC. CIK:lt,,/CI4:0)-CHOL membranes (all in a 1:1 mixture). It is our expectation that h is affected by interdigitalion. i.e., Ionji lipid tails whose ends cross the bilayer rnidsurface and protrude among the tails of the opposing hilayer leaflet (2628), as this enhances the grip between the two monolayers.

Computer simulations of membranes have provided a wealth of detailed information, as reviewed in the literature (29-32). but there are few studies focusing on the flow properties of hilayers. E number of authors have reported lateral diffusion coefficients oflipids (33-38). also in relationship to tail length (39.40), but these have not been related to the surface shear viscosity. In a recent simulation study of solvated membranes, Guigas and Weiss (41) found the diffusion coefficient of a membrane-bound tracer to decrease logarithmically with the radius of the tracer, in agreement with Saffman’s iheory. We believe thai a similar decay observed by (hese authors in simulations without solvent cannot he explained by Saffman’s theory, since its derivation crucially relies on the presence of a solvent surrounding the membrane (see (4) and Eq. 6). bui instead reflects the logarithmic radius dependence predicted by hydrodynamics for diffusion in a periodically continued two- dimensional fluid (42). The exponential relaxations of thermal undulations (43) and the relative Brownian motions of the monolayers (37,40) are, in principle, connected to the inlermonolayer friction. The first direct calculations of T;S and h have been reported only recently, using bilayers under shear (44) and at equilibrium (44- 16). Mere we extend our previous studies by using a series of coarse- grained model lipids to explore the relationship between lipid architecture and hilayer flow behavior-a relationship mat has thus far not been studied systematically in the experimental literature. Furthermore, the employed nonequilibrium simulation techniques have the advantage that they provide unequivocal access to the How characteristics, whereas the current interpretations of experiments are rather involved and nontriviul. A final motivation for this study is to investigate the origin of the discrepancy, by several orders of magnitude, between the TJS and h as obtained in previous course-grained simulations versus the values deduced I’rom experiments.

The outline of this article is as follows: The techniques used Io calculate surface shear viscosity and inlermonoluyer friction, from both equilibrium and rumcquilibrium molecular dynamics simulations, are briefly described in Theory. The Simulation Model is then introduced, and several basic properties of the membrane and solvent are calculated. We then present the Simulation Results, followed by Discussion and Conclusions.

THEORY

The aim of this section is to provide a concise introduction of the theory on the thermodynamics ami hydrodynamics of membranes, and to describe the simulation techniques employed to calculate the key parameters.

Static properties

FIGURE I Cannons of the simulation selup of a memhrane under shear, with the arrows indicating llie How tit-Ids. A perpendicular shear flow (left) is used to determine ihe surface shear visci)siii>C the bilayer; a parallel shear flow (rit;lin yields lhe intennonolayer friction coefticicnt.

The left-hand Mile is exact, with Ap = p^sub zz^ – 1/2 (p^sub xx^ + p^sub yy^) and p the stress tensor. On the right-hand side, we have made the common assumptions that the projected-area dependence of the free energy is dominated by the elastic tenu, and that A[u] [asymptotically =] A^sub ||^ A for a nearly Hal hilayer. although these approximations are to be regarded with care (49). In the absence of externally imposed restraints, a freely floating hilayer of N molecules will adapt a tensionless stale with an average area equal to the equilibrium area of A^sub 0^ = Na^sub 0^/2. In our simulations we have reproduced this state as closely as possible, by varying the ground-plane area of the box until Ap = 0. lntermonolayer friction

FIGURE 2 The average particle velocity along lhc flow (.vl direction, as a function of llio height Cl in I he simulation box. lor a hilayer exposed to ;i parallel shear w ilh a ven high shear rate, y – 10 ns ‘.The two nionolavcrx. each 2(1 A thick, are seen to slide along one another at the midsurtuce, while the solvent shows u linear How Held with shear rate y*.

SIMULATION MODEL

Fully atomic simulation models of membranes place a heavy burden on available computer resources and are therefore limited to small time- and length-scales. These drawbacks can be overcome by using a coarse-grained (CG) simulation model, in which a number of atoms are grouped together into one interaction site, known as a CCi particle (31.32). Here we employ a CG model recently developed by Marri c k el al. (35) to simulate DPPC” and related amphiphiles. These authors chose a parameleri/alion in which groups of approximately four heavy atoms, and their attached hydrogens, are reduced Io a single CG particle. In this section we first present a summary of the model, referring (he reader to the original work (35) for more details and an extensive motivation. Nexl. several characleristics of the model are calculated to provide a basis for the analysis of membrane dynamics.

Force field

The model discerns four major types of particles, representing groups of atoms with different properties: charged groups (Q), polar hydrophilic groups (P). weakly polar groups (/V). and apolar hydrophohic groups (C]. Particle types Q and N are subdivided into four categories according to their hydrogenbonding capabilities, of which only the a (acceptor! and (Kno capabilities) subtypes will be used here. The head of a DPPC lipid (molecule /I44 in Fig. 3) is then represented by one Qo particle with charge tf ~ e for the choline group and one QA particle with charge q = -c for the phosphate group. The glycerol ester linkage is modeled by two particles of type /V.,, while each C e (,H u tail is reduced to a chain of four C particles. Four water molecules are lumped into one bead of type P.

FKiURE 3 Cartoons of the double-tailed lipids simulated with the coarsegrained model hy Miirrink L-I al. e35). ‘[Eth]e- shimhaml notaiiim is included below ihc lipids, the particle type o>> the IeIl uiul riglu.

The nonbonded interactions between particles /’ and /’ at u distance r-,j are described by a Lcnnard-Jones potential, > – 72 a.u. In the DlPoIy 2.0 package (53) LI Nose-Hoover thermostat is used to maintain a temperature of 323 K and in some runs a Hoover barostat was invoked to establish an isotropic pressure of I bar. The Verlet leap-frog algorithm allows a maximum time step of 20 fs.

Basic properties of the model

Before simulating LI lipid bilayer in a solvent matrix, we first assess the quality ol’ the model by separate studies of the solvent and the lipid tails, i.e., linear alkanes. To determine the dynamical properties of the aqueous solvent, a cubic box containing 6072 /Mype particles was prepared. Using a barostat at I bar and a thermostat at 323 K, the specific gravity of the solvent converged to nearly I ,y/cm . The diffusion coefficient of the CCi particles was established at D^’ – 2.OX H) s cnr/s. which lies very close to the experimentally measured value for a water molecule at these conditions. D”p = 2.3X 10~-scnr/s. Simulations under shear yield a viscosity c^[degrees] % 7XIO~4 Pas. independent of the applied shear rate and in close proximity to the experimental value c^iA = 5.5x H)~4 Pas under the prevailing conditions (54). The dynamics of bulk C4 tails, i.e.. the CG equivalent of liquid /e-hexadecane. was analyzed using a cubic box containing IiIS molecules, again at 1 bar and 323 K. The simulations yield a diffusion coefficient D^jJ.’ – 1.2X 10~5cnr/s. or nearly twice as high as the O^ = 0.7 x IO 5cm2/s obtained by extrapolating the available experimental data (55) to the current temperature. In conformity with the Stokes-Einslein expression, the simulated viscosity of c*,!;’ – Sx H)”4 Pas is approximately hall the experimental value rf.2 = 1.9X IO ‘ Pa s (56).

TABLE 1 Lennard-Jones interaction parameters t(| (in kJ/mol) between the five particle types of the coarse-grained lipid model by Marrink et al. (35)

Groot and Rabonc (57) argued thai the diffusion coefficient of a CG particle or CG molecule representing Av real molecules is related to the experimental molecular diffusion coefficient by /?cv’Cl = / >^N/A\. By inserting the above diffusivities of water in this expression, with /.” – 4, Marrink et al. (35) concluded that the dynamics of the solvent is fourtimes too fast. Acceleration factors of -2 were obtained by comparing simulated hulk CG alkanes (C”,, with c ^ 5) with the corresponding experimental linear alkanes (C4nHs11 + ^). with A11Ii1 = 1. These authors subsequently accounted for the anticipated speedup of the bilayer dynamics by simply scaling every nanosecond of simulation time into 4 ns of real time. Although the diffusivity mapping should obviously hold true for a CG panicle or molecule representing one real molecule (Ax – I ), its validity for a compound (Ax > I ) CG particle is less evident: correlations between the molecules constituting the CG particle seriously complicate the picture. Since the major purposes of the solvent particles are to provide a hydrophilic and viscous environment to the bilayer. it appears more natural to match the viscosity rather than the diffusion coefficient of the solvent. Based on the aforementioned viscosities, we are thus led to the conclusion that the model yields an adequate description of the dynamics of hulk water and bulk alkanes. which are expected to cany over to the membrane simulations, without the introduction of a scaling factor for the time.

A hilayer-solvent system was prepared by expanding a smaller box, made available by Marrink (35.5S). to 256 DPPC (A44) lipids and 30(M) solvent particles. The system was equilibrated by a KM) ns simulation at constant temperature. 7 = 323 K. followed by a HK) ns simulation at constant temperature and pressure, n = I bar. As a final step in locating the equilibrium tensionless state, the tension versus strain curve of the bilayer was calculated by varying the ground-plane area around the final area of the NPT simillation. The zero tension intercept of this curve, sec Eq. 2, is reached lor an average area per aniphiphile of fi() = 0.66 nnr, which is slightly higher than the experimental value of 0.64 nnr at this temperature (5[degrees]- !. The deviation IVom the 0.64 nnr reported by Marrink et al. (35), as well as other small differences with their simulation results, are attributed lo slight deviations in the simulation setup, including a smaller time step, the reseating of the ground-plane area at constant volume rather than at constant normal pressure /J77, anil the use of different simulation packages.

The equilibration simulations yield two additional mechanical properties of the membrane, which will he needed later. From the slope of the tension versus strain curve follows an area compressibility EE = 370 ecI/m for the 25ft lipid bilayer. while a larger 6400 lipid bilayer yields 220 niN/m. This decrease results from the approximate area calculation A\n=s A\I, which renders the effective area compressibility defined by Eq. 2 system-size- dependent. /Oi1″= ^V + ((tBT/327rV MoIiV]J ‘. with Kthe intrinsic compressibility defined by Eq. I (49). The EE obtained by this system-size correction is ~- 2Ass larger than the K^ of the smaller bilayer. but some 4(W larger than the A”v” of the large bilayer. while reported experimental values for DPPC (5M) and a range of other PC-s (60) are typically -230 niN/m. The structure factors of the thermal undulations at/y =/A,,closely adhere to the predicted scaling law (see Eq. 4). confirming that we have indeed reached a stale with vanishing tension. At K = KX 10′ 1J or IS^H/”. the bending rigidity is higher tluin the -6 X 10 ~” J anticipated from experiments (60). In summary, the A44 membrane is a bit less flexible than a real DPW membrane, and we expect similar small deviations in the dynamical properties.

SIMULATION RESULTS

In this section, the dynamical properties of membranes are calculated using the equilibrium and nonequilibrium techniques outlined in Theory. The results obtained for hiluyers composed of DPPC (A44) lipids are first discussed in some detail, followed by simulations of A1.j lipid membranes to investigate the influence of tail lengths on the dynamical properties of the bilayer. We discuss asymmetric lipids. / / /, with a combined total tail length of /’ + j – S particles, symmetric lipids with two elongated tails. ; =./ > 4, and finally a lipid with one straight and one bend tail, / – ./’ – 5. Each simulated membrane contains 256 identical lipids surrounded by 3(KH) solvent particles, and is thoroughly equilibrated to a tension-less slate at I bar. All lipids have identical headgroups, and the tail particles are, as before, of the C type. DPPC membrane

The flow properties of a DPPC membrane, and their shearrate dependencies, were studied by exposing the small A44 bilayer patch of the previous section Io shear rates y ranging from U.I to 1 ns ‘. In the perpendicular sheai orientation, the off-diagonal element / )xy of the pressure tensor is used to calculate the surface shear viscosity (see Eq. 5 and Table 2), (o arrive at c,, = 1.2 x K)’ ” Pa m s, independent of the applied shear rate. Under parallel shear, the aforementioned range of shear rates induces slip velocities of 0.03-0.27 nm/ ns between the two hilayer leaflets. Equation 7 then yields an intermonolayer friction coefficient h – 2.4 x I0h Pa m ‘ s, independent of the slip velocity. It appears, therefore, that this coarse-grained membrane model underestimates the experimental values of c,, and h by one-to-two orders of magnitude, although it does an excellent job on many thermodynamic properties. A discussion of the possible sources of these discrepancies, which were also observed for a single-tail CG lipid membrane model (44). will be postponed until Discussion and Conclusions. In the remainder of this subsection, the equilibrium approaches for determining cI and h will be applied, to validate the numerical results from the nonequilibrium simulations and to ascertain the possible impact of nonzero shear on these parameters.

An independent conformation of the surface shear viscosity is provided by the satisfactory agreement between the lateral diffusion coefficient of a lipid in a quiescent bilayer as calculated by Saffman’s theory. Os.l(-tman = 1.4 X IO 6 cnr/s, and the actual value of />[11M, - 1.5 X I0~ft cnr/s determined from the mean- square displacement. The former should he regarded with some care, however, as the simulation conditions do not adequately match the assumptions underlying the Saffman theory (see Surface Shear Velocity). In applying Eq. 6, we have approximated the floppy lipid by a rigid cylinder of radius R = ^n/[eth]. and regarded the hiluyer as a continuum fluid on this lcngih scale. Furthermore, since the lipids span only one leaflet rather than the entire membrane, we followed previous experiments (5u and simulations (44) in substituting the hilaycr surface shear viscosity in Eq. 6 by the monolayer surface shear viscosity c(TM) = T)J- Considering the assumptions made, the agreement between the two diffusion coefficients is satisfactory.

TABLE 2 Surface shear viscosities >JB and lntermonolayer friction coefficients b obtained for the various coarse-grained lipld models of this study

As described in Iniermonoluyer Friction, inlcrmonolayer friction also manifests itself during the Brownian thermal undulations of a nicmhrune. Hg. 4 shows the time correlations of the six undulation modes /iq corresponding to the three smallest wavenumhers commensurate with the box dimensions. Single-exponential decays are clearly observed to set in after some 250 ps, especially for the modes with smaller wavenumhers. The decay rates agree with the slow relaxation rates y,u/i of the theory of Seiterl anil Langerthe exact curves calculated by Eq. 8 arc included in the graph as dashed lines- thus continuing the value of the intermonolayer friction coefliciem. Only one parameter in this theory could not be determined a priori from an independent simulation, to wit, the distance d between the midsurface of the bilayer and the neutral surfaces of the monolayers ( 17). We judiciously chose this distance to be equal to half the monolayer thickness, i.e., a quarter of the bilayer thickness, tl – / i/4 ss 1.1 nm, as this choice has worked out well in previous studies (45.46). In summary, the dynamical properties calculated from nonequilihrium simulations are continued by the equilibrium simulations for the DPPC (A44) membrane. Since the latter calculations consume more computer time anil are less straight forward in their interpretalion, we recommend the nonequilibrium simulations as the more practical and more reliable techniques.

FIGURE 4 The autocorrelations (sOlid lines) of the thermal undulations ol a bilayer of A^sub 44^ lipids, tor the six smallest wave vectors common simile with the hm dimension, i.e.. q = 0.069 [Angstrom]^sup -1^ (top), q = 0.097 [Angstrom]^sup -1^ (middle). and q = 0.137 [Angstrom]^sup -1^ (bottom). Dashed lines indicate the theoretical predictions by Eq. 8. where the amplitudes A^sub i^ and relaxation rates gamma^sub i^ have been calculated using the intemionoluycr friction coefficient determined in the parallel shear simulations.

FIGURE 5 The surfaire shear viscosity as a function of ihc shear rate Cur the A^sub 44^ lipid (cicles and solid line), two lipids with assymetric tails (squares and diamonds), three lipids with extended tails (triangles and dotted lines), and the A^sub 5’5^ lipid (pluses). Viscosities derived from the lipid lateral diffusion coeffecients by the Saffman-Stokes-Einstein theory are plotted at gamma = 0.

Lipids with asymmetric tails

The effect of asymmetric tail lengths on the membrane properties was studied by comparing the reference A^sub 44^ lipid to its cousins A^sub 53^ and A^sub 62^ (see Fig. 3). Two new membranes, each containing only one type of lipid. were made and thoroughly equilibrated Io iheir tensionless states. By exposing these bilayers Io perpendicular shear, using the aforementioned range of shear rates, both lipids were found to yield nearly identical surface viscosities to the A^sub 44^ lipid. which in Fig. 5 are seen to be independent of the shear rate. These viscosities are in good agreement with the values derived, by means of the Saffman expression, from the lateral diffusion coefficients of lipids in nonsheared hi layers; the latter are represented in the graph by the markers at gamma = 0. Because the equilibrium area a^sub 0^ and membrane thickness h are only slightly different for the three lipids, one may expect comparable lateral interactions in all three membranes and, hence, nearly identical surface shear viscosities.

The intemionolayer frictions obtained under parallel shear, however, rise with increasing tail length difference (as shown in Fig. 6). Since the equilibrium dimensions of the three membranes are comparable, this rise results from differences in the packing of the tails inside the hydrophobic core of the bilayer. The probability distributions of the amphiphilic particles along the normal Io the membrane, as presented in Fig. 7 for the three lipids. indicate that the tails of the opposing leaflets arc not interdigitaled for the A^sub 44^ lipid. The long tails of the A^sub 53^ lipids interdigitale by approximately one particle with their counterparts from the opposing leaflet, while the long tails of A^sub 62^ lipids interdigitiite by approximately three particles, Nole thai the peaks of lhe particle distributions have shifted with increasing asymmetry, thereby reducing the actual number of interdigilating particles Io less than the values of 2 and 4 expected on simple geometric grounds for A^sub 53^ and A^sub 62^, respectively. The interleaflet contacts created by interdigitation strengthen the inleraclion between the two nionolayers and are responsible for the growth of the inlemionolayer friction with increasing asymmetry. Intermonolayer friction coefficients were also deduced from the slow relaxation rates gamma^sub 1^ (see Eq. 8) of the two thermal hilaycr undulations with the smallest possible wavenumber. q = 2pi/L^sub ||^. Sampling deficiencies of lhe time-currelation functions were minimized by prolonging all simulations until t^sub simu^>50gamma^sub 1^^sup -1^. With increasing tail asymmetry, the idea of a smooth m id surface where monolayers gently slide pasl one another becomes obscured and the assumption d = h/4 might be less well founded. Nevertheless, the simulations strongly suggest thai the iniemionolayer friction coeflicienls are independent of the slip velocity.

FIGURE 6 The inicmionolayer friction coefficient as u funclinn of ihe slip velocity for the A^sub 44^ lIpid {circles and solid line). two lipids with asymmotric tails (squares, diamonds, and dotted lines). three lipids with extended tails (triangles). and the A^sub 5’5^ s lipid (pluses). Friction coeffecients obtained by the Seifert- Langer theory from the slow relaxation rates gamma^sub 1^ of the thermal undulations are plotted at Deltav = 0.

FIGURE 7 Probability distribulions along the bilayer normal of all 12 particles in the lipid, for A^sub 44^ (bottom) A^sub 53^ (middle). and A^sub 62^ (top), with the particles of the first (long) tail as soliil linos, lho particles of the second (short) tail as dasheii linos, and ihe hoail partifk-s as dash-ilottod linos. The lemiinal and interdigitating panifies are marked by dark linos, while light lines are used lnr the remaining tail particles. The asymmetric lipids limn a so-called partially nilcrd ig Haled bilavor, in which a long lail packs end-to-cnd with a short lail from ihe opposing leallel.

Lipids with symmetric tails

In the simulations of symmetric lipids. the reference lipid A^sub 44^ is compared against three lipids with longer lails: A^sub 55^. A^sub 66^ and A^sub 77^. Along this series of lipids. the equilibrium dimensions of the tcnsionless hi layers gradually grow according to a^sub o,i^ = (O.655 + 0.009i) nm^sup 2^ and h^sub i^ – ( 19 + 6i) [Angstrom]. with i the number of particles per tail. The ratios of bending rigidity to elastic modulus are well fitted by (kappa^sub i^/kappa^sub A,i^)1/2 = alpha(h^sub i^ – h^sub 0^), with alpha = 0.16 and h^sub 0^ = 1.2 nm, in good agreement with alpha = 1/ [the square root of]24 [asymptotically =] 0.20 and h^sub 0^ = 1 nm obtained by the experiments and theory of Rawicz et al. (60). Simulations under parallel shear reveal that the iniemionolayer friction coefficients vary by

In the tinal A^sub 5’5^ lipid model (depicted in Fig. 3), a monomisatunited tail is mimicked by reducing the equilibrium bending angle at the central particle of t he tail to theta^sub 0^ = 130[degrees] and by stiffen ing the bending potential of this particular angle to K^sub angle^ = 250 kJ/mol. Figs. 5 and ft show that the impact of the boomerang-shaped tail on the membrane properties is rather modest. The surface shear viscosity has increased slightly relative to its closest saturated counterpart, the A^sub 55^ lipid, while the intennonolayer friction coefficient shows a small decrease.

DISCUSSION AND CONCLUSIONS

The flow properties of bilayer membranes have been studied for a variety of coarse-grained (CG) lipids using equilibrium anil nonequilibrium molecular dynamics simulations. Intermonolayer friction coefficients h are obtained from simulations of membranes exposed to parallel shear, which forces the two membrane leaflets to slide past one another, and from analyzing lhe relaxation rates of thermal membrane undulations using the Seifert-Uanger theory (17): the results of both approaches are in good agreement. Surface shear viscosities eta^sub s^, have been calculated from simulations of membranes under perpendicular shear and from the lateral self- diffusion of lipids by the Saffman-Stokes-liinsiein expression (4). A majority of the simulated membranes display Newtonian behavior under shear and quantitative agreement between eta^sub s^ from sheared and nonsheared simulations. The two lipids with the longest tails, however, revealed shear-thinning behavior and a gradual departure from the Sat’t'man theory. The generally satisfying agreement between the results of equilibrium and nonequilibrium simulations may also be interpreted as support for both aforementioned theories.

The intermonolayer friction coefficient and surface shear viscosity obtained with the current double-tailed CG lipid model, introduced by Marrink el al. (35). are – 15 times and twice as high, respectively, as those reported previously (44) for the single- tailed CG lipid model of Goetz and Lipowsky (33). Since the latter model underestimates the viscosities of water and liquid alkanes, we have since compensated for these deficiencies by introducing pairwise-additive friction and random forces between all particles, following the methodology of dissipalive particle dynamics (61,62). These additional forces arise naturally in any coarse-graining procedure as representatives of the omitted internal degrees of freedom of the CG particles; for computational speed, and because they do not affect lhermodynamic properties, these forces are often ignored in CG models. After tuning the extra force parameters for solvent-solvent anil tail-tail interactions to reproduce the viscosities of bulk water and hexadecane, respectively, the resulting membrane How properties eta^sub s^ and h approach those of the Marrink model to within a factor 2 (63). This suggests that the reported values are generic for coarse-grained lipid models. Note, however, that these values underestimate the experimental b and eta^sub s^. as collected in Table 2, by approximately two orders of magnitude. We will return to this discrepancy shortly.

A thin layer of bulk liquid with viscosity eta^sub 1^ and thickness h^sub 1^, is readily shown io posses a surface shear viscosity eta^sub s^ = eta^sub 1^h^sub 1^ resisting perpendicular shear flow and a friction coefficient b = eta^sub 1^/h^sub 1^ opposing a velocity difference across the layer (19,22). By comparing the hilayers with symmetric lipids against these rules of thumb, we lind that the low shear-rate results are adequately described by eta^sub s^ [asymptotically =] 3eta^sub alk^h and b [asymptotically =] 12eta^sub alk^/h with eta^sub alk^ the viscosity of a bulk liquid of corresponding CG alkanes. The prefactors appearing in these expressions reflect that the hilayer is not a simple bulk liquid of tails, but that the lipids are straightened, oriented, and ordered in a bilayer. Experimental data yield prefactors of the order of 100-1000. with the intennonolayer friction requiring a higher prefactor than the surface viscosity (19). suggesting that the impact of ordering on eta^sub s^ and h is considerably higher for real lipids than for the coarse-grained model lipids studied here. We hypolhesize that these differences relied the weakness of coarse-grained models to accurately mimic the detailed interactions between alkanes. and hence their packing in dense phases. This is also illustrated by the viscosities of the bulk C^sub i^ liquids, which we found to he linear in the length i for chains of 4-7 particles. Experimental viscosities (56), however, rise rapidly with increasing chain length, while n-tetracosane (i = 6) and longer linear alkanes adopt a waxlike stale al the simulation temperature of 323 K. In the more ordered bilaycr structure, a similar transition is observed at an even lower tail length, with 1.2-dipalmitoyl-phosphalidylcholine (DPPC. diC 16:0, i = 4) in me liquid-crystalline phase and l.2-ilisiearoyl- pliosphatidylch(iline(DSPC.diCIK:0. i = 4.5) in the gel phase at 323 K. The coarse-grained lipids, tor which no chain-length induced transitions were observed, are apparently less sensitive to ordering and dense packing effects, which could also explain their relatively low surface shear viscosity and intermonolayer friction coefficient. Since lhe more realistic atomically detailed simulation models are capable of reproducing phase transitions, one might expect that they yield a quantitative agreement wiih experimental How properties of membranes.

The eflicient noncquilibrium simulation techniques for establishing eta^sub s^, and b have not yet been applied to an atomic membrane model, but the equilibrium methods provide a first glimpse at what may he expected from these models. Various studies (37.40) show that the lateral diffusion coefficients obtained in atomic molecular dynamics simulations arc in good agreement with experimental data, suggesting that they will also yield quantitatively satisfying surface shear viscosities. As we have seen, however, a direct translation of D into eta^sub s^ should be regarded with caution, and simulations under perpendicular shear are recommended. In a recent study, Wohlcrt and Edholm (37) estimated an intermonolayer friction coeflicienl of 0.7 x 10^sup 6^ Pa m^sup – 1^ s from the relative diffusion between the two leaflets of an atomic DMPC membrane. By reanalyzing the autocorrelations of the thermal undulations in an atomic DPPC membrane, published by Lindahl and Edholm (43), within the framework of the Seifert-Langer theory (Eq. K), we find that b ~ 10^sup 6^ Pa m^sup -1^ s fits the simulation data reasonably well. These tirsl indications suggest, therefore, lhat inlermonolayer friction coefficients by atomic membrane models are in close proximity Io those obtained with coarse- grained models, while both are two-to-three orders of magnitude smaller than lhe few reported experimental values. With the stacking deficiency of the coarse-grained model ruled out. the interpretation of lhe experimental data appears as the most likely source of lhe discrepancy. Small gel-like domains surviving in a predominantly fluidlike phase could result in a drastic increase of the resilience against flow, but these domains are expected to exist only within – 10[degrees]C above the melting transition (9). Thermal undulations of the membrane, which are relatively small in the simulations and conveniently ignored when analyzing the experiments, might also affect the flow properties. By varying lhe tail lengths of a double- tailed lipid, we observed two clear trends for lhe How properties of bilayers. First, the iniemionolayer friction coefficient is sensitive to the asymmetry of the tails, as interdigiiation of the longer tails roughens the interlace between the two membrane leaflets. Second, the surface shear viscosity is modulated by the combined lengths of the two tails, and hardly varies with the asymmetry of the tails. The experimental dala al our disposal are inconclusive to confirm these trends. Finally, we express the expeelation that this study inspires tutu re experimental and simulation studies on the How properties of membranes and their relation to the memhrane composition.

The authors lhank W. J. Uriels for stimulating discussions and A. Yeung for sharing unpublished experimental data.

This work is part of the Sol t Link research program of the Slichling voor Pundamenleel Onderzoek der MaIfHf. winch is linancialh supported by the Nederlandse Orgunisutie voor Wetenschuppclijk Qnderzoek.

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W. K. den Otter and S. A. Shkulipa

Computational Biophysics, Faculty of Science and Technology, University of Twente, Enschede. The Netherlands

Submitted January 26. 2007. and accepted for publication Man-It 27. 20117.

Address reprint requests to W. K. den Otter, Tel.: .H-53-4S9- 2441: E-mail: w.k.denolteitauiwenic.nl.

Copyright Biophysical Society Jul 15, 2007

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