Lipopolymer electrophoresis in supported bilayer membranes

Transcription

1 / Journal Homepage / Table of Contents for tis issue PAPER Soft Matter Lipopolymer electroporesis in supported bilayer membranes Huai-Ying Zang and Regan J. Hill* Received 13t June 2010, Accepted 6t August 2010 DOI: /c0sm00524j We report teory and experiments investigating te electric-field-induced drift of poly(etylene glycol) (PEG)-derivatized lipids in supported lipid bilayer (SLB) membranes. Based on classical continuum models, an analytical formula for te lipopolymer electroporetic mobility is derived, drawing upon te Debye H uckel approximation. Te model is extended to te ig membrane surface carge densities encountered in experiments by drawing upon te Guoy-Capman teory to correct te surface carge-surface potential relationsip. Tis convenient approximation is successfully tested by comparison wit numerical solutions of te electrokinetic model based on te non-linear Poisson Boltzmann equation. We use te analytical model to interpret measured electroporetic mobilities of 1,2-distearoyl-sn-glycero-3-pospoetanolamine-N-[poly(etylene glycol)2000-n 0 - carboxyfluorescein] (DSPE-PEG2k-CF) in glass-supported 1,2-dioleoyl-sn-glycero-3-pospocoline (DOPC) bilayers at DSPE-PEG2k-CF concentrations up to 5 mol%. By separating te frictional force on lipopolymers into separate lipid-tail and polymer-cain contributions, experimental trends are captured. Fitting te model to te data furnises several previously unknown model parameters, namely te ydrodynamic size of a polymer segment, and a polymer-bilayer frictional coupling coefficient. We sow tat electro-osmotic flow plays a determinative role in te lipopolymer drift. For DSPE-PEG2k-CF in DOPC, electro-osmotic flow yields significantly lower electroporetic mobilities tan expected by balancing te total lipopolymer electrical force wit te Stokes Einstein frictional force. Tis is attributed to te ydrodynamic coupling of PEG cains to te oppositely directed electroosmotic flow. Jaffe 1 first proposed lateral electroporesis in cell membranes as a possible cellular organization mecanism. At about te same time, Poo and Robinson 2 reported te redistribution of protein concanavalin-a receptors along embryonic muscle cells in an external electrical field. Since ten, electrical migration of many cell surface receptors and membrane proteins in a variety of cell types as been observed experimentally. Teoretical interpretation of te electrokinetic penomena, owever, as been indered by te complexity of native cell membranes 3 6 Recent advances in solid supported lipid bilayers (SLBs) as model cell surfaces ave enabled elaborate quantitative analysis of te effects of electric fields on membrane organization. 7 9 Extensive investigations of te electroporetic motion of lipids and proteins witin SLBs ave terefore been undertaken SLBs containing lipopolymers furnis a powerful model system to advance our understanding of cell membrane dynamics. Control over te polymer grafting density, layer tickness, and surface carge density can be acieved, 14 and transmembrane proteins can be integrated into SLBs wit lipopolymers acting as a spacer to eliminate te bilayer-substrate interaction and, tus, maintain protein mobility. Accordingly, lipopolymers in model membranes ave been proposed to mimic glycocalyx. 15 SLBs containing lipopolymers are promising in biotecnology, due, in part, to teir fluid and air stability. 16 Moreover, lipopolymer conformation can be tuned by varying te surface concentration, imparting control over ligand-receptor Department of Cemical Engineering, McGill University, Montreal, Quebec, H3A 2B2, Canada. regan.ill@mcgill.ca binding. 17,18 Electroporesis in SLBs as been demonstrated to be sensitive enoug to separate isomers of Texas red. 19 Moreover, a spatially resolved label-free analysis of composition gradients in SLBs as been acieved by electroporetic manipulation in concert wit surface micro-patterning and laminar flow. 20 Lipopolymer concentration gradients in patterned SLBs using electric fields ave potential in biosensing and protein separation applications. Accordingly, detailed studies of te electric-field-induced response may improve te design and quantitative analysis of suc diagnostic platforms. Here we undertake an experimental and teoretical study of te electric-field-induced drift of lipopolymers in SLBs. Te lipopolymer in tis work as poly(etylene glycol) (PEG) covalently grafted to a carged pospolipid ancor (DSPE) wit te free end bearing a carged fluorescent moiety (carboxyfluorescein) to permit quantitative imaging. In a mean-field approximation, were te carged ends are assumed to be uniformly distributed trougout te polymer layer, te PEG cains resemble a weakly carged polyelectrolyte. In addition to termodynamic and ydrodynamic interaction forces, te lipopolymers experience an electrical force wen subjected to a longitudinal electric field. Drawing on classical polymer pysics, ydrodynamics, and electrokinetic teory, we develop a matematical model for te electric-field-induced lipopolymer drift. Tis provides a quantitative interpretation of te experimentally measured lipopolymer electroporetic mobility m E ¼ V/E, defined as te ratio of te polymer drift velocity V to te strengt of te longitudinal electric field strengt E. Note tat V reflects not Tis journal is ª Te Royal Society of Cemistry 2010 Soft Matter, 2010, 6,

2 only te lipopolymer intrinsic carge but also te carge of te ost lipids. Moreover te ost lipids may present a counter carge tat drives electro-osmotic flow outside te bilayer in addition to producing ydrodynamic disturbances witin te bilayer. For example, in bilayers wit only 1 mol% carged lipids, eac bearing a single elementary carge e, te surface carge density s z 0.25 mc cm 2, yielding an electrostatic surface potential z z 5.3k B T/e in a very weak electrolyte wit Debye screening lengt k 1 z 100 nm at room temperature. Accordingly, in te absence of lipopolymer, te Helmoltz-Smolucowski slip velocity driven by an electric field E ¼ 10 V cm 1 is about 160 mm s 1. Note tat bilayer pospolipids attain muc lower electro-osmotic velocities, because te lipid ead protrudes only a small distance into te electro-osmotic boundary layer, and tere is significant frictional drag from te lipid tails witin te bilayer itself. 13 For lipopolymers, te polymer ydrodynamic drag makes te electroporetic mobility muc more susceptible to electro-osmotic flow. We seek to capture tese and oter influences by drawing on tractable continuum approximations of te many-body ydrodynamic, electrostatic, and steric interactions. 1 Teory Previous studies of SLBs witout lipopolymers assumed tat te bilayer and support reside on te same plane. 8,9 Te glass support was estimated to contribute a surface carge density s ¼0.25 Cm 2, and electrostatic screening in te intervening 1 nm layer of water was neglected. In our experiments, owever, te bilayersupport separation is generally considered to be comparable to te 7 nm unperturbed root-mean-squared end-to-end distance of te grafted polymer. Moreover, water trapped between te bilayer and support is tougt to maintain te relatively ig ionic strengt of te electrolyte buffer tat is present during bilayer syntesis. 21 Tus, because te 1 nm Debye lengt is sorter tan te 7 nm bilayer-support separation, electrostatic interactions between te bilayer and support may be neglected. Indeed, we assume tere are negligible influences of te substrate on te (upper) leaflet. Te obstructed diffusivity of lipids and proteins by absorbed polymers or lipopolymers in one leaflet as been sown to equally inder dynamics in te oter leaflet. It is terefore generally considered tat te two leaflets are strongly coupled. 22,23 Suc coupling is assumed for electro-migrating lipopolymers, and, accordingly, we interpret our experiments as two strongly coupled leaflets. Tus, wile our calculations explicitly address te upper leaflet, coupling wit te bottom leaflet is implicitly accounted for troug a lipid-tail friction coefficient g t. Moreover, we assume lipopolymers maintain a uniform surface mole fraction c wile subjected to a longitudinal electric field E. According to Monte Carlo lattice calculations, 24 tis is reasonable wen c < 10 mol%. Te single negative carge of te lipopolymer at te PEG-lipid junction contributes a surface carge density s b ¼ z eb, were te valence z ¼1and te lipopolymer surface number density b ¼ c/a l wit A l z nm 2 te average lipid cross-sectional area. 14 Anoter carge wit valence z t ¼1 resides at te free end of te PEG cain. We assume tese carges uniformly sample te layer wit tickness L, tereby endowing te polymer layer wit a uniform carge density r p ¼ z t eb/l. We let L ¼ 2R ¼ 2lN 1/2 z 7.52 nm, were R is te PEG-cain unperturbed rootmean-squared end-to-end distance wit N z 28 te number of statistical segments per cain and l z 0.71 nm te statistical segment lengt. 25 Counterions in te diffuse double layer are assumed to be Boltzmann distributed in te y-direction perpendicular to te bilayer. It follows tat te contribution of te N counterion species to te diffuse carge is r c ðyþ ¼ XN z j en N zj ejðyþ j exp (1) k B T j¼1 were n jn is te bulk concentration of te jt ion wit valence z j, e is te fundamental carge, and k B T is te termal energy. Te equilibrium electrostatic potential j(y) satisfies te wellknown Poisson Boltzmann equation, wic we linearize in te usual manner (Debye H uckel approximation) for j <k B T/e, giving d 2 j dy 2 k2 j ¼ r p (2) 33 o were r t (y) ¼ r p + r c (y) is te total (volume) carge density, and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 k B T33 o ¼ 2e 2 I is te Debye lengt wit I te ionic strengt. Eqn (2) is accompanied by Gauss s law applied as a boundary condition 33 o dj/dy y¼0 ¼ s b, wic, consistent wit te foregoing Debye H uckel approximation, is equivalent to setting 33 o kj(y ¼ 0) ¼ s b. Recall, in our experiments, s b ¼ z eb is attributed only to te lipopolymers, since te supporting DOPC lipids are zwitterions wit zero net carge. In te absence of a longitudinal pressure gradient, fluid momentum conservation (in te absence of fluid inertia) demands 0 ¼ d2 u dy þ r tðyþe g 2 s n s ðu VÞ (3) were is te (water) sear viscosity, n s ¼ Nb/L is te average statistical segment number density in te (uniform) polymer layer, g s ¼ 6pa s is te statistical-segment friction coefficient wit Stokes ydrodynamic radius a s, u(y) is te (longitudinal) fluid velocity, and V is te polymer drift velocity. We assume a no-slip boundary condition u ¼ 0aty¼ 0, wic, in turn, is motivated by an assumption tat tere is zero net bilayer drift. Tus, wile carged lipids migrate under te influence of te electric field, a counter-flow of uncarged lipids and an accompanying longitudinal bilayer pressure gradient is generated spontaneously to maintain a constant lipid density. I ¼ 1/2 P z 2 jc j, were c is te bulk concentration of te jt ion wit valence z j. Te buffer in electroporesis experiments contains 0.5 mm Na 2 HPO 4 and 0.25 mm NaH 2 PO 4, wit ph ¼ 7.4 at 295 K. From te 2 acid dissociation constants (pk a s) of H 3 PO 4,H 2 PO 4 and HPO 4, and te buffer ph (ph canges due to electroporesis are neglected), te 3 concentrations of H 3 PO 4, PO 4 and H + are negligible compared to 2 tose of HPO 4 and H 2 PO 4, so we ave 1.25 mm Na +, 0.5 mm 2 HPO 4, and 0.25 mm H 2 PO Soft Matter, 2010, 6, Tis journal is ª Te Royal Society of Cemistry 2010

3 Te lipopolymer equation of motion is Table 1 Summary of lipopolymer electroporesis model parameters 0 ¼ (z t + z )ee g t V b 1Ð N 0 g s n s (V u)dy (4) were g t is te lipid-tail drag coefficient. Note tat te total frictional force on te lipopolymer comprises te friction on te lipid (ydrocarbon) tail witin te bilayer and te ydrodynamic drag on its polymer cain; tese are te second and tird terms, respectively, on te rigt-and side of eqn (4). Eqn (2) (4) are easily solved using liner superposition to furnis an explicit formula for te lipopolymer electroporetic mobility m E ¼ V/E (5) Details of te analytical solution, including explicit expressions for te fluid and polymer velocities, are provided in te Appendix, and a typical set of model parameters is listed in Table 1. Note tat our calculations accommodate electrical and ydrodynamic boundary conditions at te top surface of te electroporesis cannel. However, tese boundary conditions do not influence m E in tis work, because te cannel eigt H z 500 mm is significantly greater tan L + k 1 z 20 nm. Tus, we simply demand a vanising equilibrium electrostatic potential j / 0 and vanising ydrodynamic sear stress vu/vy / 0as y / N. Because our model is quasi-steady, it is valid only wen a uniform membrane is subjected to a steady, uniform electric field. Wit lateral confinement, membranes eventually become inomogeneous, approacing an equilibrium state wit Boltzmann distributed energy were te lipopolymer diffusion flux balances electro-migration. Tis parallels te pioneering studies of Boxer and coworkers involving carged bilayers witout lipopolymers. 11,12 Te time scale to reac equilibrium is te time L c /V for lipopolymers to drift across teir lateral confinement distance L c at velocity V ¼ m E E. We typically syntesized lipid bilayers wit L c > 15 mm. However, to ensure te bilayers under observation are continuous, is is reasonable to state tat L c $ 500 mm, wic is te maximum field of view. Te accompanying equilibrium time for lipopolymers wit drift velocity V 100 nm s 1 is terefore $ s, wic is considerably longer tan te several seconds required to measure te electric-field-induced drift. Accordingly, te membranes can be reasonably considered as uniform on te measurement time scale. We also neglect te dynamics of establising a quasi-steady ydrodynamic profile witin te electroporesis cannel. Tis is te H 2 /n 0.04 s momentum diffusion time scale, were n 10 6 m 2 s 1 is te fluid (water) kinematic viscosity. Clearly, tis is muc sorter tan te 1 s imaging time, so te electro-osmotic flow can indeed be considered quasi-steady. Note tat te foregoing linear relationsip between surface potential and surface carge density breaks down at ig lipopolymer concentrations, because te accompanying carge densities produce surface potentials greater tan k B T/e. Under We solved a sub-problem were te lipopolymers are translated at a specified velocity V in te absence of an electric field (E ¼ 0), and anoter sub-problem were te lipopolymers are fixed (V ¼ 0) in te presence of an electric field. Tese solutions are superposed to satisfy te polymer equation of motion (5) and, tus, determine m E. PEG-cain molecular weigt M 2 kg mol 1 PEG-cain statistical segments per N 28 cain PEG-cain statistical segment l 0.71 nm lengt Lipid ead cross-sectional area A l 0.65 nm 2 PEG cain mean-squared end-toend R ln 1/2 z 3.76 nm distance PEG layer tickness L 2R Lipopolymer mole fraction c Lipopolymer number density or b c/a l PEG-cain grafting density Bilayer surface carge density s b z eb Lipid-tail friction coefficient g t Lipopolymer area fraction f bl 2 Hydrodynamic segment size a s Solvent dielectric constant Solvent viscosity 8.89 mpa s Absolute temperature T 295 K Electroporesis cannel (top) s t 0 surface carge density Electroporesis cannel eigt H 500 mm Electrolyte ionic strengt I 1.75 mm Electrolyte Debye lengt k nm tese conditions, we adopt te well-known Gouy-Capman equation 26 jðy ¼ 0Þ ¼ 2k BT ze asin sb ze (6) 2k33 o k B T to calculate a surface potential j(y ¼ 0) corresponding to te specified surface carge density s b ¼ z eb. Tis implicitly furnises an effective or corrected surface carge density s e ¼ 33 s k j(y ¼ 0) < s b. To test tis approximation, we solved te full model wit te non-linear Poisson Boltzmann equation numericallyx on te intervals [0, L] and [L,50L]. Boundary conditions at y ¼ 0 and L are te same as for te analytical solution, but we applied te analytical solution evaluated at y ¼ 50L to furnis a far-field boundary condition for te numerical calculations. Fig. 1 confirms tat te analytical solution based on te linearized Poisson Boltzmann equation is accurate wit lipopolymer concentrations c # 1 mol%. However, it overestimates te electro-osmotic drag at iger concentrations due to overestimating te surface potential from te specified surface carge density. Neverteless, by correcting te surface potential wit eqn (6), te analytical solution acieves satisfactory agreement wit te numerical calculations at all lipopolymer concentrations. Accordingly, we interpreted our experimental data using te analytical solution wit j(y ¼ 0) specified according to eqn (6) wit s b ¼ z eb. 2 Experimental We conducted experiments wit glass cover-slip supported bilayers doped wit various concentrations of 1,2-distearoyl-snglycero-3-pospoetanolamine-N-[poly(etylene glycol)2000- N 0 -carboxyfluorescein] (DSPE-PEG2k-CF, Avanti Polar Lipids) in 1,2-dioleoyl-sn-glycero-3-pospocoline (DOPC, Avanti x Te governing partial differential equations were solved using te Matlab function bvp4c. Tis journal is ª Te Royal Society of Cemistry 2010 Soft Matter, 2010, 6,

4 Fig. 1 (a) Teoretical calculations of lipopolymer (DSPE-PEG2k-CF) electroporetic mobility m E ¼ V/E as a function of its mole fraction c: g t ¼ D t /(k B T) wit D t ¼ 3.5 mm 2 s 1 and a s ¼ 0.74 A. Oter parameters are detailed in te text. Te dased line is te analytical solution wit te Debye H uckel approximation; te dotted line is te analytical solution corrected wit te Gouy-Capman equation; and te solid line is a numerical solution wit te non-linear Poisson Boltzmann equation. (b) Scematics of DOPE-NBD doped DOPC (top) and DSPE-PEG2k- CF doped DOPC (bottom) igligting te negative (red) carges fixed to DOPE-NBD and DSPE-PEG2k-CF, and positive (blue) counterions. DOPC lipids are zwitterions wit one positive and one negative carge (neiter sown). Polar Lipids). Bilayers were syntesized as described by Zang & Hill. 27 Briefly, a mixture of lipids containing 2 mg DOPC and a specified concentration DSPE-PEG2k-CF in cloroform was dried under a stream of nitrogen gas, deep dried under vacuum for 2 and reconstituted in buffer (10 mm pospate, 100 mm NaCl, ph 7.4) to 2 mg ml 1. Te lipid mixture was extruded first troug a 100 nm and ten a 50 nm polycarbonate membrane (Avanti Polar Lipids) to form small unilamellar vesicles tat we deposited on pre-treated cover-slips to form lipid bilayers by vesicle fusion. Te cover-slips were first boiled in a detergent solution for 30 min, rinsed excessively in reverse osmosis (RO) water, dried under a stream of nitrogen gas, and finally Pirana etced for 20 min in a solution of 3 : 1 (v/v) concentrated sulfuric acid (H 2 SO 4 ) and 30% ydrogen peroxide (H 2 O 2 ) followed by rising in RO water and drying under a stream of nitrogen gas. Fluid cannels wit dimensions 30 mm 6mm 0.5 mm were constructed from polydimetylsiloxane (PDMS, Sylgard 184, Dow Corning) using a metal mold and mounted onto a precleaned cover-slip used as te bilayer support. Two platinum (Pt) electrodes were placed in PBS buffer reservoirs at te ends of te cannel. A low ionic strengt buffer containing 0.5 mm of Na 2 HPO 4 and 0.25 mm of NaH 2 PO 4 was used during electroporesis to reduce Joule eating. Te ph is 7.4 and, recall, te ionic strengt is 1.75 mm. For an electric field strengt less tan 100 V cm 1, te temperature increase was calculated to be less tan 0.5 C min 1, and can terefore be neglected. Te electric field strengt was calculated from knowledge of te current (available from te power supply, PS325, Stanford Researc Systems), te conductivity of te diluted buffer (measured using a Zetasizer Nano Series instrument, Malvern Instruments Ltd.), and te cannel crosssectional area (estimated from te te mold used to cast te PDMS cannel). Bilayers were imaged using a Zeiss LSM 5 confocal microscope fitted wit a 63 /1.4 oil-immersion objective and a 488 nm argon ion laser operating at % intensity. Te lipopolymer drift velocity was obtained by imaging a poto-bleaced stripe 28 wit te sample carefully aligned to ensure orizontal electromigration across te field of view. Te stripe was poto-bleaced at 100% laser intensity in less tan 100 ms, and its electromigration recorded and analyzed using te Zen software (Carl Zeiss) to obtain fluorescence intensity time series. Te stripe center was accurately identified by fitting a Gaussian function to te vertically averaged fluorescence intensity line scans. Plotting te position versus time yielded a straigt line wose slope is te DSPE-PEG2k-CF lipopolymer electromigration velocity V. Control experiments were undertaken wit 0.5 mol% 1,2-dioleoyl-sn-glycero-3-pospoetanolamine-N-(7-nitro-2-1,3- benzoxadiazol-4-yl) (ammonium salt) (DOPE-NBD, Avanti Polar Lipids) in DOPC wit various concentrations of 1,2-distearoyl-sn-glycero-3-pospoetanolamine-N-[metoxy(polyetylene glycol)-2000] (ammonium salt) (DSPE-PEG2k, Avanti Polar Lipids). 3 Results and discussion 3.1 Measured lipopolymer electroporetic mobility A representative time series of bilayer images is sown in Fig. 2 (a). Lipopolymer self-diffusion broadens te vertical potobleaced stripe. 28,29 Wile te self-diffusion coefficient can be ascertained from te broadening dynamics, tis yields iger self-diffusion coefficients tan obtained from fluorescence recovery after potobleacing (FRAP) experiments wit a stationary, circular bleaced spot. Tis is because te ig illumination intensities required to accurately track te electricfield-induced drift yield (non-linear) potobleacing artifacts. Neverteless, because poto-bleacing does not break te foreaft symmetry, te observed drift velocity is independent of poto-bleacing. Representative poto-intensity profiles are sown in Fig. 2 (b). Least-squares Gaussian fits [e.g., panel (c)] accurately identify te translating mid-point, wose time dependence yields te lipopolymer electroporetic drift velocity V [panel (d)]. We selected te initial poto-bleacing widt w z 5 mm and dept to optimize te signal-to-noise ratio. Note tat te caracteristic recovery time is te w 2 /D s 30 s diffusion time, wic is considerably longer tan te several seconds required to accurately track te drift. Plotting te drift velocity V as a function of te electric field strengt E yields a straigt line wose slope is te electroporetic mobility m E ¼ V/E. In Fig. 3, for example, m E z (mms 1 )/(V cm 1 )isobtained wit c z 0.5 mol% DSPE-PEG2k-CF in DOPC. Electroporetic mobilities m E of lipopolymer DSPE-PEG2k- CF in DOPC are plotted as a function of its mole fraction c in Fig. 4 (a) (circles). Also sown are te mobilities of 0.5 mol% DOPE-NBD in DOPC wit various DSPE-PEG2k mole fractions c (squares). Te negative sign corresponds to te negative lipid valences: z ¼2 for DSPE-PEG2k-CF and z ¼1 for DSPE-PEG2k. Moreover, te monotonic decrease in te DSPE- PEG2k-CF mobility magnitude wit increasing concentration indicates tat te migration is indered, eiter because of direct 5628 Soft Matter, 2010, 6, Tis journal is ª Te Royal Society of Cemistry 2010

5 Fig. 2 Lipopolymer electroporesis in supported lipid bilayers. (a) Representative fluorescence images of DSPE-PEG2k-CF in DOPC under te influence of a orizonal electric field at times t z 0, 1.0, 4.4, 7.5, 11.1, 14.5, 17.9 s (left-to-rigt). Te scale is set by te initial widt w z 5 mm of te (vertical) poto-bleaced strip. (b) Vertically averaged poto-intensity profiles at times t z 0, 3.4, 6.5, 10.1, 13.5 and 16.9 s after potobleacing. Intensity is normalized wit te intensity immediately before potobleacing [first image in panel (a)]. (c) A representative Gaussian fit to te vertically averaged poto-intensity profile at t z 10.1 s after potobleacing. (d) An electroporetic drift velocity V z 0.12 mm s 1 is deduced from te linear dependence of position on time wit c z 0.5 mol% and E z 14 V cm 1 ). Fig. 3 Lipopolymer DSPE-PEG2k-CF electroporetic drift velocity V as a function of te electric field strengt E wit lipopolymer concentration c z 0.5 mol% in DOPC. Error bars are te standard deviation from tree measurements of V at eac value of E. Here, te electroporetic mobility m E z (mm s 1 )/(V cm 1 ). ydrodynamic and termodynamic interactions or because of coupling to te oppositely directed electro-osmotic flow. Fitting an empirical formula m E ¼ m E0 /(1 + ac) (7) to te DSPE-PEG2k-CF mobility data furnises m E0 ¼ mms 1 /(V cm 1 ) and a ¼ 44 (solid line) wit c expressed as a mole fraction. Fig. 4 (a) Electroporetic mobilities m E of lipopolymer DSPE-PEG2k- CF in DOPC as a function of te lipopolymer mole fraction c (circles). Also sown are te electroporetic mobilities of 0.5 mol% DOPE-NBD in DOPC wit DSPE-PEG2k mole fractions c (squares). (b) Self-diffusion coefficients D s of DSPE-PEG2k-CF in DOPC as a function of its mole fraction c (circles). 27 Also sown are self-diffusion coefficients of 0.5 mol% DOPE-NBD in DOPC wit DSPE-PEG2k mole fractions c (squares). Error bars are te standard deviation of two or tree data sets at eac lipopolymer concentration, and te solid lines in panels (a) and (b) are, respectively, eqn (8) and (9) fitted to te DSPE-PEG2k-CF data. Te mobilities of te 0.5 mol% DOPE-NBD control ave a muc weaker dependence on lipopolymer concentration at DSPE-PEG2k concentrations c # 4 mol%. Te sligt increase wit DSPE-PEG2k concentration at low concentrations suggests tat te DOPE-NBD mobility migt be enanced by coupling to DSPE-PEG2k disturbances witin te bilayer. Alternatively, te increase migt be due to DSPE-PEG2k in te bottom leaflet Tis journal is ª Te Royal Society of Cemistry 2010 Soft Matter, 2010, 6,

6 increasing te bilayer-support separation, tereby reducing ydrodynamic coupling to te solid support. At iger DSPE- PEG2k concentrations, te mobility decreases, as migt be expected from coupling to electro-osmotic flow driven by te increasing DSPE-PEG2k counter carge density. Interestingly, te electroporetic mobilities of 0.5 mol% DOPE-NBD and DSPE-PEG2k-CF are comparable in te limit of vanising lipopolymer concentration, even toug DSPE- PEG2k-CF bears twice te carge as DOPE-NBD. Tus, te terminal carge on DSPE-PEG2k-CF seems to ave a negligible influence on te lipopolymer mobility. Surprisingly, te DOPE- NBD and DSPE-PEG2k-CF mobilities reflect a dominant balance of te electrical force on te lipid ead (z ¼1 for bot lipids) and te drag force on te (ydrocarbon) lipid tail. Evidently, te electrical force on te polymer cain is balanced by ydrodynamic coupling to electro-osmotic flow. Self-diffusion coefficients D s of lipopolymer DSPE-PEG2k- CF in DOPC are plotted as a function of its mole fraction c in Fig. 4 (b) (circles). Tese data are taken from Zang & Hill, 27 wo attributed te indered self-diffusion coefficient at small, but finite, concentrations to soft repulsive termodynamic interactions between te PEG cains. Also sown are te selfdiffusion coefficients of 0.5 mol% DOPE-NBD in DOPC wit various DSPE-PEG2k mole fractions c (squares). As sown by Zang & Hill, 27 te DSEP-PEG2k-CF self-diffusion coefficient is well represented by te empirical formula D s ¼ D 0 /(1 + ac) (8) were D 0 z 3.36 mm 2 s 1 and a z 56 wit c expressed as a mole fraction. By te well-known Stokes Einstein relationsip, tis corresponds to a drag coefficient g ¼ k B T/D s (c) tat increases linearly wit lipopolymer concentration (see Fig. 7). Accordingly, if te electrical force is assumed constant and balanced by only frictional drag, ten te electroporetic mobility would ave te same concentration dependence as te self-diffusion coefficient. Tis is qualitatively confirmed by te DSPE-PEG2k-CF electroporetic mobilities and self-diffusion coefficients sown in Fig. 4 (a) and (b), respectively. However, we will see tat te quantitative differences can be attributed to electro-osmotic flow. Similarly to te 0.5 mol% DOPE-NBD electroporetic mobilities in panel (a), te 0.5 mol% DOPE-NBD self-diffusion coefficient as a weak dependence on DSPE- PEG2k concentration. Wile te DOPE-NBD self-diffusion coefficient increases sligtly wit DSPE-PEG2k concentration at low lipopolymer concentrations, te self-diffusion coefficient is sligtly indered at iger DSPE-PEG2k concentrations. At least qualitatively, tese variations can be attributed to te same mecanisms tat influence te electroporetic mobility. To expedite a quantitative relationsip between m E as D s, let us consider te electrical force zee balancing te drag force Vk B T/D s furnised by te te Stokes Einstein relationsip. Tis yields an electroporetic mobility m D ¼ zed s (9) k B T tat is not necessarily equal to te actual electroporetic mobility m E ¼ V/E. Indeed, Stelzle et al. 13 reported m E /m D z 0.6 for DOPE-NBD in te absence of lipopolymer, attributing te deviation from unity to electro-osmotic flow. Similarly, McLauglin & Poo 4 found tat concanavalin-a receptors on embryonic muscle cells tat normally accumulate at te catode side of te cell can be induced to accumulate eiter at te anode side by reducing te negative carge (decreasing te electroporetic force) or at te catode side by adding negative carge to te cell surface (increasing te electroosmotic force). Since te bulk lipids (DOPC) in our SLBs ave zero net carge, te negatively carged lipopolymers (DSPE-PEG2k-CF and DSPE- PEG2k) endow te layers wit a surface carge density s < 0 wose magnitude is proportional to te respective lipopolymer concentration. Terefore, te accompanying z-potentials are negative, and te accompanying electroosmotic flow slows te electroporetic drift of DSPE-PEG2k-CF, DSPE-PEG2k, and 0.5 mol% DOPE-NBD. Te DSPE-PEG2k-CF electroporetic mobility m E (circles) is plotted wit m D (squares) in Fig. 5 (a). Also sown are m E (uptriangles) and m D (rigt-triangles) for 0.5 mol% DOPE-NBD in DOPC wit various DSPE-PEG2k mole fractions c. Wile te DSPE-PEG2k-CF electroporetic mobilities m E are ostensibly lower tan teir respective m D, te 0.5 mol% DOPE-NBD mobilities m E are somewat iger teir respective m D. Te ratios m E /m D are plotted in Fig. 5 (b), were te significant deviations from unity indicate tat electro-osmosis, wic is absent in te specification of m D, plays an important role. Te ratio m E /m D for DSPE-PEG2k-CF is approximately independent of concentration, suggesting tat te dominant influence of electro-osmotic flow is independent of lipid interactions. Tus, as expected, te DSPE-PEG2k-CF mobility m E is indered by electro-osmotic flow. Te nominal value m E /m D 0.7 confirms inferences above tat te electrical force on te free end of te PEG cain is balanced by ydrodynamic coupling of te PEG cains to electro-osmotic flow driven by te mobile counter carges. If tis coupling were perfect, i.e., te electrical force were exactly balanced by te ydrodynamic retardation force, ten te effective DSPE-PEG2k-CF valence would be reduced to z ¼1; rater, te effective valence z z 1.4 suggests an 60% reduction in te apparent electrical force. Fig. 5 Interpreting electroporetic mobilities wit knowledge of selfdiffusion coefficient. (a) Actual electroporetic mobilities m E (circles) and electroporetic mobilities m D (squares) of DSPE-PEG2k-CF in DOPC wit DSPE-PEG2k-CF mole fractions c. Also sown are m E (up-triangles) and m D (rigt-triangles) of 0.5 mol% DOPE-NBD in DOPC wit DSPE-PEG2k mole fractions c. (b) m E /m D for DSPE-PEG2k-CF (circles) and 0.5 mol% DOPE-NBD (squares) in DOPC wit lipopolymer mole fractions c. Lines are to guide te eye Soft Matter, 2010, 6, Tis journal is ª Te Royal Society of Cemistry 2010

7 In striking contrast, m E /m D 1.2 > 1 for 0.5 mol% DOPE-NBD, so electrical influences evidently enance te electroporetic mobility. Tis migt be due to te accompanying DSPE-PEG2k electro-migration. Since te DOPE-NBD ead presents a muc smaller ydrodynamic size tan te PEG2k cains on DSPE-PEG2k, it is likely to be more susceptible to ydrodynamic disturbances witin te bilayer rater tan above it. Tus, te collective electro-migration of DSPE-PEG2k tails witin te leaflet migt enance DOPE-NBD electro-migration. Note tat tis can only occur if tere exists a positive correlation between DOPE-NBD and DSPE-PEG2k concentration fluctuations, tereby demanding an attractive contribution to te interaction between DOPE-NBD and DSPE-PEG2k. Te electrostatic interaction is obviously repulsive over separation distances on te order of te Bjerrum lengt ( 0.7 nm), so an attractive interaction would ave to arise from te PEG2k interaction wit NBD. Alternatively, suc correlations migt be facilitated by local fluctuations of bilayer curvature. 3.2 Teoretical interpretation Te teoretical model, evaluated wit a constant lipid-tail drag coefficient, was demonstrated in Fig. 1 to yield a lipopolymer electroporetic mobility tat decreases monotonically wit increasing lipopolymer concentration. Tis is clearly in good qualitative agreement wit te foregoing experiments wit lipopolymer DSPE-PEG2k-CF in DOPC. Our model assigns a lipid-tail drag coefficient and separately calculates te drag force on te PEG cains via a Brinkman (continuum) model of te polymer layer wit uniform carge density and ydrodynamic permeability. Even wit tese approximations, tere are several model parameters tat must be specified to expedite a quantitative comparison of te teory and experiment. Tese parameters include te lipid-tail drag coefficient g t, polymer layer tickness L, and ydrodynamic permeability l B2, all of wic may vary wit lipopolymer concentration. To assess te layer tickness and density, we performed selfconsistent mean-field calculations furnising te segment-density distributions sown in Fig. 6. Even at te igest lipopolymer concentration c z 5 mol%, te layers ave Gaussian profiles rater tan te parabolic profiles tat are caracteristic of bruses wen te area fraction f b(2r) 2 $ 1. Tus, te caracteristic layer tickness is independent of te lipopolymer concentration, and is well approximated by twice te unperturbed root-mean-squared end-to-end distance, i.e., L ¼ 2R ¼ 2lN 1/2 z 7.52 nm, as adopted in section 1. Accordingly, te uniform layers in our electrokinetic model ave a segment density n s ¼ bn/l ¼ cn/(la l ). For PEG2k, te number of statistical segments N ¼ 28, and te segment lengt l ¼ 0.71 nm. 25 Having specified te polymer layer tickness, we compute te ydrodynamic permeability by assigning a Stokes ydrodynamic radius a s to eac statistical segment. Tus, te ydrodynamic 2 drag force per unit volume, 6pa s n s [u(y) V] l B [u(y) V], yields a ydrodynamic permeability (square of te Brinkman screening lengt) l 2 B ¼ 1 L ¼ pa s n s 6pa s bn ¼ l (10) 3pa s bn 1=2 Accordingly, by specifying l, N, and b ¼ c/a l, we consider te Stokes radius a s as a fitting parameter. Fig. 6 Scaled polymer segment density distributions n s L/(Nb) for various scaled polymer grafting densities (area fractions) f ¼ bl 2, were L ¼ 2R ¼ 2lN 1/2, according to self-consistent mean field teory wit Flory parameter c ¼ Note tat te dimensional segment density n s is scaled wit Nb/L, te value for a uniform profile wit tickness L ¼ 2R ¼ 2lN 1/2, and distance from te (flat) grafting surface y is scaled wit L. Lines identify scaled grafting densities (area fraction) f ¼ bl 2 z 0.43 (solid), 1.74 (dased), 4.24 (das-dotted), and 6.95 (dotted). Now consider te lipid tail drag coefficient g t. Again, we turn to te Stokes Einstein relationsip. However, because our electrokinetic model accounts for frictional forces on te polymer, te lipid-tail drag coefficient must be ascertained from te DOPE-NBD self-diffusion coefficient, since te drag force on tese lipids is predominantly from te ydrocarbon tail. Neverteless, even te DOPE-NBD drag coefficient may be influenced by bilayer-polymer ydrodynamic coupling. We terefore assess te lipid-tail drag coefficient from te Stokes Einstein interpretation of te 0.5 mol% DOPE-NBD self-diffusion coefficient in te presence of DSPE-PEG2k wit mole fraction c. Tis data, wic is presented in Fig. 4, confirms tat te drag coefficient g t ¼ k B T/D s is a muc weaker function of te lipopolymer concentration tan te drag coefficient of te lipopolymer itself. Fig. 7 sows te drag coefficients of DSPE-PEG2k-CF and 0.5 mol% DOPE-NBD as a function of te lipopolymer concentration, eac normalized by its respective value g 0 at infinite lipopolymer dilution. From te empirical form of te selfdiffusion coefficient proposed by Zang & Hill, 27 te drag coefficient increases linearly wit lipopolymer concentration, wit a slope tat reflects termodynamic and ydrodynamic interactions. For DSPE-PEG2k-CF, te self-diffusion coefficient correlated by Zang & Hill 27 gives g/g 0 z 1+56c, were c te DSPE- PEG2k-CF mole fraction. Note tat an excellent fit of our electrokinetic model for te DSPE-PEG2k-CF electroporetic mobility is furnised by a concentration dependent lipid-tail drag coefficient g t /g 0 z 1+5c (11) were g 0 ¼ k B T/D 0 z kg s 1 and c is expressed as a mole fraction. Here, D 0 ¼ 3.5 mm 2 s 1 is a reasonable estimate of DOPE-NBD and DSPE-PEG2k-CF self-diffusion coefficients at infinite lipopolymer dilution. Recall, lipopolymer Tis journal is ª Te Royal Society of Cemistry 2010 Soft Matter, 2010, 6,

8 Fig. 7 Scaled drag coefficients g/g 0 ¼ D 0 /D s of DSPE-PEG2k-CF in DOPC wit DSPE-PEG2k-CF mole fractions c (circles), and 0.5 mol% DOPE-NBD in DOPC wit DSPE-PEG2k mole fractions c (squares). Te solid line is g/g 0 ¼ 1+56cobtained directly from DSPE-PEG2k-CF self-diffusion coefficients, 27 and te dased line is g/g 0 ¼ 1+5c obtained indirectly from te DSPE-PEG2k-CF electroporetic mobilities interpreted wit te continuum electroporesis model. self-diffusion coefficients at infinite dilution are practically independent of te grafted PEG2k cains. 27 As sown in Fig. 7, eqn (11) (dased line) is compatible wit te lipopolymer concentration dependence of te 0.5 mol% DOPE-NBD friction coefficient (squares) inferred from te self-diffusion coefficient in SLBs wit varying concentrations of DSPE-PEG2k. Fig. 8 Te electroporetic mobility m E of lipopolymer DSPE-PEG2k- CF as a function of te scaled lipopolymer concentration f ¼ cl 2 /A l ¼ bl 2. Circles are te experimental data; te dotted line is te teory wit PEG-cain statistical-segment Stokes radius a s ¼ 0.86 A and a constant lipid-tail drag coefficient g t ¼ k B T/D 0 wit D 0 ¼ 3.5 mm 2 s 1 ; te solid line is te teory wit a s ¼ 0.74 A and a lipid-tail drag coefficient g t ¼ (1 + 5c)k B T/D 0 wit D 0 ¼ 3.5 mm 2 s 1 ; and te upper dased line is te electroporetic mobility m D predicted from te concentration dependence of te lipopolymer self-diffusion coefficient. 27 Te das-dotted line is a teoretical prediction of te electroporetic mobility of DSPE-PEG2k in DOPC, and te lower dased line is te electroporetic mobility m D predicted from te concentration dependence of te DSPE-PEG2k-CF self-diffusion coefficient. 27 Fig. 8 compares te DSPE-PEG2k-CF electroporetic mobility predicted by our electroporesis model (solid and dotted lines) wit te experimental data (circles). Te parameters adopted for tese calculations are listed in Table 1. Te teoretical calculations sown as te dotted line ave a constant lipid-tail drag coefficient g t ¼ k B T/D 0 wit D 0 ¼ 3.5 mm 2 s 1, and a s ¼ 0.86 A as a single fitting parameter. Tere is a good correspondence at low lipopolymer concentrations, but an erroneous increase at iger concentrations suggests tat te lipid-tail drag coefficient actually increases wit lipopolymer concentration, as corroborated by Fig. 7. Accordingly, calculations wit a lipid-tail drag coefficient g t ¼ (1 + 5c)k B T/D 0 wit D 0 ¼ 3.5 mm 2 s 1, and a s ¼ 0.74 A te single fitting parameter, produce an excellent fit over te entire range of lipopolymer concentrations. For convenient reference, Fig. 8 also sows te electroporetic mobility m D predicted using te self-diffusion coefficient reported by Zang & Hill 27 (dased line). As discussed above, tis does not account for electro-osmotic flow, and, tus, overpredicts te mobility at low lipopolymer concentrations, and under-predicts te mobility at ig concentrations. In fact, te teoretically predicted mobility m E at ig concentrations appears to approac a finite value at ig lipopolymer concentrations, wereas te electroporetic mobility m D vanises. It is not clear weter te correlation of Zang and Hill can be extrapolated beyond c z 5 mol%, since te PEG-cains are expected to adopt increasingly extended, brus-like configurations at ig enoug lipopolymer concentrations, wic may furnis a non-linear dependence of te drag coefficient on lipopolymer concentration. Neverteless, te finite lipopolymer electroporetic mobility at ig concentrations can be attributed to te linear increase of te electrical and ydrodynamic forces on te polymer layer wit concentration. Recall, wit fixed layer tickness, te electrical carge density and segment concentration are eac proportional to te lipopolymer concentration (in te mean-field approximation). On te oter and, te electroporetic mobility m D comes from balancing te constant electrical force per lipopolymer wit a lipopolymer drag coefficient tat increases approximately linearly wit lipopolymer concentration. 27 Tus, our teoretical model and experiments identify a fundamental difference between te drag forces operating on a lipid as predicted by te Stokes Einstein interpretation of te self-diffusion coefficient and te drag forces operating during collective electro-migration under te influence of electrical forces and electro-osmotic flow. Finally, we ave plotted in Fig. 8 teoretical predictions of te DSPE-PEG2k electroporetic mobility in DOPC (das-dotted line). Tese calculations were undertaken wit te same parameters as for DSPE-PEG2k-CF, but wit zero carge at te free PEG end. Here, te electro-osmotic flow is driven entirely by te counter carge of te lipid-peg junction. As expected, te mobility m E is muc closer to te electroporetic mobility m D predicted from te lipid valence z ¼1and friction coefficient from te self-diffusion coefficient. 27 Interestingly, suc a close correspondence prevails even toug te calculation of m E does not draw upon explicit empirical knowledge of ow te selfdiffusion coefficient is modulated by te grafted PEG cain. Surprisingly, lipopolymer self-diffusion coefficients, wic are successfully calculated on te basis of termodynamics in te dilute limit, are reasonably well approximated by te drag 5632 Soft Matter, 2010, 6, Tis journal is ª Te Royal Society of Cemistry 2010

9 coefficient for collective translation, as furnised by te Brinkman model in our study. 4 Conclusions We developed a continuum electroporesis model for lipopolymers in supported lipid bilayer membranes, and used tis model to interpret experimental measurements of te electroporetic mobility of lipopolymer DSPE-PEG2k-CF in DOPC. Similarly to our earlier report of te concentration dependence of te self-diffusion coefficient, te experiments revealed an electroporetic mobility tat decreases monotonically wit increasing concentration. To a first (qualitative) approximation, te electroporetic mobility can be estimated by balancing te electrical force zee wit te ydrodynamic drag force Vk B T/D s furnised by te Stokes Einstein interpretation of te selfdiffusion coefficient D s (c). However, quantitative comparison of teory and experiment at finite lipopolymer concentrations demands tat electroosmotic flow and ydrodynamic coupling are accounted for. We obtained excellent agreement between te continuum electroporesis model and experiments by calculating te ydrodynamic drag on te polymer cains according to a Brinkman model. Tis accounts for polymer layer tickness and polymersegment density, and allows te lipid-tail drag coefficient to increase linearly wit DSPE-PEG2k-CF concentration, i.e., g t ¼ (1 + 5c)k B T/D s, were D 0 ¼ 3.5 mm 2 s 1 and c is te lipopolymer mole fraction. Finally, we complemented lipopolymer electroporesis data wit control experiments measuring te self-diffusion coefficients and electroporetic mobilities of DOPE-NBD (a non-pegylated, carged, fluorescent lipid) at low concentration (0.5 mol%) in DOPC bilayers wit varying lipopolymer DSPE-PEG2k concentrations. Tese data support several interpretations of te lipopolymer experiments and modeling, and reveal many weaker and muc more subtle interactions tat prevail in tese intriguing quasi-two-dimensional fluids. A Analytical solution of te electroporesis model Te electroporesis cannel comprises a domain inside te polymer layer, extending from te bilayer surface at y ¼ 0 to te polymerlayer edge at y ¼ L; and a domain outside te polymer layer, extending from y ¼ L to te cannel top surface at y ¼ H. Inte fluid domain, te polymer segment density n s ¼ 0. We neglect te finite bilayer tickness, because it is small compared to te polymer layer tickness L 10 nm and cannel eigt H 500 mm. As detailed in te main text, tree coupled equations must be solved: d 2 j dy 2 k2 j ¼ r p 33 o (12) 0 ¼ d2 u dy 2 þ r tðyþe g s n s ðu VÞ (13) 0 ¼ (z t + z )ee g t V b 1Ð N 0 g s n s (V u)dy (14) Te general solution of eqn (12) in te bilayer domain (wit r p ¼ z be/l) is j ¼ C 1 e ky þ C 2 e ky þ z eb (15) 33 o Lk 2 and in te fluid domain (wit r p ¼ 0) j ¼ C 3 e ky + C 4 e ky (16) Te constants C 1 C 4 are fixed by te following boundary conditions. At y ¼ 0, dj dy j y¼0 ¼s b (17) 33 o were s b is te bilayer surface carge density. At y ¼ H, dj dy j y¼h ¼s t (18) 33 o were s t is te electroporesis cannel surface carge density. Finally, at y ¼ L, and j(l + d) ¼ j(l d) (19) dj dy j y¼lþd ¼ dj dy j y¼ld (20) as d / 0. Tese give C 3 ¼ 1 1 z ebe kl 33 o k e 2kH 1 2kL z ebe kl 2kL þ s s t b e kh e kh C 1 ¼ C 3 z ebe kl 33 o 2k 2 L C 2 ¼ C 1 þ s b 33 o k (21) (22) (23) C 4 ¼ C 2 þ z ebe kl (24) 33 o 2k 2 L Next, te solutions of eqn (14) and (15) are obtained from te linear superposition of te two simpler sub-problems detailed below. In te E-problem, an electric field wit strengt E s 0is applied wit fixed lipopolymers, i.e., V ¼ 0. Accordingly, 0 ¼ d2 u E dy þ r pðyþe g 2 s n s u E (25) giving a force on eac lipopolymer (linear in E) f E ¼ (z t + z )ee + b 1Ð N 0 g s n s u E dy (26) In te V-problem, lipopolymers are translated wit a velocity V s 0 in te absence of an electric field, i.e., E ¼ 0. Accordingly, Tis journal is ª Te Royal Society of Cemistry 2010 Soft Matter, 2010, 6,

10 0 ¼ d2 u V dy 2 g sn s u V V (27) A 4 ¼ 33 oc 3 e kh 33 oc 4 e kh (41) giving a force on eac lipopolymer (linear in V) Te solution of eqn (27) in te polymer layer is f V ¼g t V b 1Ð N 0 g s n s (V u V )dy (28) Finally, superposing tese solutions to satisfy f V + f E ¼ 0 (29) gives te lipopolymer electroporetic mobility zt þ z Ð e þ b 1 N g 0 s n s ðu E =EÞdy m E ¼ V=E ¼ g t þ b Ð 1 N g 0 s n s ð1 u V =VÞdy Te solution of eqn (25) in te polymer layer is u E E ¼C 5e ly þ C 6 e ly 33 o þ k 2 l 2 C 1e ky þ k2 33 o k 2 l 2 C 2e ky þ eb l 2 L were te reciprocal Brinkman permeability Outside te polymer layer, k2 (30) (31) l 2 ¼ g s n s / (32) u E ¼ C 7 y þ C 8 þ 33 o C 3e ky þ 33 o C 4e ky (33) Te constants C 5 C 8 are specified by continuity of u E and its gradient (sear stress) at y ¼ L, and u E ¼ 0 (no slip) at y ¼ 0 and H. Te results are u V V ¼ C 9e ly þ C 10 e ly þ 1 (42) and outside te polymer layer u V V ¼ C 11y þ C 12 (43) Te constants C 9 C 12 are specified from te continuity of u V and its gradient (sear stress) at y ¼ L, and u V ¼ 0 (no slip) at y ¼ 0 and H. Te results are 1 e ll ð1 þ lh llþ C 10 ¼ (44) e ll ð1 ll þ lhþe ll ð1 þ ll lhþ C 9 ¼C 10 1 (45) C 11 ¼lC 10 (e ll + e ll ) le ll (46) Te electroporetic mobility becomes were C 12 ¼C 11 H (47) m E ¼ V E ¼ zt þ z e þ b 1 g s n s D E g t þ b 1 g s n s D V (48) C 6 ¼ ell ðlh ll þ 1ÞA 1 þ e kl ðkl kh 1ÞA 2 þ e kl ðkh kl 1ÞA 3 þ A 4 e=ðng s Þ e ll ðll lh 1Þ þ e ll ðll lh þ 1Þ (34) were C 5 ¼(C 6 + A 1 ) (35) C 7 ¼lC 6 (e ll + e ll ) le ll A 1 + ke kl A 2 + ke kl A 3 (36) C 8 ¼C 7 H + A 4 (37) A 1 ¼ 33 ok 2 C 1 k2 l 2 þ 33 ok 2 C 2 k2 l 2 þ e (38) Ng s and D E ¼ C 5 l e ll 1 þ C 6 1 e ll þ 33 okc 1 l k2 l 2 e kl 1 þ 33 okc 2 k2 l 2 1 e kl þ el (49) Ng s D V ¼ C 9 l 1 e ll þ C 10 e ll 1 (50) l A 2 ¼ 33 ok 2 C 1 k2 l 2 33 oc 3 A 3 ¼ 33 ok 2 C 2 k2 l 2 33 oc 4 (39) (40) Acknowledgements R.J.H. gratefully acknowledges support from te Natural Sciences and Engineering Researc Council of Canada and te Canada Researc Cairs program; H.-Y.Z. tanks McGill University and te Faculty of Engineering, McGill University, for generous financial support troug a Ricard H. Tomlinson 5634 Soft Matter, 2010, 6, Tis journal is ª Te Royal Society of Cemistry 2010

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