pubs.acs.org/Langmuir © 2009 American Chemical Society
Mechanical Behavior of a Supported Lipid Bilayer under External Shear Forces :: :: :: Peter Jonsson,† Jason P. Beech,† Jonas O. Tegenfeldt,† and Fredrik Hook*,†,‡ †
Division of Solid State Physics, Lund University, SE-22100 Lund, Sweden, and ‡Department of Applied Physics, Chalmers University of Technology, SE-41296 Gothenburg, Sweden Received December 22, 2008. Revised Manuscript Received March 24, 2009
Shear forces from a pressure-driven bulk flow in a microfluidic channel can be used to induce and control the motion of a supported lipid bilayer (SLB) formed on the walls of the channel. We here present a theoretical model that relates the experimentally observed drift velocities of an egg yolk phosphatidylcholine (egg PC) SLB to the hydrodynamic drag force from the bulk flow, the intermonolayer friction coefficient, b, of the bilayer, and the friction coefficient, bls, between the lower leaflet of the bilayer and the supporting substrate. The drift velocity and diffusivity of the lipids in the SLB were obtained by photobleaching a delimited area of fluorescently labeled lipids and subsequently monitoring the recovery and convective motion of the bleached spot. A striking observation was that the drift velocity of the lipids was observed to be nearly 6 orders of magnitude smaller than the bulk velocity at the center of the channel. This predicts a value for bls that is at least 25 times as high as predicted by the traditional model with the SLB and the support spaced by a homogeneous 1 nm thick film of water. In addition, the intermonolayer friction coefficient was estimated to 2 107 Pa s/m, a value that increased after addition of glycerol to the bulk solution. This increase was accompanied by an equal decrease in the lipid diffusivity, with both observations indicating an increased viscous drag within the bilayer when glycerol was added to the bulk solution.
Introduction The lipid bilayer surrounding living cells can be described as a two-dimensional liquid within which the dominating mode of lateral transport is diffusion.1,2 Supported lipid bilayers (SLBs) are today one of the most common model systems used for investigating the biophysical properties of lipid bilayers.3,4 In a recent publication, we introduced a method where all molecules in an SLB could be transported in a controllable manner.5 This was accomplished by forming the SLB on the walls of a microfluidic channel and having a pressure-driven bulk flow in the channel. The bulk flow will exert a shear force on the SLB which results in a controlled motion of the SLB and its constituents. An interesting observation was that the lower leaflet of the SLB was moving markedly slower than the upper leaflet, in these experiments, indicating a high frictional coupling between the SLB and the supporting glass substrate. In this work, we present a theoretical model that relates the drift velocity of the SLB to the strength of the hydrodynamic drag force acting on the bilayer, the intermonolayer friction coefficient, b,6 and the frictional coupling between the SLB and the supporting substrate.7,8 This model is then used to estimate b and the friction coefficient, bls, between the bilayer and the support, from experimental data on the drift velocity of the SLB. Thus, the work presented here illustrates a straightforward method to determine different mechanical *To whom correspondence should be addressed. E-mail: fredrik.hook@ chalmers.se. (1) Singer, S. J.; Nicolson, G. L. Science 1972, 175, 720–731. (2) Saffman, P. G.; Delbruck, M. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 3111– 3113. (3) Tanaka, M.; Sackmann, E. Nature (London) 2005, 437, 656–663. (4) Sackmann, E. Science 1996, 271, 43–48. :: :: :: (5) Jonsson, P.; Beech, J. P.; Tegenfeldt, J. O.; Hook, F. J. Am. Chem. Soc. 2009, 131, 5294–5297. (6) den Otter, W. K.; Shkulipa, S. A. Biophys. J. 2007, 93, 423–433. (7) Evans, E.; Sackmann, E. J. Fluid Mech. 1988, 194, 553–561. (8) Merkel, R.; Sackmann, E.; Evans, E. J. Phys. (Paris) 1989, 50, 1535–1555.
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parameters of SLBs, as, for example, the intermonolayer friction coefficient, parameters that previously have relied on complex interpretations of the analyzed data, resulting in a wide range of estimated values when using different methods.9 Experiments were made using an egg yolk phosphatidylcholine (egg PC) SLB that was formed inside a microfluidic channel by vesicle adsorption and subsequent rupture.10 To visualize the lipids in the SLB, a fraction of them (1 wt %), was exchanged with the fluorescently labeled lipid rhodamine-DHPE (Lissamine rhodamine B 1,2-dihexadecanoyl-sn-glycero-3-phosphatidyl-ethanolamine). Rhodamine-DHPE was chosen as the fluorescent lipid probe, since it was previously shown to be essentially excluded from the lower leaflet of the SLB in similar studies.5 Thus, only a single moving fraction of fluorescent lipids, moving at the velocity of the lipids in the upper leaflet of the SLB, will be present in the experiments. This simplifies the analysis of the drift velocity and makes it possible to simultaneously monitor the diffusivity in the SLB at various bulk flow rates.
Theory The Drift Velocity of an SLB under Shear Forces. The flow of a bulk solution through the microfluidic channel will exert a drag force on the SLB. This force will lead to a collective movement of the bilayer with different drift velocity for the upper and lower leaflet of the bilayer. The viscous drag force exerted on the top of the bilayer, by the pressure-driven bulk flow in the channel, σub, can be estimated by solving Navier-Stokes equations for the flow in a rectangular duct with a no-slip boundary condition at the walls of the duct and at the interface between the bulk solution and the upper leaflet of the bilayer.6,11 Since the (9) Shkulipa, S. A.; den Otter, W. K.; Briels, W. J. Biophys. J. 2005, 89, 823–829. (10) Brian, A. A.; McConnell, H. M. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 6159–6163. (11) Cross, B.; Steinberger, A.; Cottin-Bizonne, C.; Rieu, J. P.; Charlaix, E. Europhys. Lett. 2006, 73, 390–395.
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thickness of the supported bilayer (∼5 nm) is very small, compared with the dimensions of the microfluidic channel (∼100 μm), and since the velocity of the SLB is many orders of magnitude lower than the flow velocity of the bulk liquid in the microfluidic channel (see below), it is reasonable to assume that the velocity profile of the bulk flow in the channel will not be noticeably influenced by the presence of the SLB at the surface of the channel, except of course in close proximity to the SLB. The following approximate expression (valid to within 0.2% under the current experimental conditions12) can be derived for the bulk flow velocity, vf, in the microfluidic channel depicted in Figure 1a ¥ X 48Q 1 coshðnπz=hÞ 1 sinðnπy=hÞ νf ðy, zÞ≈ hðw -0:630hÞπ3 n, odd n3 coshðnπw=2hÞ ð1Þ where Q is the flow rate, h is the height, and w is the width of the duct.12 The shear force per unit area exerted on the top of the bilayer by the bulk flow, σub, corresponds to the force per area at the floor of the channel at y = 0. Thus, from the definition of vf in eq 1, it follows that σ ub ðzÞ ¼ 2 3 ¥ X Dvf 6ηQ 8 1 coshðnπz=hÞ 41 5 ð2Þ η ≈ 2 π2 n, odd n2 coshðnπw=2hÞ Dy y ¼0 h ðw -0:630hÞ
Figure 1. (a) Schematic illustration of the microfluidic channel, with the bulk flow along the x-direction. (b) Plot of σub versus z at the bottom (y = 0) of the channel using the expression in eq 2, with h = 103 μm, w = 200 μm, η = 0.95 mPa s, and Q = 400 μL/min.
where η is the viscosity of the bulk liquid. From eq 2, it is clear that the shear force will vary in a characteristic way as a function of the distance to the center of the channel (see also Figure 1b). To describe the velocity within the SLB, each monolayer is modeled as an incompressible fluid with the surface viscosity ηm.7 If all movement out of the plane of the SLB is neglected, then each leaflet can be described using the two-dimensional NavierStokes equations with the external forces at the interface of each monolayer presented in Figure 2 -
DΠu ðxÞ D2 vu ðzÞ þ ηm ¼ bðvu ðzÞ -v1 ðzÞÞ -σ ub ðzÞ Dx Dz2
DΠ1 ðxÞ D2 v1 ðzÞ þ ηm ¼ b1s v1 ðzÞ -bðvu ðzÞ -v1 ðzÞÞ Dx Dz2
ð3aÞ ð3bÞ
where Πu/l is the surface pressure and vu/l is the velocity of the upper/lower leaflet.7,13,14 The different terms in eq 3 originate from (1) A frictional force between the bilayer and the glass surface: σ1s ¼ -b1s v1 where bls is the friction coefficient between the lower leaflet of the bilayer and the substrate and vl is the velocity of the lower leaflet.7 (2) Shear forces within each leaflet of the SLB: D2 vu=1 ðσ m Þu=1 ¼ ηm Dz2 However, an order of magnitude estimation of σm yields that this force can be neglected under the current experimental situation (see the Appendix). (12) Bruus, H. Theoretical Microfluidics; Oxford University Press: Oxford, U.K., 2008; p 48-51. (13) Stone, H. A. Phys. Fluids 1995, 7, 2931–2937. (14) Saffman, P. G. J. Fluid Mech. 1976, 73, 593–602.
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Figure 2. Schematic illustration of the SLB and the forces acting at the interfaces of each monolayer.
(3)
Gradients of the surface pressure in the upper and lower leaflet of the SLB:13 ðσΠ Þu=1 ¼ -3Πu=1
(4)
These forces can be strong near the front of a moving bilayer, due to the forces arising at the intersection of the bilayer-glass-water phase. It is this force that is responsible for the self-spreading of an SLB when no bulk flow is present.15,16 However, when moving the SLB in a channel which is already covered with a bilayer, the magnitude of these forces can be shown to be negligible under the current experimental conditions (see the Appendix). A term accounting for the frictional coupling between the two leaflets: σ ul ¼ ( bðvu -v1 Þ where b is the intermonolayer friction coefficient.6
(15) Nissen, J.; Gritsch, S.; Wiegand, G.; Radler, J. O. Eur. Phys. J. B 1999, 10, 335–344. :: (16) Radler, J.; Strey, H.; Sackmann, E. Langmuir 1995, 11, 4539–4548.
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If the surface pressure terms and the terms due to the internal shear forces are neglected in eq 3, the steady state velocity of the upper and lower leaflet of the SLB can be approximated as 1 σ ub ðzÞ b1s 1 1 þ σ ub ðzÞ vu ðzÞ≈ b1s b v1 ðzÞ≈
ð4aÞ ð4bÞ
The proportionality constant bls, that is, the friction coefficient between the lower leaflet and the supporting substrate, depends on the interactions between the lipids in the lower leaflet and the glass surface as well as the properties of the solute in this region. This frictional force may arise due to (1) A drag force from a thin viscous layer between the bilayer and the supporting glass surface, yielding7,8,15,16 b1s ¼ η=hW ð5aÞ The parameter hw is the thickness, and η is the viscosity of the liquid film between the lower leaflet and the substrate. (2) Frictional forces within the SLB due to local pinning sites where the lipids are immobilized.15,16 The frictional coefficient can in this case be written as b1s ¼
ckB T D
ð5bÞ
where c is the number of immobile sites per area, kB is Boltzmann’s constant, T is the temperature, and D is the diffusion coefficient of the lipids in the monolayer.
Materials and Methods Fabrication of Microfluidic Channels. Microfluidic channels made of polydimethylsiloxane (PDMS) were created by replica molding.17 A master made of SU8 (MicroChem, Newton, MA), defining the channel, was made with conventional optical lithography, after which the master was treated with an antiadhesive layer of tridecafluoro-(1,1,2,2)-tetrahydrooctyl-trichlorosilane (CAS [78560-45-9], Sigma-Aldrich, St. Louis, MO) to facilitate demolding. The microfluidic channel is schematically depicted in Figure 3. The average height of the channel was measured with a profilometer (Veeco Dektak 6M, Vecco Instruments Inc., Tucson, AZ) and found to be 103 μm, with a measured variation of less than 1% along the length of the channel. There is a 3 3 mm connection area, with periodically spaced pillars to prevent the roof from collapsing, at the beginning and end of the channel. These areas also act as lipid reservoirs. Access holes were made to the inlet and outlet connection areas with a 1 mm hollow steel tube. Finally, two silicone tubes (inner diameter 1.5 mm) were glued to the access holes using silicone adhesive (Elastosil AO7, RTV-1 silicone rubber, Wacker Silicones, Munich, Germany) constituting the inlet and outlet of the device. The PDMS replicas were made of a mixture of 10:1 Sylgard 184 and curing agent (Dow Corning, Midland, MI), which was allowed to cure for at least 1 h at 80 C. The glass slides (0.13:: 0.16 mm in thickness, Menzel-Glaser, Braunschweig, Germany) were first cleaned with piranha solution (3:1 mixture of concentrated sulfuric acid and 30% hydrogen peroxide) (VWR International, Stockholm, Sweden) for 15 min and then thoroughly (17) McDonald, J. C.; Duffy, D. C.; Anderson, J. R.; Chiu, D. T.; Wu, H. K.; Schueller, O. J. A.; Whitesides, G. M. Electrophoresis 2000, 21, 27–40.
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Figure 3. (a) Schematic illustration of the microfluidic setup. (b) Dimensions of the microfluidic channel, as seen from above. rinsed with deionized Milli-Q water (Millipore, Billerica, MA) and dried at 150 C for 30 min. To obtain a strong seal between the PDMS and the glass slide, the contacting surfaces were first treated with oxygen plasma (Plasma Preen II-862, Plasmatic Systems, Inc., North Brunswick, NJ) before bonding the PDMS to the glass. The channels were filled with Milli-Q water (through capillary forces) immediately after bonding, to ensure that the surfaces in the microfluidic channel remained hydrophilic. Vesicle Preparation. Vesicles were prepared according to :: the procedure described by Jonsson et al.18 Lipid vesicles of egg yolk phosphatidylcholine (egg PC) from Avanti Polar Lipids (Alabaster, AL) were prepared by extrusion first through a 100 nm membrane and then through a 30 nm membrane (Whatman, Maidstone, U.K.) using an Avanti miniextruder (Avanti Polar Lipids). In addition to egg PC, the vesicles had 1 wt % Lissamine rhodamine B 1,2-dihexadecanoyl-sn-glycero-3-phosphatidylethanolamine (rhodamine-DHPE) from Invitrogen (Carlsbad, CA) as the fluorescent probe. The buffer solution used to prepare the vesicles was a mixture of 100 mM NaCl, 10 mM tris[hydroxymethyl]aminomethane (TRIS), and 1 mM ethylenediaminetetraacetic acid disodium salt dihydrate (EDTA) from Sigma-Aldrich, at a pH of 8.0. The lipid vesicles were diluted with the buffer solution to a total lipid concentration of 100 ng/μL before each experiment. To study the effect of the viscosity of the bulk solution, glycerol (BDH Laboratory Supplies, Poole, England) was added at different weight fractions (25, 35.8, and 42.5 wt %) to the vesicle suspension and the buffer solution to change the bulk viscosity, η, to 2.0, 3.0, and 4.0 mPa s, respectively.19 A bulk viscosity of 0.95 mPa s, corresponding to no glycerol added, is assumed for the bulk solution in the experiments unless otherwise stated. Bilayer Formation in the Microfluidic Channel. The microfluidic device was glued to a microscope slide with a hole underneath the channel, to ensure stability without compromising the resolution of the images, using a UV optical adhesive (Norland Products Inc., Carnbury, NJ). To access the device, poly (tetrafluoroethylene) (PTFE) tubing (1/1600 outer diameter, 0.17 mm inner diameter, VWR International) was inserted into the silicone inlet and outlet tubes of the channel. The frictional :: :: :: (18) Jonsson, P.; Jonsson, M. P.; Tegenfeldt, J. O.; Hook, F. Biophys. J. 2008, 95, 5334–5348. (19) http://www.dow.com/glycerine/resources/table18.htm (November 23, 2008).
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force was sufficient to hold the tubing in place. A six-way injection valve (Upchurch Scientific, Oak Harbor, WA) was used to switch between the buffer solution and the vesicle suspension. After filling the channel with buffer solution, the lipid vesicles were injected into the microfluidic channel. After the bilayer started to form, the flow was turned off and the bilayer formation was allowed to continue for 30-60 min, after which the channel was rinsed with an excessive amount of buffer solution to remove vesicles from the bulk. In all cases, a seemingly flawless bilayer was formed over the entire channel, characterized using fluorescence recovery after photobleaching (FRAP),18 with an immobile lipid fraction typically in the range of 1-3%. Microscopy Setup. The motion of the fluorescently labeled lipids in the SLB was studied with an inverted Nikon Eclipse TE2000-U microscope (Nikon Corporation, Tokyo, Japan), using an Andor iXon EMCCD camera (Andor Technology, Belfast, Northern Ireland). To minimize the influence of stray light from the PDMS ceiling of the channel, and thus only monitor the bottom glass surface, a 100 magnification (NA = 1.49) oil immersion objective (Nikon Corporation) was used. The acquired images consisted of 512 512 pixels with a pixel size of 0.158 0.158 μm. A Kr-Ar mixed-gas ion laser (Stabilite 2018, SpectraPhysics Lasers, Mountain View, CA) was used to create an approximately Gaussian bleaching spot (∼10 μm diameter, 50-70% bleaching). The wavelength used for bleaching was 531 nm. To monitor the recovery of the fluorescent molecules, a super-high-pressure mercury lamp (Nikon Corporation) was used with a TRITC filter cube (Nikon Corporation). The images were acquired at 1 s intervals, using time-lapse acquisition, with the exposure time set to 50 ms. The illumination from the lamp was controlled by a shutter (Ludl, Hawthorne, NY) in order to minimize bleaching between frames.
Analysis of the Convective Drift Velocity and the Diffusivity. Measurements were made on the fluorescent lipid probes in the SLB at different bulk flow rates, including zero flow. The bulk flow will exert a drag force on the SLB, moving the entire bilayer in the direction from the inlet to the outlet of the channel. However, due to the large size of the connection areas at the inlet and outlet (6.6 mm2) compared to the total area of the SLB being moved during a typical experiment (0.3 mm2), the accumulation/ depletion of lipids at the outlet/inlet is negligible in the current experiments. The flow rate was controlled by using a syringe pump (Aladdin-1000, World Precision Instruments, Aston, U.K.). At least three measurements were made at each flow to confirm that the drift velocity of the SLB had stabilized. Furthermore, two independent measurements were made for each bulk viscosity (three for the 0.95 mPa s and 3 mPa s case) to estimate the spread from different experiments. The bleached spot was positioned at the center (z = 0, see Figure 1a) of the channel, and the flow rate was allowed to settle for a couple of minutes before conducting the measurements. All measurements were made at a temperature of 22.5 ( 0.5 C, monitored with a thermocouple at the outlet of the device. In a previous study we presented an improved method for FRAP analysis of a bleached spot with circular symmetry and implemented the method in MATLAB 2007b (The Mathworks, Natick, MA).18 This method first finds the center of mass of the bleached spot and reduces the two-dimensional diffusion problem to a one-dimensional problem by circular averaging. The recovery profile is then analyzed using a Hankel transform which has the advantage that no prior knowledge of the shape of the bleached area is needed. However, in the earlier method, the center of mass is determined for the first frame after photobleaching only, which assumes that there is no net drift velocity of the labeled molecules. In the present work, we extend the method to be able to track the center of mass of the bleached spot for all frames, thus effectively moving the reference frame for each analyzed image. Under the assumption that the velocity profile is approximately constant over the area of the bleached spot, this reduces the description of the recovery profile to that
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of ordinary diffusion.20 The value of the diffusion coefficient can then be determined using the same method of analysis as previously presented by us.18 The analysis of the data was made in the following order: (1) The diffusion coefficient, D, and the fraction of immobile fluorophores, γ0, were determined at zero flow, with the center of mass kept constant. (2) The average value of γ0 was used to compensate for fluorophores that are stuck to the surface. Since an immobile fraction will break the required circular symmetry of the problem, we used the following expression to only monitor the mobile fluorophores: Imobile ðr, tÞ ¼ Iall ðr, tÞ þ Iall ð¥, 0Þ -Iall ðr, 0Þ Æγ0 æ ð6Þ
(3)
where Imobile(r,t) is the intensity of the mobile fluorphores and Iall(r,t) is the intensity of both mobile and immobile fluorophores at the position r and time t. Iall(¥,0) corresponds to the intensity at the edge of the image at the first frame after photobleaching, and Æγ0æ is the average value of γ0 obtained from step 1. The center of mass of the bleached spot was then determined for each frame, typically over a total of 20 frames at an interval of one frame per second. This span gives enough data while at the same time avoiding the uncertainty when analyzing images where the bleached spot is nearly fully recovered.
The diffusion coefficient was finally determined using a curve fit to a single diffusion coefficient without an immobile fraction of molecules (since this fraction is compensated for in step 2). The maximum value of k was chosen to be the value where the Hankel transform changes the most from the first to the :: last frame used in the analysis (see Jonsson et al. for more 18 details ).
Results and Discussion Simultaneous Drift Velocity and Diffusivity Measurements. Figure 4 shows the recovery of a bleached area of fluorophores in the SLB without (a) and with (b) a pressuredriven bulk flow in the channel. The two spots have a similar diffusive behavior, but the recovery without a bulk flow is stationary, whereas the spot in Figure 4b moves in the direction of the applied bulk flow. Note also that only a single spot is observed in Figure 4b. This is in agreement with previous observations we made on a similar system, which indicated that rhodamine-DHPE was preferentially located at the upper leaflet of the bilayer.5 It should be noted that this a characteristic feature when using rhodamine-DHPE as the fluorescent probe. When using the tail-labeled lipid probe NBD C12-HPC (2-(12-(7-nitrobenz-2oxa-1,3-diazol-4-ylamino)dodecanoyl-1-hexadecanoyl-sn-glycero3-phosphocholine), two recovering mobile populations were observed (data not shown).5 The drift velocity, v, and the diffusion coefficient, D, of the labeled lipids can be determined simultaneously from an analysis of the photobleached area at each frame. Figure 5a shows the position of the center of mass of the bleached spot as a function of time at a bulk flow of 400 μL/min. In this case, we measured a drift velocity of the bleached fluorophores of v = 1.0 μm/s and an angle of their trajectories relative to the walls of the channel of -0.5 degrees. Thus, the labeled lipids move in the direction of the bulk flow parallel to the walls of the channel. Note further that while the drift velocity of rhodamine-DHPE in Figure 5a is in the (20) Jain, R. K.; Stock, R. J.; Chary, S. R.; Rueter, M. Microvasc. Res. 1990, 39, 77–93.
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Figure 4. Recovery of a photobleached area of rhodamine-DHPE for the following two cases: (a) there is no bulk flow in the channel and (b) there is a 200 μL/min bulk flow in the channel from left to right. The viscosity of the bulk was 4 mPa s, but a similar behavior is seen for other bulk viscosities as well.
Figure 5. (a) Position of the center of mass at each frame as a function of time. 1 corresponds to the x-position (parallel to the bulk flow), whereas O corresponds to the z-position (perpendicular to the bulk flow) in the channel. The solid lines are linear fits to the data points, which are given for a bulk flow of 400 μL/min. (b) Drift velocity, v, as a function of the bulk flow rate, Q. The solid line is equal to Æv/QæQ with Æv/Qæ = 154 m-2. The error bars correspond to ( one standard deviation.
order of 1 μm/s, the maximum bulk velocity in the channel is about 0.65 m/s, nearly 6 orders of magnitude larger. The drift velocity of rhodamine-DHPE as a function of the bulk flow rate is shown in Figure 5b. The drift velocity v increases Langmuir 2009, 25(11), 6279–6286
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linearly with the bulk flow rate Q, in agreement with the expression given in eq 4, with σub given by eq 2. In addition, no intensity gradient along the length of the channel was observed in these experiments, which would have been indicative of a changing lipid density along the channel.15,21,22 It is also worthwhile to point out that there was no visible drift in the velocity when measuring at a fixed bulk flow and neither did the SLB move backward when the bulk flow was stopped. We thus conclude that the surface pressure terms in the SLB may be neglected when analyzing the current experiments (see also the Appendix). The determined value of the diffusion coefficient for rhodamine-DHPE in the egg PC bilayer is presented in Figure 6 for the various bulk flow rates. The value D = 3.18 μm2/s, obtained at zero bulk flow, is comparable to previously published values by us18 and others.23,24 There was a trend that D increased by on average 6%, with a standard deviation of 1%, obtained from at least three different locations on three different samples, when comparing the diffusivity at the highest bulk flow (400 μL/min) with the value at zero flow. This behavior was seen regardless of the order of the measurements, that is, if D was measured first at a high bulk flow and followed by a measurement at zero flow or vice versa. Neither could any correlation with the temperature of the bulk solution be seen that explained this behavior. Furthermore, no change in the fluorescence intensity over the length of the channel was observed, or between the different bulk flows, which could indicate that the mobility of the lipids was affected by a change in lipid density in the bilayer.8,21 A plausible explanation to this observation is instead that there are inhomogeneities, such as immobile molecules, in the SLB (the fraction of immobile molecules, γ0, was typically 1-3%). Fluorescently labeled lipids, moving along the flow direction, that encounter these obstacles will be slowed down, thus spreading the bleached spot over a wider area. This would then make the lipids appear to diffuse faster than they actually are. Note also that the bleached fluorophores will be mixed by a fraction of immobile fluorescent molecules when the bleached spot moves away from the initial bleached area, which would also appear as an apparent increase in the recovery rate. Evaluating the Drag Coefficients at the SLB Interfaces. The mechanical properties of the studied SLB can be quantified using the theoretical model summarized in eq 4 together with the measured data on the drift velocity. Since the photobleaching is made at the center of the channel, all data have been evaluated at z = 0, where 6ηQ 8 1 ð7Þ 1- 2 σub ¼ σub ð0Þ≈ 2 h ðw -0:630hÞ π coshðπw=2hÞ is the drag force from the bulk flow at the center of the channel (see eq 2, where the terms with n > 1 in the sum have been neglected, since these terms only correspond to 0.002% of the final value of the sum in this case). In reality, the bleached spot has a finite size, which means that the outer parts of the spot will move slightly slower than the central part. However, the force on the SLB at the center of the channel does not change significantly with z (see Figure 1b), and even at distances 20 μm from the middle of the channel the drag force will only be around 2% lower than the value in the center of the channel. Thus, the effect of a finite size of the bleached spot can be neglected in the analysis. Since the (21) Nabika, H.; Takimoto, B.; Murakoshi, K. Anal. Bioanal. Chem. 2008, 391, 2497–2506. (22) Vaz, W. L. C.; Melo, E. J. Fluoresc. 2001, 11, 255–271. (23) Schutz, G. J.; Schindler, H.; Schmidt, T. Biophys. J. 1997, 73, 1073–1080. (24) Zhang, L. F.; Granick, S. J. Chem. Phys. 2005, 123, 211104.
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Figure 6. (a) Representative fit (solid line) of the fluorescence recovery (circles) using the Hankel transform method to determine the diffusion coefficient of the labeled lipids.18 (b) Value of the diffusion coefficient, D, as a function of the bulk flow rate, Q. The error bars correspond to ( one standard deviation.
measured drift velocity, v, of rhodamine-DHPE corresponds to the velocity of the upper leaflet,5 eq 4b can be written v 1 1 ð8Þ ¼ þ σ ub b1s b where σub is given by eq 7. Using the measured values for the drift velocity, this corresponds to v/σub ≈ 4.1 10-8 (Pa s/m)-1. From eq 8, this means that both b and bls are at least 2.4 107 Pa s/m. However, since it was previously observed that the lower leaflet of the bilayer moves markedly slower than the upper leaflet,5 it is expected that b ∼ 2.4 107 Pa s/m whereas bls . 2.4 107 Pa s/m (see eq 4). The value obtained for b is in agreement with previously published data by Merkel et al. who obtained values for b in the range 107-108 Pa s/m for the frictional coupling between two bilayer leaflets,8 even though literature values for b vary somewhat depending on the system studied and the method used to estimate the frictional coupling.6,9 If the frictional coupling between the lower leaflet and the substrate is dominated by a frictional force from a viscous layer of fluid between the SLB and the substrate (see eq 5a), the relation bls . 2.4 107 Pa s/m corresponds to an upper limit of hw = 0.039 nm for a film with the same viscosity as the bulk solution. Several groups have reported that the average film thickness between the substrate and the SLB should be in the order of 0.53 nm.25-28 This is about 1-2 orders of magnitude larger than the upper limit of 0.04 nm obtained here. However, due to the roughness of the support and the protrusion of lipid head groups into the underlying liquid film, this layer will be a mixture of the (25) Johnson, S. J.; Bayerl, T. M.; McDermott, D. C.; Adam, G. W.; Rennie, A. R.; Thomas, R. K.; Sackmann, E. Biophys. J. 1991, 59, 289–294. (26) Tanaka, M.; Hermann, J.; Haase, I.; Fischer, M.; Boxer, S. G. Langmuir 2007, 23, 5638–5644. (27) Bayerl, T. M.; Bloom, M. Biophys. J. 1990, 58, 357–362. (28) Koenig, B. W.; Kruger, S.; Orts, W. J.; Majkrzak, C. F.; Berk, N. F.; Silverton, J. V.; Gawrisch, K. Langmuir 1996, 12, 1343–1350.
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substrate, the buffer, and the lipid head groups.25,28 It should also be noted that the viscous properties of a liquid on these small length scales may have different properties than those in bulk solution.29 Thus, the effective viscosity for the liquid between the SLB and the substrate may actually be substantially higher than the viscosity in the bulk solution. If instead a mean thickness of 1 nm is assumed for the distance between the SLB and the substrate, the measured value for the frictional coupling corresponds to an effective viscosity of at least 24 mPa s for the liquid film between the SLB and the substrate. This value is between the bulk viscosity of water and lipids in a bilayer,2,30 thus indicating the possibility of the liquid film being more complex than just a plane viscous film of the same viscosity as the bulk solution. If on the other hand the frictional coupling between the lower leaflet and the substrate is dominated by friction within the SLB due to a certain number, c, of pinning sites where the lipids are immobile, then using eq 5b and bls . 2.4 107 Pa s/m a lower limit of c = 19 000 μm-2 is obtained. This is roughly 1 order of magnitude :: higher than the value estimated by Radler et al. from self16 spreading bilayer studies on glass. However, if the mean area per lipid is 0.63 nm2, as stated by Smaby et al.,31 19 000 pinning sites/μm2 would correspond to ∼1% of the lipids in the lower leaflet being pinned. This number is of the same order as the number of immobile molecules determined from the FRAP data. Since only a lower limit of c is determined, another possibility is that the entire lower leaflet is pinned to the substrate. However, it should be mentioned that when having NBD C12-HPC as the fluorescently labeled lipid in the SLB, which is present in both leaflets of the bilayer, the lipids in the lower leaflet are seen to recover in the FRAP experiments (even though these lipids have a substantially lower drift velocity compared to the lipids in the upper leaflet of the bilayer when subjected to a bulk flow).5 This suggests that not all lipids in the lower leaflet are pinned. Effects on the Lipid Mobility When Changing the Bulk Viscosity. Figure 7 shows how the drift velocity of rhodamineDHPE varies with the bulk flow rate for different bulk viscosities, obtained by varying the concentration of glycerol in the bulk solution. To avoid the risk of deformation of the microfluidic channel, the bulk flows used in the experiments were decreased when measuring at higher bulk viscosities.32 A linear increase in drift velocity with increasing bulk flow was observed for all viscosities, but the slope v/Q changed to a higher value when the viscosity of the bulk solution was increased (See Figure 7a). Figure 7b shows how D and v/σub (see eqs 7 and 8) change with the bulk viscosity. The values of D were all obtained from measurements with zero bulk flow to avoid the effect of inhomogeneities in the bilayer (see discussion above), even though the same principal behavior was observed when the measurements were made at different bulk flows. Interestingly, both D and v/σub change with the bulk viscosity in a similar way. From the discussion in the previous section, it was noted that v/σub ∼ 1/b, and thus the frictional coupling between the two bilayer leaflets seems to increase with the viscosity of the bulk solution according to the result in Figure 7b. This indicates a structural change in the SLB when adding glycerol to the buffer solution. A decrease in the bilayer thickness has previously been observed when adding (29) Major, R. C.; Houston, J. E.; McGrath, M. J.; Siepmann, J. I.; Zhu, X. Y. Phys. Rev. Lett. 2006, 96, 177803. (30) Evans, E.; Yeung, A. Chem. Phys. Lipids 1994, 73, 39–56. (31) Smaby, J. M.; Momsen, M. M.; Brockman, H. L.; Brown, R. E. Biophys. J. 1997, 73, 1492–1505. (32) Gervais, T.; El-Ali, J.; Gunther, A.; Jensen, K. F. Lab Chip 2006, 6, 500– 507.
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where kB is Boltzmann’s constant, T = 295.65 K is the temperature, K0 and K1 are modified Bessel functions of the second kind of order zero and one, respectively, a is the effective radius of a lipid which corresponds to 0.45 nm for an area of 0.63 nm2 per lipid,31 and b = 2.4 107 Pa s/m is the intermonolayer friction coefficient (here for a bulk solution without glycerol). Inserted into eq 9, this yields a surface viscosity of ηm = 2 10-10 Pa s m. This corresponds to a surface viscosity of 4 10-10 Pa s m for the entire bilayer, which is in agreement with results obtained in other studies.6,8,30
Conclusions
Figure 7. (a) Drift velocity versus bulk flow rate for the four different bulk viscosities used in the experiments, where the error bars correspond to ( one standard deviation. The linear curves correspond to Æv/QæQ at the various viscosities, where Æv/Qæ is the average value of v/Q. (b) Values of D (O) and v/σub (1) for the different bulk viscosities normalized to the value obtained without glycerol in the bulk solution.
glycerol to the studied system.33 This can in turn have the effect of increasing the interdigitation of the two leaflets of the bilayer, which is expected to increase b.6 Since D is inversely proportional to the drag force felt by a lipid, an increase in b will also make the lipids experience a stronger drag force from the opposite leaflet, yielding a lower value of D. An alternative, or additional, explanation is that the viscosity of the lipid bilayer increases when adding glycerol to the bulk solution, thus increasing the frictional drag on the lipids from within the bilayer. If the viscosity within each bilayer leaflet is increased, then it is likely that also the frictional coupling between the two bilayer leaflets increases. All together, this may explain why the relative changes in v/σub and D, when adding glycerol to the bulk solution, are similar. The determined values of D and b can also be used to estimate the surface viscosity of the upper lipid monolayer, ηm, using the relation7 D ¼
kB T λT
ð9aÞ
with λT ¼ 4πηm and
1 2 εK1 ðεÞ ε þ 2 K0 ðεÞ
sffiffiffiffiffiffi b ε ¼a ηm
ð9bÞ
Appendix Estimating the Magnitude of the Forces Acting on a Moving SLB. The governing equations describing the velocity of the upper and lower leaflet of the bilayer are given in eq 3 in the main manuscript but are summarized here for convenience: DΠu ðxÞ D2 vl ðzÞ ¼ bðvu ðzÞ -v1 ðzÞÞ -σub ðzÞ ð10aÞ þ ηm Dx Dz2 -
ð9cÞ
(33) Johansson, L. B. A.; Kalman, B.; Wikander, G.; Fransson, A.; Fontell, K.; Bergenstahl, B.; Lindblom, G. Biochim. Biophys. Acta 1993, 1149, 285–291.
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The bulk flow in a microfluidic channel will result in a viscous shear force on the walls of the channel, which can be accurately described using Navier-Stokes equations. These forces can be used to move an SLB, formed on the walls of the channel, with a characteristic drift velocity that, in addition to the bulk flow rate, depends on the frictional coupling between the two bilayer leaflets and the friction coefficient between the lower leaflet of the bilayer and the supporting substrate. For an egg PC bilayer on glass, it was observed that the coupling between the SLB and the substrate was at least 25 times as high as predicted by the traditional model with the SLB and the support spaced by a 1 nm thick film of water.7,34 This observation indicates that the effective viscosity of a thin liquid film between the SLB and the support must be much higher than that of the bulk liquid. Increasing the viscosity in the bulk solution by adding glycerol was observed to increase the frictional coupling between the two bilayer leaflets, which without glycerol was characterized by an intermonolayer friction coefficient of approximately 2 107 Pa s/ m. This increase in frictional coupling between the bilayer leaflets was also accompanied by a proportional decrease in the lipid diffusivity, indicating an increased viscous drag within the bilayer when glycerol was added. The presented technique opens up a new and versatile approach to estimate the mechanical properties of SLBs, something that previously has been a complex procedure.9 We also emphasize that different membrane-associated components may have different drift velocities in the SLB due to the size and structure of these molecules and their coupling to the bilayer and the surrounding environment. Being able to quantify the mechanical properties of the SLB and to predict how the bilayer reacts to a specific bulk flow is therefore of importance, not only from a fundamental point of view but also for future applications of the technique aimed at controlled separation of membraneassociated molecules.
DΠ1 ðxÞ D2 vu ðzÞ þ ηm ¼ b1s v1 ðzÞ -bðvu ðzÞ -v1 ðzÞÞ Dx Dz2
ð10bÞ
where Πu/l is the surface pressure and vu/l is the velocity of the upper/lower leaflet. σub is the shear force from the bulk flow on the upper leaflet, ηm is the surface viscosity of a monolayer, b is the (34) Groves, J. T.; Ulman, N.; Boxer, S. G. Science 1997, 275, 651–653.
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intermonolayer friction coefficient, and bls is the friction coefficient between the lower leaflet of the bilayer and the substrate. The terms in eq 10 including the surface viscosity, ηm, arise due to shear forces between the lipids within a monolayer. However, the magnitude of these terms compared to the magnitude of the other terms in eq 10 is negligible as can be seen when compared to the frictional force between the two monolayers: ! D2 vu=1 ðzÞ η =Oðbvu=1 ðzÞÞ ¼ m2 e10 -9 O ηm Dz2 bw where literature values of ηm (10-10-10-9 Pa s m) and b (107-109 Pa s/m) have been used.8,30,35 The reason why the frictional term within the SLB can be neglected, in comparison to the other terms in eq 10, is that the velocity changes slowly in the z-direction in comparison to the abrupt change in velocity at the interface between the upper and the lower leaflet of the bilayer. Furthermore, the narrow thickness of the SLB in comparison to the height of the channel and the fact that the drift velocity of the SLB is nearly 6 orders of magnitude smaller than the maximum bulk flow velocity in the channel also contribute to that the magnitude of the shear forces within each monolayer can be neglected. To estimate the magnitude of the term including the surface pressure we use that the additional surface pressure, ΔΠ, required to change the mean area per lipid, A, an amount ΔA, is given by ΔΠ ð11Þ Ka ¼ -A ΔA (35) Raphael, R. M.; Waugh, R. E. Biophys. J. 1996, 71, 1374–1388.
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where Ka is the compressibility modulus, which for a typical phosphatidylcholine monolayer in the fluid phase is in the order of 100 mN/m.31,36,37 The difference in surface pressure between the inlet and outlet of the channel should not be expected to be larger than the surface pressure difference required to change the area per lipid in the outlet and inlet reservoirs an amount (ΔA. In reality, the difference in surface pressure may be much smaller than this, since the SLB may be pushed up onto the walls of the reservoir and also pushed out into the outlet tubing. However, this approach will yield an upper limit of the surface pressure drop in the SLB over the channel length. For this purpose, it is used that the free area of the reservoirs are roughly 6.6 mm2. During a typical measurement, the SLB in the channel moves approximately 1.5 mm in the direction from the inlet to the outlet, corresponding to a change in lipid area of 0.3 mm2. Thus, the area change per lipid then corresponds to ΔA/A ∼ (0.05, where the plus sign is for the inlet and the minus sign is for the outlet. Inserted into eq 11, this yields O(∂Πu/l/∂x) e 1 Pa, assuming a constant pressure drop over the 10 mm long channel, which is less than 5% of the force exerted on the SLB when using a bulk flow of 400 μL/min in the channel. Acknowledgment. This work was financially supported by the Swedish Research Council for Engineering Sciences, Contract Number 2005-3140, and the INGVAR grant from the Strategic Research Foundation. (36) Rawicz, W.; Olbrich, K. C.; McIntosh, T.; Needham, D.; Evans, E. Biophys. J. 2000, 79, 328–339. (37) Koenig, B. W.; Strey, H. H.; Gawrisch, K. Biophys. J. 1997, 73, 1954–1966.
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