Effects of Surface Pressure and Internal Friction on the Dynamics of

Dec 10, 2010 - Peter Jönsson* and Fredrik Höök*. Department of Applied Physics, Chalmers University of Technology, SE-41296 Gothenburg, Sweden...
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Effects of Surface Pressure and Internal Friction on the Dynamics of Shear-Driven Supported Lipid Bilayers† Peter J€onsson* and Fredrik H€oo€k* Department of Applied Physics, Chalmers University of Technology, SE-41296 Gothenburg, Sweden Received October 4, 2010. Revised Manuscript Received November 10, 2010 Supported lipid bilayers (SLBs) are one of the most common model systems for cell membrane studies. We have previously found that when applying a bulk flow of liquid above an SLB the lipid bilayer and its constituents move in the direction of the bulk flow in a rolling type of motion, with the lower monolayer being essentially stationary. In this study, a theoretical platform is developed to model the dynamic behavior of a shear-driven SLB. In most regions of the moving SLB, the dynamics of the lipid bilayer is well explained by a balance between the hydrodynamic shear force arising from the bulk flow above the lipid bilayer and the friction between the upper and lower monolayers of the SLB. These two forces result in a drift velocity profile for the lipids in the upper monolayer of the SLB that is highest at the center of the channel and decreases to almost zero at the corners of the channel. However, near the front of an advancing SLB a very different flow behavior is observed, showing an almost constant drift velocity of the lipids over the entire bilayer front. In this region, the motion of the SLB is significantly influenced by gradients in the surface pressure as well as internal friction due to molecules that have accumulated at the front of the SLB. It is shown that even a modest surface fraction of accumulated molecules (∼1%) can drastically affect the behavior of the SLB near the bilayer front, forcing the advancing lipids in the SLB away from the center of the channel out toward the sides.

Introduction Detailed studies of the membrane around a living cell are often difficult because of the complexity of the cell membrane. For in-depth studies of the biophysical properties of cell membranes and their specific functions, much work has been focused on various means of forming cell-membrane mimics. A commonly used cellmembrane mimic is the supported lipid bilayer (SLB), which consists of a lipid bilayer formed on a solid support, typically a glass slide or a polymer cushion.1-3 This type of system mimics many of the properties of a living cell membrane but is less fragile and has a controllable molecular composition. Furthermore, it is much easier to study a planar SLB than the curved membrane surrounding a living cell, and several very accurate surface-based techniques are currently available for studies of molecules on planar surfaces.2,3 We recently found that when a liquid flows over an SLB the hydrodynamic force arising from the flowing liquid will drive the lipid bilayer and the molecules within the bilayer in the direction of the liquid flow (see Figure 1, which shows a phosphatidylcholine SLB containing 0.2 wt % of the fluorescently labeled lipid lissamine rhodamine B 1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine (R-DHPE)).4 Since this discovery, shear-driven lipid bilayers have been used in the following applications. (i) To estimate the mechanical parameters of SLBs such as the intermonolayer friction coefficient and the frictional coupling between an SLB and the underlying support.5 (ii) To drive an SLB over a † Part of the Supramolecular Chemistry at Interfaces special issue. *Corresponding authors. E-mail: [email protected]; fredrik.hook@ chalmers.se.

(1) Castellana, E. T.; Cremer, P. S. Surf. Sci. Rep. 2006, 61, 429–444. (2) McConnell, H. M.; Watts, T. H.; Weis, R. M.; Brian, A. A. Biochim. Biophys. Acta 1986, 864, 95–106. (3) Sackmann, E. Science 1996, 271, 43–48. (4) Jonsson, P.; Beech, J. P.; Tegenfeldt, J. O.; Hook, F. J. Am. Chem. Soc. 2009, 131, 5294–5297.

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surface with embedded submicrometer wells.6 By adjusting the pH of the bulk solution, the lipid bilayer could be made to follow the contour of the surface or to span the wells, forming a sealing lipid bilayer. (iii) To accumulate and separate membrane-bound proteins.7 The anchored proteins were first driven toward the edge of the lipid bilayer by hydrodynamic forces, where the molecules accumulate. Reversing the flow drives the protein molecules backwards with a drift velocity characteristic of each type of protein molecule. To develop this experimental technique further, it is important to understand the parameters that control the dynamic behavior of a shear-driven SLB. As a first step in this direction, we have previously developed a theoretical model relating the drift velocity of the SLB to the bulk flow of liquid in the microfluidic channel.5 However, this model was developed to describe the behavior of an SLB covering the entire microfluidic channel and not the flow behavior of the lipids close to the front of an advancing SLB. In particular, the question of why the entire front of the lipid bilayer seems to move at a constant velocity across the microfluidic channel has not yet been answered.4 The aim of this study was to derive a more general theoretical platform that can also describe the dynamic behavior of the SLB close to the front of the lipid bilayer. The proposed model takes into account variations in the surface pressure and the internal friction within the advancing lipid bilayer. Special attention is paid to the situation where molecules are accumulated at the front of the lipid bilayer (see Figure 1, which shows the accumulation of R-DHPE molecules at the front of the advancing SLB), a common situation occurring when some molecules in the upper monolayer of the (5) Jonsson, P.; Beech, J. P.; Tegenfeldt, J. O.; Hook, F. Langmuir 2009, 25, 6279–6286. (6) Jonsson, P.; Jonsson, M. P.; Hook, F. Nano Lett. 2010, 10, 1900–1906. (7) Jonsson, P.; Gunnarsson, A.; Hook, F. Anal. Chem. DOI: 10.1021/ ac102979b.

Published on Web 12/10/2010

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Figure 1. Fluorescence micrographs of an SLB with 0.2 wt % R-DHPE under a bulk flow of 200 μL/min from left to right. The dashed lines indicate the walls of the microfluidic channel.

Figure 2. Rectangular microfluidic channel with bulk flow along the x direction.

SLB are completely or partially prevented from entering the lower monolayer at the front of the advancing lipid bilayer.

Theoretical Background The SLB moves in the direction of the bulk flow as a result of the hydrodynamic shear force, σhydro, arising from the flowing liquid in the microfluidic channel. In this study, a microfluidic channel with a rectangular cross section was used (Figure 2). The height of the microfluidic channel, h, was 110 μm, and the width, w, was 150 μm. The shear force on an SLB adsorbed on the floor of the channel at y = 0 can then be described by the following expression:5,8 σ hydro

! ¥ Δp h 8 X 1 coshðkπz=hÞ 1- 2 ¼ Δx 2 π k odd k2 coshðkπw=2hÞ

ð1Þ

The pressure drop over the length of the channel, Δp/Δx, is given approximately by the bulk flow rate in the channel, Q, according to Δp 12ηQ  - 3 Δx h wð1 - 0:630h=wÞ

ð2Þ

where η is the viscosity of the bulk liquid (1 mPa s in this case).5,8 The value of σhydro on the floor of the microfluidic channel is shown in Figure 3 for a bulk flow rate of Q = 200 μL/min. The hydrodynamic shear force gives rise to a rolling motion of the SLB such that the velocity of the front of the lipid bilayer is given by the average velocity of the upper and lower monolayers of the lipid bilayer (Figure 4). For an egg yolk phosphatidylcholine SLB on a glass support, it was found that the velocity of (8) Bruus, H. Theoretical Microfluidics; Oxford University Press: Oxford, U.K., 2008; pp 48-51.

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Figure 3. Hydrodynamic shear force, σhydro, acting on the floor of the microfluidic channel at a bulk flow rate of 200 μL/min.

Figure 4. SLB moving with a rolling motion, where the velocity of the bilayer front is given by the average value of the velocity of the upper and lower monolayers of the SLB.

the bilayer front, vfront, corresponds to approximately half the average velocity of the lipids in the upper monolayer of the SLB, vupper.4 Because the SLB exhibits a rolling motion, the velocity of the lower monolayer of the SLB, vlower, is therefore much smaller than vupper and can be approximated to zero in this context.4 Because the lower monolayer of the SLB is approximately stationary, only the upper monolayer of the SLB needs to be modeled to describe the full motion of the lipid bilayer. For this purpose, the upper monolayer is treated as an incompressible fluid with a surface viscosity denoted by ηm.5,9-11 The motion of (9) Evans, E.; Sackmann, E. J. Fluid. Mech. 1988, 194, 553–561. (10) Hughes, B. D.; Pailthorpe, B. A.; White, L. R. J. Fluid. Mech. 1981, 110, 349–372. (11) Saffman, P. G. J. Fluid. Mech. 1976, 73, 593–602.

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the upper monolayer in the plane of the SLB can then be described by the 2D Navier-Stokes equation for creeping flow

SLB. This difference in velocity will result in an internal frictional force on the lipids, Fob, which for each obstacle can be written as

0 ¼ - rΠ þ ηm r2 v þ σ hydro ex - bv

ð3aÞ

Fob ¼ - λob ðv - vob Þ

r3v ¼ 0

ð3bÞ

In the limit of an infinitely low surface fraction of obstacles, the drag coefficient, λob, will be given by

where Π is the surface pressure, v is the velocity of the upper monolayer of the SLB, and ex is a unit vector in the x direction. Parameter b is a constant characterizing the frictional coupling between the two monolayers of the SLB and is called the intermonolayer friction coefficient.5,12 An order-of-magnitude estimate of the different terms in eq 3a shows that the viscous term ηmr2v will make a significant contribution to the sum in eq 3a only if the velocity changes over a length scale of ∼10 nm or less. However, the hydrodynamic force, σhydro, changes over a length scale of ∼100 μm (Figure 3). This means that ηmr2v can be neglected when describing the macroscopic velocity of the SLB, which changes over the same length scale as σhydro. Velocity Profile Far from the Bilayer Front. Equation 3a can be simplified when describing the flow in the SLB far from the front of the advancing lipid bilayer. In this region of the SLB, the gradients in surface pressure can be expected to be negligible compared to the hydrodynamic force, σhydro.5 The macroscopic velocity of the upper monolayer of the lipid bilayer can thus be written as   σ hydro ex ð4Þ v ¼ b

λob ¼

kB T Dob

ð7Þ

ð8Þ

where kB is Boltzmann’s constant (1.381  10-23 J/K), T is the temperature, and Dob is the diffusivity of the molecules that accumulate in the SLB.9 The macroscopic velocity, v, is defined as the average velocity around an ensemble of obstacles at the same macroscopic position in the SLB. The same definition applies to the macroscopic pressure, Π. In addition to the drag force, Fob, the obstacles also have the effect of reducing the amount of lipids per unit area of the SLB. To account for this effect, the macroscopic version of eq 3a for an SLB with nanometer-sized obstacles can be written as 0 ¼ - rΠ þ σhydro ð1 - φÞex - bð1 - φÞv -

  φλob ðv - vob Þ Aob

ð9Þ

where φ is the surface fraction of obstacles and Aob is the crosssectional area of an obstacle in the SLB. Equation 9 can be rewritten in a form similar to eq 5, yielding v ¼ -

rΠ σ eff þ beff beff

ð10Þ

which means that the lipids in the upper monolayer move in the direction of the bulk flow with a velocity that is highest at the center and falls to zero at the corners of the channel walls (Figure 3).5 Velocity Profile near the Bilayer Front. For the situation in which the gradients in surface pressure are comparable to the hydrodynamic force arising from the bulk flow, the macroscopic drift velocity from eq 3a is   rΠ σhydro ð5Þ þ ex v ¼ b b

where

Note that, in this case, the drift velocity does not have to be in the same direction as the bulk flow. Inserting the expression for v, given by eq 5, into eq 3b results in

Numerical Procedure There are no general analytical solutions to the equations presented in the previous section. To obtain solutions of the surface pressure and velocity profile in the SLB, numerical methods must be used. Simulations were made here using COMSOL Multiphysics 3.5a (COMSOL AB, Stockholm, Sweden), a program that solves partial differential equations using the finite element method. Three separate cases are presented below. Velocity Profile in the Bulk of the SLB. In this section, the numerical procedure followed to solve the equation for the motion of a bleached band of R-DHPE molecules, far away from the front of the lipid bilayer, is given. The bleached band is perpendicular to the flow direction and stretches over the width of the channel. The relative concentration of R-DHPE in the lipid bilayer, c, was modeled using   ∂c ∂ cσhydro ¼ Dr2 c ð14Þ ∂t ∂x b

r2 Π ¼ 0

ð6Þ

where b is assumed to be constant over the entire SLB. Thus, to obtain the velocity of the upper monolayer of the SLB the Laplace equation for the surface pressure Π must first be solved for the lipid bilayer under the appropriate boundary conditions, after which Π is inserted into eq 5 to give v. Effect of Obstacles. The presence of obstacles in the upper monolayer of the SLB will affect the local flow profile of the lipids and the surface pressure in the surroundings of the obstacle. The type of obstacles that will be treated in this study are membraneassociated molecules that are reluctant to enter the lower monolayer of the SLB at the edge of the advancing lipid bilayer. Because of the rolling motion of the SLB,4 these molecules accumulate in a region near the front of the lipid bilayer where they have a drift velocity, vob, that is different from the macroscopic velocity, v, of the main type of lipids in the advancing upper monolayer of the (12) den Otter, W. K.; Shkulipa, S. A. Biophys. J. 2007, 93, 423–433.

1432 DOI: 10.1021/la103959w

σ eff ¼ ð1 - φÞσ hydro ex þ beff ¼ ð1 - φÞb þ

  φλob vob Aob

ð11Þ

φλob Aob

ð12Þ

In addition, the macroscopic version of the conservation of mass equation (eq 3b) is r 3 ½ð1 - φÞv þ φvob  ¼ 0

ð13Þ

where D is the diffusivity of R-DHPE in the SLB (set to 2.3 μm2/s) and the expression in eq 4 has been used to represent the convective drift velocity. Equation 14 was solved numerically Langmuir 2011, 27(4), 1430–1439

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where γ = 10-11 N is the line tension14-16 and Rf = 76 μm from Figure 5. The boundary condition at the front of the lipid bilayer can also be modeled using the condition that the entire bilayer front should move at a constant velocity, vfront, in the x direction. This is equivalent to v n ð17Þ vfront ¼ 3 2nx Figure 5. Geometry used for the simulations near the front of the SLB.

using COMSOL Multiphysics in a rectangular geometry with the sides given by: -w/2 < z < w/2 and 0 < x < 500 μm, where w = 150 μm. The concentration at t = 0 was defined as ! ðx - x0 Þ2 ð15Þ cðt ¼ 0Þ ¼ 1 - exp w2 where x0 = 200 μm and w = 10 μm were used as typical values for the width of a bleached band of R-DHPE molecules in the SLB. The value of σhydro used in the simulation was taken from Figure 3. The intermonolayer friction coefficient was determined from the average drift velocity of the R-DHPE molecules in the SLB. With Ævæ = 0.31 μm/s and Æσhydroæ = 12.5 Pa, eq 4 gives b = 4  107 Pa s/m. This value is comparable to previously published values of b for phosphatidylcholine lipid bilayers.5,13 As boundary conditions, the concentration was set to c = 1 at x = 0 and 500 μm. For the boundaries at z = (w/2, the derivative of c normal to the boundary was set to zero. A mesh consisting of ∼20 000 triangles was used for the geometry, and eq 14 was solved with the time-dependent solver in COMSOL Multiphysics with relative and absolute tolerances in the time step of 10-4 and 10-5, respectively. Repeating the simulation for a geometry stretching over all four walls of the channel had only a marginal effect on the motion of the bleached lipids on the floor of the microfluidic channel. Simulations near the Front of the SLB. To model the flow and surface pressure close to the front of the lipid bilayer, the geometry shown in Figure 5 was used. Here, the front of the lipid bilayer was assumed to have a fixed curvature equal to 1/Rf = 1/76 μm-1, which was estimated from the experiments performed in this study. In the simulations, the value of σhydro in Figure 3 was used, and b was set to 4  107 Pa s/m (see above). The geometry shown in Figure 5 corresponds to an ∼500-μmlong lipid bilayer patch that ends at the walls of the microfluidic channel with the bilayer front at boundary 3. In reality, the SLB will stretch further back than 500 μm, but a longer geometry did not change the results of the simulations significantly. The boundary condition at boundaries 1, 2, and 4 was set to Π = 0 in the simulations. The actual boundary condition at boundaries 2 and 4 depends somewhat on the situation studied, for example, whether an SLB is adsorbed on these walls. However, the principal behavior of the SLB near the front of the lipid bilayer will be the same. The surface pressure at the front of the lipid bilayer can be estimated from the pressure drop over the SLB due to the curvature of the bilayer front, 1/Rf, according to Π ¼

γ ¼ 10 - 7 Pa m Rf

ð16Þ

(13) Merkel, R.; Sackmann, E.; Evans, E. J. Phys. (Paris) 1989, 50, 1535–1555. (14) Chernomordik, L. V.; Kozlov, M. M.; Melikyan, G. B.; Abidor, I. G.; Markin, V. S.; Chizmadzhev, Y. A. Biochim. Biophys. Acta 1985, 812, 643–655. (15) Evans, E.; Heinrich, V.; Ludwig, F.; Rawicz, W. Biophys. J. 2003, 85, 2342– 2350.

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at boundary 3, where n = (nxex þ nzez) is a unit normal to the bilayer front. Thus, the velocity of the bilayer front depends on both the velocity in the x direction, vx, and the velocity in the z direction, vz, of the lipids in the upper monolayer. The influence of the z velocity will be zero in the center of the channel but will increase gradually toward the sides of the bilayer front where nz increases. This can also explain why the stationary curvature of the bilayer front is generally observed to increase when the bulk flow rate in the channel increases because the hydrodynamic contribution to the velocity of the bilayer front is highest in the center of the channel and proportional to the bulk flow rate, Q. This results in a progressively more parabolic bilayer front as the bulk flow rate is increased, until the value of nz on the sides of bilayer front is large enough for vz to make a contribution to vfront with a magnitude similar to the hydrodynamic contribution in the center of the channel. The drift velocity of the bilayer front is approximately half the average velocity of the lipids far from the bilayer front, which from eq 4 yields vfront ¼

Æσ hydro æ 2b

ð18Þ

where Æσhydroæ is the average value of σhydro over the width of the channel. With these boundary conditions, the flow profile in the SLB was solved using the stationary Direct (UMFPACK) solver in COMSOL Multiphysics for a mesh with ∼14 000 triangular elements. Two scenarios were considered in this study: (i) an SLB without obstacles in the upper monolayer of the lipid bilayer and (ii) an SLB with obstacles in the upper monolayer of the SLB. In the first scenario, the surface pressure is given by the solution to eq 6. In this case, the boundary condition in eq 17 can be written as rΠ 3 n ¼ σ hydro nx - Æσ hydro ænx

ð19Þ

In the second scenario, it is eq 13, with v given by eq 10, that must be solved. For this purpose, the surface fraction of obstacles, φ, in the SLB must be given, together with the values of λob, Aob, and vob. For an SLB in which the obstacles that accumulated at the front of the lipid bilayer are R-DHPE molecules, Aob = 0.63 nm2,17 and vob can be approximated by the velocity of the bilayer front, vob = vfrontex. For an SLB with a known number of R-DHPE molecules, the surface fraction of obstacles can be estimated from the fluorescence intensity of the bilayer front. For an SLB with 0.2 wt % R-DHPE molecules in the lipid bilayer, the surface fraction of R-DHPE molecules in the bulk of the SLB will be ∼0.23%, assuming that the R-DHPE molecules are located predominantly in the upper monolayer of the SLB4,18 and that the molecular masses of R-DHPE and POPC are 1334 and 760 g/mol, respectively. The value of φ near the front of the lipid bilayer can then be obtained by dividing the intensity near the bilayer front by the intensity in the bulk of the SLB and (16) Zhelev, D. V.; Needham, D. Biochim. Biophys. Acta 1993, 1147, 89–104. (17) Smaby, J. M.; Momsen, M. M.; Brockman, H. L.; Brown, R. E. Biophys. J. 1997, 73, 1492–1505. (18) Ajo-Franklin, C. M.; Yoshina-Ishii, C.; Boxer, S. G. Langmuir 2005, 21, 4976–4983.

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Figure 6. Geometry of a hexagonal unit cell containing a circular obstacle in the center.

multiplying by 0.23%. The value for λob as a function of the surface fraction, φ, was determined from numerical simulations. (See below.) Flow around an Obstacle in the SLB. To estimate the drag coefficient, λob, due to obstacles in the lipid bilayer, eqs 3a and 3b were solved with COMSOL Multiphysics in a hexagonal unit cell, as shown in Figure 6. In the model, b = 4  107 Pa s/m and ηm = 2  10-10 Pa s m were used.5,13 Because of the small size of the unit cell compared to the size of the microfluidic channel, the value of σhydro can be assumed to be constant over the unit cell and was set to 10 Pa in the simulations. Each obstacle is treated as a cylindrical object in the SLB, with an effective radius a located in the center of the unit cell. The surface fraction of obstacles, φ, is related to the area of a unit cell, Acell, by φ ¼ where Acell

πa2 Acell

pffiffiffi sh 2 3 3 ¼ 2

ð20Þ

The liquid flowing around the obstacle will give rise to a net force in the x direction on the obstacle given by Z ð22Þ Fob ¼ - σx ds where the integral is over the boundary of the circular obstacle and     ∂vx ∂vx ∂vz n x þ ηm nz ð23Þ þ σ x ¼ - Π þ 2ηm ∂x ∂z ∂x where nx and nz are the x and z components of the unit normal to the inner circular boundary in Figure 6. The force Fob scales linearly with the average bulk flow in the x direction of the unit cell, Ævxæ, yielding Ævx æ ð24Þ λob ¼ Fob where λob is the drag coefficient first appearing in eq 7.

Materials and Experimental Methods ð21Þ

and sh is the length of a side in the hexagonal unit cell (Figure 6). For the scenario in which the obstacles correspond to accumulated R-DHPE molecules, the cross-sectional area of the obstacle was set to πa2 = 0.63 nm2,17 corresponding to an effective radius of a = 0.45 nm. The drag force on the lipids caused by the obstacle is proportional to the difference between the average velocity of the lipids in the unit cell and the velocity of the obstacle (eq 7). To estimate λob, the boundary condition for the velocity at the circular obstacle was set to zero, which means that eqs 3a and 3b are solved for the relative velocity, v - vob. The boundary conditions on the outer boundaries were chosen to be periodic in v and Π such that the velocity and surface pressure at boundary 1 (boundary 2) are identical to the values at boundary 4 (boundary 5). At boundaries 3 and 6, the velocity was set to vz = 0 and ∂vx/∂z = 0, where vx and vz are the x and z components of v, respectively. This corresponds to a situation where the macroscopic change in Π over a unit cell is negligible and the main driving force for the lipids in the SLB is the hydrodynamic shear force. Both the surface pressure and the drift velocity in the unit cell will then scale linearly with σhydro. 1434 DOI: 10.1021/la103959w

Figure 7. Schematic of the 110-μm-high microfluidic channel, seen from above.

Fabrication of the Microfluidic Channel. The microfluidic channel was made of polydimethylsiloxane (PDMS) using the technique of replica molding.19,20 A clean glass slide (0.13-0.16 mm in thickness, Menzel-Gl€aser, Braunschweig, Germany) constituted the floor of the channel. The channel had four arms in the shape of a 1 cm  1 cm cross (Figure 7), where the width w (150 μm) and height h (110 μm) of each channel arm was determined using an optical surface profiler (Wyko NT 1100, Veeco Instruments Inc., Tucson, AZ). Details of the fabrication of the microfluidic channel have been published elsewhere.7 Vesicle Preparation. Lipid vesicles were prepared by extrusion through a 30 nm membrane (Whatman, Maidstone, U.K.) using an Avanti Mini-Extruder (Avanti Polar Lipids, Alabaster, AL). The vesicles consisted of 1-palmitoyl-2-oleoyl-sn-glycero-3phosphocholine (POPC) from Avanti Polar Lipids with either 0.2 or 1 wt % of the R-DHPE fluorescently labeled lipid probe (Invitrogen, Carlsbad, CA). The buffer solution used in the experiments was a mixture of 100 mM NaCl (Sigma-Aldrich, Stockholm, Sweden), 10 mM tris(hydroxymethyl)aminomethane (TRIS, VWR International, Stockholm, Sweden), and 1 mM ethylenediaminetetraacetic acid disodium salt dihydrate (EDTA, Sigma-Aldrich) with a pH of 8.0. The lipid vesicles were diluted (19) 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. (20) Xia, Y. N.; Whitesides, G. M. Annu. Rev. Mater. Sci. 1998, 28, 153–184.

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J€ onsson and H€ oo€k with the buffer solution to a total lipid concentration of 100 ng/μL before each experiment. Forming and Driving the Lipid Bilayer. To access the microfluidic device, PTFE tubing (1/16 in. outer diameter and 0.17 or 0.5 mm inner diameter, VWR International) was inserted into the silicone inlet and outlet tubes of the channel. The frictional force was sufficient to hold the tubing in place. A six-way selection valve (Upchurch Scientific, Oak Harbor, WA) was connected to the tube on arm 1 in order to be able to inject and switch between different solutions. An SLB was formed in the left part of the channel by injecting the vesicle solution into arm 1 and the buffer solution into arm 4. The vesicle solution will thus flow between the inlet on arm 1 and the outlets on arms 2 and 3, selectively forming an SLB in the left-hand part of the device only. After the formation of the SLB and the removal of excess vesicles from the bulk solution by rinsing, valves connected to the tubing on arms 2 and 3 were closed. The SLB could now be moved into the empty arm, 4, by applying a bulk flow of buffer solution from the inlet of arm 1 to the outlet of arm 4. The bulk flow rate was controlled with a syringe pump (NE-1000, New Era Pump Systems Inc., Wantagh, NY) and was set to 200 μL/min in the experiments. Microscopy Setup. Fluorescently labeled SLB was studied with an inverted Nikon Eclipse Ti-E microscope (Nikon Corporation, Tokyo, Japan) using an Andor iXonþ EMCCD camera (Andor Technology, Belfast, Northern Ireland) and a 60 magnification (NA = 1.49) oil-immersion objective (Nikon Corporation). The acquired images consisted of 512 pixels  512 pixels with a pixel size of 0.38 μm  0.38 μm. To monitor the fluorescent R-DHPE, a mercury lamp connected to the microscope using an optical fiber (Intensilight C-HGFIE, Nikon Corporation) was used together with a TRITC filter cube (Semrock, Rochester, NY). A series of images were acquired using time-lapse acquisition with exposure times of 50-100 ms. All measurements were made at ambient temperature, ∼22 C. Photobleaching Experiments. Photobleaching was used to determine the diffusivity and drift velocity of R-DHPE molecules in an SLB with 1 wt % R-DHPE in total. A diode-pumped solidstate laser operating at 532 nm (BWN-532-100E; B&W Tek Inc., Newark, DE) was used to bleach the labeled lipids. A band of R-DHPE molecules was bleached across the width of the channel by moving the channel relative to the focused laser spot, in steps of 10 μm every 100 ms, using the xy stage of the microscope. Images of the bleached molecules were taken at 1 s intervals to monitor the diffusive and convective motion of the lipids in the SLB. The diffusivity of the bleached lipids was determined from the recovery of the bleached band of R-DHPE molecules. For this purpose, a procedure similar to that of the Hankel transform method, previously developed by us to evaluate the recovery of a circular bleach spot in the SLB,21 was followed. In the following description, it is assumed that the dark count value of the EMCCD camera has been subtracted from each image. First, a set of 10 images was acquired before photobleaching and averaged, yielding the prebleach intensity Ipre. A new intensity I2 was then defined as P ðIð1Þ=Ipre Þ IðiÞ ð25Þ I2 ðiÞ ¼ 1 P ðIðiÞ=Ipre Þ Ipre for each frame i after photobleaching, where the sum is over all pixels in the image (delineated by the sides of the channel). This assures that the total intensity from the SLB is constant for all image frames, which minimizes the effects of intensity drift and postbleaching in the analysis. The value of I(1)/Ipre was also divided by the average intensity of I(1)/Ipre at the edge of the image to ensure that I2 approached zero at the edge of the image. A Fourier transform of the intensity I2 along a line at the center of (21) Jonsson, P.; Jonsson, M. P.; Tegenfeldt, J. O.; Hook, F. Biophys. J. 2008, 95, 5334–5348.

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Article the channel floor, averaged over 40 pixels, was then made. The absolute value of the Fourier transform, denoted here as F, can be shown to be a function of the spatial frequency, k, and the diffusivity of the lipids, D, according to22,23 Fðk, tÞ ¼ Fðk, 0Þ expð - 4π2 k2 DtÞ

ð26Þ

The reason for using the absolute value of the Fourier transform is that a net translation of the system, due to convective motion, appears as a phase change in the Fourier transform,22 whereas the magnitude given by eq 26 is independent of the drift velocity as long as the velocity can be assumed to be constant over the area being studied. When fitting the experimental data, only values of k < 0.03 μm-1 were used because F is less accurate at higher values of k. This resulted in D = 2.30 ( 0.02 μm2/s, where the spread in data is the standard deviation obtained from four measurements. To track the position of the bleached band of R-DHPE molecules, the center-of-mass position of I2 was determined for each row in the image. First, the position of the maximum intensity was determined for each row, after which a 10th order polynomial was fitted to the line intensity (30 μm around the intensity maximum. Next, the two x positions of the polynomial fit at which the intensity had decreased to 50% of the maximum intensity were determined by interpolation, after which the position of the center of mass of the fitted curve between these two x positions was determined. This was repeated for all rows of the image within the channel.

Results and Discussion Velocity in the Bulk of the SLB. Figure 8 shows three fluorescence micrographs of an SLB labeled with 1 wt % R-DHPE molecules under a bulk flow of 200 μL/min in the channel. The flow is from left to right in the Figure. At t = 0, a band of R-DHPE molecules perpendicular to the flow direction is bleached using a laser. At subsequent times, the band of bleached lipids is seen to move in the direction of the bulk flow, with the highest velocity in the center of the microfluidic channel. At the same time, the band becomes less distinct because of the diffusion of the bleached molecules. The observation that the bleached lipids move fastest at the center of the channel is in agreement with eq 4, according to which the magnitude of the drift velocity should be proportional to the hydrodynamic shear force σhydro, which is largest in the center of the channel (Figure 3). The center-of-mass position of the bleached lipids, Δxcm, for each horizontal line in the image is given in Figure 9 at two different times after photobleaching. The corresponding results from the simulations of eq 14 in COMSOL Multiphysics are shown as solid lines. Values of the parameters used in the simulation, D = 2.3 μm2/s and b = 4  107 Pa s/m, were estimated from the experimental data. The convective motion of the bleached band was determined from eq 4 with the value of σhydro given by Figure 3. Note that despite the fact that σhydro tends to zero at the corners of the walls (Figure 3), the change in the position of the center of mass at z = (w/2 is not zero in Figure 9. The reason for this is the diffusion of molecules perpendicular to the net flow direction. Molecules near the corners can diffuse away, thus obtaining a higher drift velocity, and then diffuse back after having traveled a net distance in the x direction. However, diffusion does not affect the average velocity over the width of the channel, which was 0.31 μm/s for the data in Figure 9. The same value is obtained (22) Berk, D. A.; Yuan, F.; Leunig, M.; Jain, R. K. Biophys. J. 1993, 65, 2428– 2436. (23) Tsay, T. T.; Jacobson, K. A. Biophys. J. 1991, 60, 360–368.

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Figure 8. Fluorescence micrographs of bleached R-DHPE molecules under a bulk flow of 200 μL/min from left to right. The dashed lines show the center of mass of the bleached band at t = 0.

Figure 9. Simulated (solid lines) and experimental (dots) values of the center-of-mass position, Δxcm, of a bleached band of R-DHPE molecules in the SLB under a bulk flow of 200 μL/min. Δxcm = 0 at t = 0.

when using eq 4 with b = 4  107 Pa s/m and Æσhydroæ = 12.5 Pa from Figure 3. It should also be noted that the drift velocity in the center of the channel (z = 0) is only marginally affected by diffusion. Using the peak value of σhydro = 16.6 Pa from Figure 3 and b = 4  107 Pa s/m gives the theoretical distance traveled by the bleached R-DHPE molecules in the center of the channel, 60 s after photobleaching, of 25 μm. This value is in good agreement with the value of 24 to 25 μm observed in Figure 9. Behavior at the Front of the Advancing SLB. In contrast to the lipids in the bulk of the SLB, the lipids at the front of the bilayer seem to move at a constant velocity over almost the entire width of the channel (Figure 10A-C). This behavior cannot be explained by the expression in eq 4, which predicts that the drift velocity should be proportional to σhydro and would thus vary across the SLB. Instead, there seem to be internal forces within the SLB that make the lipid bilayer front move at a constant velocity across the width of the channel. Figure 11A shows the surface pressure in a simulation of the flow in the upper monolayer of the SLB using eq 6, in which the drift velocity was held constant over the entire bilayer front using the boundary condition at the front of the lipid bilayer given by eq 19. The streamlines were calculated from the surface pressure using eq 4. From the simulation in Figure 11A, it can be seen that the surface pressure approaches a constant value to the left in the simulated geometry. This indicates that in the bulk of the SLB the 1436 DOI: 10.1021/la103959w

Figure 10. (A-C) Fluorescence micrographs of the front of the

lipid bilayer at 100 s intervals under a bulk flow of 200 μL/min from left to right. The dashed lines show the position of the bilayer front at t = 0. (D) Line profile of the intensity through the center of the channel, converted into the surface fraction, φ, of R-DHPE. The SLB contains 0.2 wt % R-DHPE in total.

velocity of the lipids is mainly determined by the shear force from the hydrodynamic bulk flow and the frictional force between the two monolayers of the SLB (eq 4). However, the gradients in surface pressure near the front of the lipid bilayer are important in achieving a constant velocity over the entire bilayer front (Figure 11A). The increase in surface pressure in the center of the channel has the effect of forcing the lipids out of this region toward the sides of the channel. The magnitude of the surface pressure at the bilayer front is of the same order as the adhesion energy, Wa, of the SLB.24,25 In fact, for self-spreading lipid bilayers Wa is the driving force of the SLB.25,26 However, in the case of self-spreading lipid bilayers, there is a lipid reservoir from which the spreading takes place and where the surface pressure is Wa higher than at the bilayer front. In the systems described here, there is no lipid reservoir because the entire SLB is already adsorbed onto the supporting surface. As the front of the SLB advances, the rear of the SLB simultaneously retreats, conserving the total area of the SLB. Hence, (24) Anderson, T. H.; Min, Y. J.; Weirich, K. L.; Zeng, H. B.; Fygenson, D.; Israelachvili, J. N. Langmuir 2009, 25, 6997–7005. (25) Radler, J.; Strey, H.; Sackmann, E. Langmuir 1995, 11, 4539–4548. (26) Nissen, J.; Gritsch, S.; Wiegand, G.; Radler, J. O. Eur. Phys. J. B 1999, 10, 335–344.

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Figure 11. Simulated values of the surface pressure, Π, near the front of the SLB for a system without obstacles and where (A) the velocity is constant over the bilayer front and (B) the surface pressure at the bilayer front is Π = 10-7 Pa m, resulting from the curvature of the bilayer front. The streamlines are given by the velocity profile of the lipids in the upper monolayer of the SLB.

assuming that the adhesion energy of the SLB is the same on all parts of the surface, this effect should not result in a driving force on the bilayer front. Furthermore, the magnitude of the surface pressure at the bilayer front is 3 orders of magnitude larger than that accounted for by the curvature of the bilayer front. This can be explicitly seen in Figure 11B, where the surface pressure and velocity of the lipids have been simulated without fulfilling the constant-velocity condition at the bilayer front but where the surface pressure at the front of the lipid bilayer has instead been set to the value resulting from the curvature of the bilayer front, Π = 10-7 Pa m. This results in hardly any change in the actual flow profile in the SLB (streamlines in Figure 11B), suggesting that there must be some additional braking force at or close to the front of the lipid bilayer that gives rise to this behavior. One possible source of the positive surface pressure at the front of the lipid bilayer is molecules adsorbed onto the surface in front of the SLB. In this case, the bilayer front will act as a plow pushing the molecules in front of the lipid bilayer edge. However, it is not clear whether such an effect could give rise to the surface pressure shown in Figure 11A, which is needed to obtain a constant velocity at the front of the lipid bilayer. An exhaustive study of different factors that could cause the pressure increase at the front of the lipid bilayer was beyond the scope of this work. However, one effect that can significantly alter the flow profile near the front of the SLB is the accumulation of molecules at the front of the lipid bilayer. Effect of Molecules Accumulating at the Front of the Lipid Bilayer. In Figure 10, it can be seen that the fluorescence intensity increases near the front of the lipid bilayer, corresponding to an accumulation of R-DHPE molecules. This is transformed into a surface fraction, φ, of R-DHPE molecules in the upper monolayer of the SLB in Figure 10D. The surface fraction of R-DHPE molecules increases by roughly a factor of 10 near the front of the SLB. The reason for this increase is that the edge of the lipid bilayer seems to act as a sieve through which the R-DHPE molecules are reluctant to pass.4 The passage of R-DHPE molecules over the edge of the lipid bilayer into the lower monolayer of the SLB is also expected to be energetically unfavorable because R-DHPE molecules, with their relatively large headgroup and net negative charge, are known to be preferentially located in the upper monolayer of the SLB.4,18 A similar accumulation of molecules Langmuir 2011, 27(4), 1430–1439

at the front of the lipid bilayer was also observed when hydrodynamic forces were used to drive an SLB with fluorescently labeled protein molecules bound to the SLB via receptor molecules in the lipid bilayer.7 Although laterally mobile, the accumulated molecules will have a different velocity from that of the surrounding lipids. As a consequence, they will act as obstacles that perturb the flow profile of the lipids in the SLB. Figure 12A shows the results of a simulation of the flow around an obstacle at the center of a hexagonal unit cell in the SLB. The size of each obstacle was set to 0.63 nm2, roughly corresponding to the cross-sectional area of an R-DHPE molecule,17 and the surface fraction of obstacles was set to φ = 1%. The perturbed flow profile and surface pressure around an obstacle will, in turn, result in a net force on the lipids in the SLB, depending on the velocity difference between the lipids and the obstacles (eq 7). Simulated values of λob at different surface fractions, φ, of obstacles in the SLB are given in Figure 12B. The drag coefficient is approximately constant for surface fractions smaller than 0.1% and approaches a value of λob = 1.4  10-9 Pa m s for small φ. From eq 8, we find that this would correspond to a diffusivity of Dob = 2.9 μm2/s, which is comparable to values of the diffusivity of R-DHPE molecules in a phosphatidylcholine SLB obtained by us5,21 and others27. At higher surface fractions, the value of λob increases. The reason for this is that neighboring obstacles also affect the flow profile around each obstacle when the obstacles are closer together. The accumulated molecules will also affect the macroscopic flow profile and surface pressure in the SLB. Figure 13 shows the results of a simulation similar to that in Figure 11B but including the effect of R-DHPE molecules accumulated at the front of the lipid bilayer. The number of accumulated R-DHPE molecules, φ, was estimated from the fluorescence intensity of the R-DHPE molecules in Figure 10. The R-DHPE molecules accumulated at the front of the lipid bilayer have the effect of directing the flow of lipids out toward the sides of the SLB, in contrast to the situation in Figure 11B where no obstacles are present in the SLB. In fact, the velocity of the bilayer front for the simulation in Figure 13 is considerably closer to a constant velocity across the width of the channel than is the (27) Zhang, L. F.; Granick, S. J. Chem. Phys. 2005, 123, 211104.

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Figure 12. (A) Surface pressure, Π, and streamlines of the velocity, v, around an obstacle in the SLB. (B) Drag coefficient, λob, as a function of the surface fraction, φ.

Figure 13. Surface pressure, Π, and streamlines of the velocity for an SLB with R-DHPE molecules accumulated at the front of the lipid bilayer. (For φ, see Figure 10.) The boundary condition at the front of the lipid bilayer was Π = 10-7 Pa m.

Figure 14. Velocity of the bilayer front, vfront, according to eq 17 for the simulation in Figure 11B (dashed curve) and the simulation in Figure 13 (solid curve).

corresponding velocity for the simulation in Figure 11B (Figure 14). It can also be observed that the surface pressure arising from the obstacles is of the same magnitude as the surface pressure required for a constant drift velocity of the entire bilayer front for a system without obstacles, as shown in Figure 11A. The simulated velocity profile at the front of the lipid bilayer will depend on factors such as the surface fraction of accumulated molecules at the front of the SLB and the curvature of the bilayer front. The general observation is that increasing the surface fraction of accumulated molecules at the center of the channel causes the velocity profile of the bilayer front to increase on the sides and decrease in the center of the channel (data not shown). The same trend was observed when increasing the curvature of the bilayer front. A high curvature generally leads to an increase in the velocity of the bilayer front on the sides of the channel. 1438 DOI: 10.1021/la103959w

The results in Figures 13 and 14 show that the accumulation of R-DHPE molecules, or other molecules at the front of the lipid bilayer, will have a significant effect on the behavior of the lipid bilayer at the front of the advancing SLB. It should also be noted that here the concentration of R-DHPE molecules near the front of the lipid bilayer was used as a measure of the number of obstacles in the upper monolayer of the SLB. However, it cannot be ruled out that contaminants in the SLB or contaminants in the microfluidic channel could be accumulated at the front of the lipid bilayer, affecting the profile of the lipid bilayer front. Thus, even for an SLB without any R-DHPE molecules, there may still be molecules in the SLB that accumulate at the front of the lipid bilayer and affect the motion of the SLB. This may be one explanation of why the lipid bilayer front in previous work also seems to move at a constant drift velocity when the SLB is labeled with only NBD C12-HPC molecules, which do not accumulate at the front of the lipid bilayer.4

Conclusions The hydrodynamic shear force from a flowing liquid above an SLB can be used to drive the lipid bilayer in the direction of the flow. The motion of the bilayer can be controlled by modulating the magnitude and direction of the bulk flow. This provides a controlled way of handling and transporting molecules solubilized in or associated with SLBs. In this work, a theoretical platform was developed that can be used to describe the motion of the lipid bilayer under these hydrodynamic shear forces. Experimental observations and theoretical simulations show that the drift velocity of the lipids far away from the front of the SLB is directly proportional to the hydrodynamic shear force arising from the bulk liquid flowing in the microfluidic channel. Because the hydrodynamic shear force is highest in the center of the channel, this means that the lipids move fastest at the center of Langmuir 2011, 27(4), 1430–1439

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the channel. However, it is also experimentally observed that the front of the advancing lipid bilayer moves with a constant drift velocity over the entire width of the channel. To explain the difference in motion near the front of the lipid bilayer and in the bulk of the SLB, the effect of surface gradients in the lipid bilayer must be included. It was found that for the bilayer front to have a constant velocity across the channel there should be an increase in surface pressure on the order of ∼10-4 Pa m near the center of the bilayer front, which forces the lipids in the SLB toward the sides of the channel. This local increase in surface pressure may have different origins. However, an effect that cannot be neglected is the accumulation of molecules such as R-DHPE or membrane-bound proteins at the front of the advancing SLB. The accumulated molecules

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reside in the upper monolayer of the SLB and act as obstacles for the lipid molecules, which affects not only the local flow profile around each obstacle but also the macroscopic flow profile of the lipids in the SLB. Simulations showed that at a surface coverage of ∼1 wt % of accumulated R-DHPE molecules the velocity of the bilayer front approaches a constant velocity over the width of the channel, as observed experimentally. Acknowledgment. This work was financially supported by the Swedish Research Council for Engineering Sciences (contract number 2005-3140), an INGVAR grant from the Swedish Strategic Research Foundation, and a research grant (M€arta and Erik Holmberg’s donation) from the Royal Physiographic Society in Lund.

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