J. R. Soc. Interface (2011) 8, 127–143

*Author for c

doi:10.1098/rsif.2010.0200Published online 11 June 2010

Received 3 AAccepted 5 M

Concentration dependence oflipopolymer self-diffusion in supported

bilayer membranesHuai-Ying Zhang and Reghan J. Hill*

Department of Chemical Engineering, McGill University, Montreal, Quebec,Canada H3A 2B2

Self-diffusion coefficients of poly(ethylene glycol)2k-derivatized lipids (DSPE-PEG2k-CF) inglass-supported DOPC phospholipid bilayers are ascertained from quantitative fluorescencerecovery after photobleaching (FRAP). We developed a first-order reaction–diffusionmodel to ascertain the bleaching constant, mobile fraction and lipopolymer self-diffusion coef-ficient Ds at concentrations in the range c � 0.5–5 mol%. In contrast to control experimentswith 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(7-nitro-2-1,3-benzoxadiazol-4-yl)(ammonium salt) (DOPE-NBD) in 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), thelipopolymer self-diffusion coefficient decreases monotonically with increasing concentration,without a distinguishing mushroom-to-brush transition. Our data yield a correlation Ds ¼

D0/(1 þ ac), where D0 � 3.36 mm2 s21 and a � 0.56 (with c expressed as a mole percent).Interpreting the dilute limit with the Scalettar–Abney–Owicki statistical mechanicaltheory for transmembrane proteins yields an effective disc radius ae � 2.41 nm. On theother hand, the Bussell–Koch–Hammer theory, which includes hydrodynamic interactions,yields ae � 2.92 nm. As expected, both measures are smaller than the Flory radius of the2 kDa poly(ethylene glycol) (PEG) chains, RF � 3.83 nm, and significantly larger than thenominal radius of the phospholipid heads, al � 0.46 nm. The diffusion coefficient at infinitedilution D0 was interpreted using the Evans–Sackmann theory, furnishing an inter-leafletfrictional drag coefficient bs � 1.33 � 108 N s m23. Our results suggest that lipopolymer inter-actions are dominated by the excluded volume of the PEG-chain segments, with frictionaldrag dominated by the two-dimensional bilayer hydrodynamics.

Keywords: lipopolymers; phospholipid bilayer membranes; self-diffusioncoefficient; fluorescence recovery after photobleaching

1. INTRODUCTION

The well-defined and easily controlled composition ofreconstituted phospholipid bilayers makes them excel-lent models for studying cellular membranes. Lateraldiffusion is of particular interest because it plays animportant role in cellular processes such as cell signal-ling [1] and adhesion [2,3]. Consequently, there hasbeen a concerted effort in the literature to understandthe connection between lateral diffusion and membranephysics. A detailed discussion of the functional role oflateral diffusion is found in a review by Saxton [4].

Saffman & Delbruck [5] first modelled lateral diffu-sion of a single molecule in membranes usingcontinuum hydrodynamics. The Saffman–Delbruckequation successfully describes transmembrane proteindiffusion at vanishingly small concentrations [6]. Atfinite concentrations, however, the diffusion coefficientis ostensibly concentration dependent [6]. This concen-tration dependence is due to non-specific protein–protein interactions, including direct (thermodynamic)and hydrodynamic interactions [7,8].

orrespondence (reghan.hill@mcgill.ca).

pril 2009ay 2010 127

In interacting systems, self-diffusion and gradientdiffusion must be distinguished. Self-diffusion—alsotermed tracer diffusion—quantifies the mean-squareddisplacement of a single molecule. In two dimensions,the self-diffusion coefficient Ds is defined by kr2l ¼4DsDt, where kr2l is the mean-squared displacement ina time interval Dt under Brownian forces. Gradient dif-fusion—also termed mutual diffusion—is themacroscopic flux of particles due to Brownian forcesand a concentration gradient. The gradient diffusioncoefficient Dg appears in Fick’s law [9].

Theoretical modelling of self- and gradient diffusionis extensive. Direct interactions, including hard-corerepulsion and soft interactions, were analysed byScalettar et al. [7] and Abney et al. [8,10] using the stat-istical mechanical theory of fluids. The influence ofexcluded volume on self-diffusion from the continuumapproach is in close agreement with the lattice modelsof Pink [11] and Saxton [12]. In addition to thermodyn-amic interactions, Bussell et al. [13–15] studiedhydrodynamic interactions, achieving better agreementwith the measured concentration dependence of proteinself-diffusion coefficients.

This journal is q 2010 The Royal Society

128 Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill

Recently, Deverall et al. [16] experimentally observedself-diffusion of lipids and proteins obstructed in lipopoly-mer-containing bilayers. The obstacles were hydrophobiclipopolymers (lipid-mimicking dioctadecylamine moi-eties) in the bottom leaflet of model membranes, withbulk lipids (without polymer attached) in the upper leaf-let. Theoretical interpretation of the results suggestedthat lipopolymers act as immobile obstacles to lipid andprotein diffusion. Soong and coworkers, however, haveshown (using NMR spectroscopy) that the mobility ofpoly(ethylene glycol) (PEG)-lipids in magneticallyaligned bicelles decreases with increasing lipopolymerconcentration [17]. Most recently, Albertorio et al. [18]studied mobile PEG-lipids in planar supported lipidbilayers (SLBs). Fluorescence recovery after photobleach-ing (FRAP) experiments revealed a lipopolymerself-diffusion coefficient that decreases with increasingconcentration, but only at concentrations above themushroom-to-brush transition.

In contrast to transmembrane protein diffusion, theor-etical understanding of lipopolymer diffusion is poor, andfew systematic experimental studies have been under-taken. Such studies may help to understand thedynamics of phospholipid-anchored membrane proteinsthat bear large head groups, which are known to obstructmembrane fluidity [19]. Such knowledge would also guidethe engineering of technological devices based on lipopoly-mer-grafted bilayers. For example, phospholipids bearingPEG chains can be used to prolong vesicle circulationtime in drug delivery [20], enhance bio-compatibility onthe surfaces of implantable materials [21], provide airand fluid stability of bilayers used in microfluidic diagnos-tics [18,22], eliminate bilayer–substrate interactions thatimmobilize proteins in solid supported bilayers [23], andmimic glycocalyx in cell adhesion [24] and ligand–recep-tor binding [25].

FRAP is the most widely adopted technique formeasuring self-diffusion coefficients, mainly because itcan be conducted using readily available confocalmicroscopes. Other techniques, such as fluorescence cor-relation spectroscopy (FCS) and single particle tracking(SPT), require more specialized instrumentation [26].FRAP is also used for studying transport in cellularmembranes [27–29], protein dynamics [30–33],mRNA mobility [34], signal transduction [35,36] andin vivo cellular binding [37–39].

Qualitative assessment of FRAP is often simple andstraightforward, but quantitative interpretation requiresspecial care. For example, failing to meet the requirementsfor accurate FRAP has hindered understanding dynamicsin biological systems, and has even lead to erroneousresults (see [39–41], for examples). Accordingly, mucheffort has been devoted to improve FRAP accuracy. Forexample, FRAP fitting models have been extended froma uniformly bleached circle [27,42] to a Gaussian bleachedspot [27,43], a uniformly bleached rectangle [44] and anarbitrary geometry with an arbitrary initial profile [45].Moreover, violating the assumption of an infinite fluoro-chrome reservoir when working with cells has beencorrected by adopting a finite sized reservoir in the fittingmodel [38,39,44]. Other sources of uncertainty, such asfinite bleaching time, cell movement and bleaching rever-sibility, have been addressed to various extents [38,46].

J. R. Soc. Interface (2011)

One important issue—the intensity loss during acqui-sition—is not fully resolved. Acquisition bleaching mustbe distinguished from intentional FRAP bleaching, andshould be minimized by carefully choosing experimentalparameters, such as imaging time and illuminationintensity.

Experimental parameters must also achieve an accepta-ble signal-to-noise ratio. This makes acquisition bleachingdifficult to avoid and, thus, necessitates correcting rawdata [46]. Several corrective methodologies have beenused in the literature, but, according to Mueller et al.[39], none are completely satisfactory. Indeed, Muelleret al. [39] recently devised new approach, which has beentheoretically corroborated [47], to correct acquisitionbleaching. Nevertheless, the authors still find acquisitionbleaching artefacts in the apparent binding rates ofsite-specific transcription factors to chromatin [39].

To address the foregoing gaps in the literature, weundertook a systematic study of the concentrationdependence of the self-diffusion coefficient of lipopoly-mer DSPE-PEG2k-CF (1,2-distearoyl-sn-glycero-3-phosphoethanolamine-N-[poly(ethylene glycol) 2000-N0-carboxyfluorescein) in glass-supported DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine) bilayers. To rig-orously analyse FRAP, we developed a reaction–diffusion FRAP model that accounts for acquisitionbleaching and a finite fluorochrome mobile fraction.This furnished a correlation that quantifies how lipopo-lymer interactions hinder self-diffusion. Interestingly,our experiments do not reveal a distinct mushroom-to-brush transition. Rather, we observe a smooth tran-sition from the dilute to semi-dilute regimes, echoingthe smooth variation in spreading pressure [48]. Note-worthy is that theoretical interpretation of our dataat small, but finite, lipopolymer concentrations fur-nishes reasonable PEG-chain dimensions and otherphysical characteristics of the lipids. Overall, dragforces are dominated by the two-dimensional hydrodyn-amics of the lipid tails in their respective leaflets, withinteractions in the dilute limit dominated by theexcluded-volume of the PEG-chains in random coilconfigurations with weak hydrodynamic interactions.

The paper is set out as follows. Section 2 describesbilayer synthesis and data collection. In §3, we set outthe FRAP model for handling acquisition bleachingand an immobile fraction. Section 4.1 uses simulatedFRAP data to test existing methodologies for correctingacquisition bleaching. We apply the FRAP model in§4.2 to extract from the experiments the self-diffusioncoefficient, mobile fraction and bleaching constant.The concentration dependence of the self-diffusion coef-ficient is examined in §4.3. We compare our results withthe available literature (§4.4) and theoretically inter-pret these data to furnish PEG-chain dimensions(§4.5) and inter-leaflet friction coefficient (§4.6).Section 5 provides a concluding summary.

2. EXPERIMENTAL

2.1. Bilayer synthesis

SLBs were prepared by vesicle fusion following litera-ture procedures [49] with minor modifications. First, a

Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill 129

mixture of lipids containing 2 mg 1, 2-dioleoyl-sn-gly-cero-3-phosphocholine (DOPC, Avanti PolarLipids, Alabaster, AL, USA) and a desired concen-tration of lipopolymer 1,2-distearoyl-sn-glycero-3-phosphoethanolamine-N-[poly(ethylene glycol)2000-N0-carboxyfluorescein] (DSPE-PEG2k-CF, AvantiPolar Lipids, Alabaster, AL, USA) in chloroform wasdried under a stream of nitrogen gas, followed bydesiccation under vacuum for 2 h before reconstitutingin buffer (10 mM phosphate, 100 mM NaCl, pH 7.4) to2 mg ml21. The lipid mixture was extruded 20 timesthrough a 100 nm polycarbonate membrane, andthen another 20 times through a 50 nm polycarbonatemembrane (Avanti Polar Lipids, Alabaster, AL, USA)to form small unilamellar vesicles (SUVs). The SUVswere deposited on pre-cleaned glass cover-slips(VWR, Ville Mont-Royal, QC, Canada) to form lipidbilayers by vesicle fusion. The cover-slips were firstboiled in 7X solution (MP Biomedical, Solon, OH,USA) for 30 min, rinsed excessively with reverse osmo-sis (RO) water, dried under a stream of nitrogen gas,and further cleaned by piranha etching for 20 min ina solution of 3 : 1 (v/v) concentrated sulphuric acid(H2SO4) and 30 per cent hydrogen peroxide (H2O2).The cover-slips were then rinsed excessively with ROwater, dried under a stream of nitrogen gas and usedimmediately. Control experiments were performedwith 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(7-nitro-2-1,3-benzoxadiazol-4-yl) (ammonium salt)(DOPE-NBD) (Avanti Polar Lipids, Alabaster, AL,USA) in DOPC. All chemicals, except where otherwisestated, were used as purchased (Sigma Aldrich,Oakville, Ontario, Canada).

2.2. Fluorescence intensity and fluorochromeconcentration

FRAP interpretation is based on an assumption thatfluorescence intensity is proportional to fluorochromeconcentration. This is valid only when the fluorochromeconcentration is sufficiently low. Otherwise, photons aresubject to quenching by multiple absorption and emis-sion [50]. Generally, the fluorescence intensity can beexpressed as I ¼ QS0 (1 2e2A), where S0 is the illumi-nation intensity, Q is the fluorochrome quantum yieldand A is the absorbance [51]. The amount of lightabsorbed by the fluorphores is proportional to their con-centration, so A ¼ kc, where k is a constant and c is theconcentration. Accordingly, I ¼ QS0 (1 2 e2kc) � QS0

kc when kc�1.Epifluorescence images of DOPC bilayers doped

with c � 0, 0.5, 1, 2, 3 and 5 mol% DSPE-PEG2k-CF are shown in figure 1a. These wereobtained using a Nikon 2000-U inverted microscopewith a 10 � /0.25 air-immersion objective. Thespecimen was illuminated with a mercury lamp fil-tered through two neutral filters (ND8 and ND4).Fluorescence was imaged with a green filter(C82465, Chroma Technology, Brattleboro, VT,USA) and a digital CCD camera (CoolSNAP ES,Photometrics, Tucson, AZ, USA). Intensity in thecentral circular region with radius 8.5 mm of eachimage was collected, and intensity in the image

J. R. Soc. Interface (2011)

for c ¼ 0 mol% was subtracted as a backgroundlevel for all images of bilayers containing DSPE-PEG2k-CF. In figure 1b, an exponential fit givesI/I0 � 1 2 e20.19c (solid line), suggesting that alinear approximation is acceptable when c �5 mol%. Similar measurements for DOPE-NBD inDOPC bilayers with the same filters give I/I0 �1 2 e20.11c. Based on these data, subsequent self-dif-fusion coefficient measurements were limited toconcentrations c�5 mol%.

2.3. FRAP data collection

Bilayers were imaged with a Zeiss LSM510 confocallaser scanning microscope using a 63 � /1.4 oil-immer-sion objective and a 488 nm argon ion laser (25 mW)with intensities in the range 0.1—0.5%. The imagesize was 68 � 68 mm with resolution 256 � 256 pixels.FRAP spots (circles with radius r0 � 5.13 mm) wereproduced using laser intensities in the range 10–100%. Two images were captured immediately beforeFRAP bleaching to ascertain the initial average inten-sity I1. To minimize the FRAP bleaching time, asingle bleach of approximately 45 ms duration wasadopted. The scan time to acquire the first post-bleach-ing image is approximately 500 ms if the entire image isscanned. However, this provides an initial conditionthat is closer to a Gaussian profile, which is problematicwhen fitting FRAP models for uniformly bleached cir-cles [40,52]. Thus, for the diffusion coefficientsreported in §4.3, only the FRAP spot and a circularreference spot were imaged (e.g. figure 5). This achievesa threefold reduction of the imaging time to approxi-mately 150 ms, there by achieving a nearly uniforminitial bleach.

2.4. FRAP data analysis

Raw images (16 bits) were transferred to tag imagefile format (TIFF), which guarantees lossless com-pression, using the IMAGEJ software (W. Rasband,National Institutes of Health, Bethesda, MD, USA),with quantitative analysis undertaken using Matlab(The MathWorks, Natick, MA, USA). Numerical sol-utions of a cylindrically symmetric reaction–diffusionmodel were fitted to the FRAP time series to obtainthe self-diffusion coefficient Ds, mobile fraction fmand a photobleaching rate constant k. Details ofthe reaction–diffusion model are provided in §4.3.Fitting the model to experimental data was under-taken using the Matlab function ‘lsqcurvefit’.Further details of the fitting procedure are providedin §4.2.

3. REACTION–DIFFUSION FRAP MODEL

We model fluorescence intensity by defining n as theconcentration (number per unit area) of active fluoro-chromes in the bilayer. Assuming the fluorescenceintensity I is proportional to n, the intensity of auniform, uniformly illuminated bilayer satisfies

@I@t¼ �kI ; ð3:1Þ

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Figure 1. (a) Epifluorescence images (592 � 442 mm) of DOPC bilayers with DSPE-PEG2k-CF at concentrations c ¼ 0, 0.5, 1, 2,3 and 5 mol%. (b) Normalized intensity I/I0 in the central circular regions with radius 8.5 mm versus DSPE-PEG2k-CF concen-tration c with fit I/I0 � 1 2 e20.19c (solid line). Error bars are the standard deviation from three replicates.

130 Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill

where k is the photobleaching rate constant. This yieldsan exponential decay in intensity, with a time constanttk ¼ k21. Although the photobleaching mechanism ismuch more complicated [53], the first-order model faith-fully captures the photobleaching kinetics in ourexperiments. For example, time series for c � 1 mol%fluorescent lipopolymer DSPE-PEG2k-CF in DOPCbilayers with uniform illumination at two laser intensi-ties are shown in figure 2. Fitting exponential decays tothese data furnishes tk � 6.3 and 18.5 s with high- andlow-power illumination, respectively.

We approximate the bilayer as a large circle withradius r2 and uniform initial flurochrome concentrationn1. A smaller concentric circle with radius r1 can beimaged, with FRAP bleaching within an even smallerconcentric circle—termed the FRAP spot—withradius r0. The FRAP spot is uniformly bleached to con-centration n0.

For a bilayer with mobile fraction fm, the reaction–diffusion equation governing the concentration ofmobile fluorochromes is

@nm

@t¼ Dsr2nm � kðrÞnm; ð3:2Þ

where nm is the mobile fluorochrome concentration andDs is the self-diffusion coefficient. The bleaching rate kis a constant when r � r1 and k(r) ¼ 0 in the regionr . r1 where the bilayer is not imaged.

With cylindrical symmetry, equation (3.2) is easilysolved numerically using the Matlab function ‘pdepe’.Calculations furnishing the spatial and temporal

J. R. Soc. Interface (2011)

evolution nm(r, t) were undertaken on a domain withradial distances 0 , r � r2. The initial conditions are

nmðr; 0Þ ¼ n0fm; 0 , r � r0 ð3:3Þ

and

nmðr; 0Þ ¼ n1fm; r0 , r � r2; ð3:4Þ

with zero-flux boundary conditions @nm/@r ¼ 0 at r ¼ 0and r2; and continuous radial diffusion fluxes 2Ds@nm/@r at r ¼ r0 and r1. Note that situations where only theFRAP spot is imaged correspond to setting r1 ¼ r0.

The concentration of immobile fluorochromes is

niðr; tÞ ¼ niðr; 0Þe�kt ð3:5Þ

with initial conditions

niðr; 0Þ ¼ n0ð1� fmÞ; 0 , r � r0 ð3:6Þ

and

niðr; 0Þ ¼ n1ð1� fmÞ; r0 , r � r2: ð3:7Þ

Note that ni(r, t) ¼ n1(1 2 fm) when r1 , r � r2,since k(r) ¼ 0 when r1 , r � r2.

It is expedient to scale the concentrations with n1,radial position with r0 and time with tD ¼ r0

2/(4Ds).The dimensionless reaction–diffusion equation for themobile fluorochromes then presents a single dimension-less parameter

e ¼ ktD ¼kr2

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ð4DsÞ: ð3:8Þ

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Figure 2. First-order kinetics of photobleaching. (a) Representative confocal laser scanning images of a circular domain (radius102 mm) with c � 1 mol% fluorescent lipopolymer DSPE-PEG2k-CF in a DOPC bilayer. (b) Circles and squares are averageintensities in a small concentric circle (radius 4 mm) within the images illuminated using low- and high-power laser intensities,respectively. Solid lines are exponential fits (first-order kinetics) giving decay constants tk ¼ k21 � 6.3 and 18.5 s.

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Figure 3. FRAP-spot intensity time series with e ¼ ktD ¼

0.03, fm ¼ 0.7, n0/n1 ¼ 0.4, r1/r0 ¼ 2 and r2/r0 ¼ 100. Thesolid line is the average intensity of fluorochromes in theFRAP spot, and the dashed and dash-dotted lines are,respectively, the contributions from the mobile and immobilefractions.

Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill 131

The other independent dimensionless parameters,which arise from the initial and boundary conditions,are the mobile fraction fm, initial bleaching intensityn0/n1, imaging size r1/r0 and boundary size r2/r0 �r1/r0.

A time series of the FRAP intensity I, defined as theaverage intensity within the FRAP spot, i.e.

I ¼Ð r0

0 2prnðrÞ drÐ r0

0 2pr dr; ð3:9Þ

is shown in figure 3. Here, the intensity of mobilefluorochromes (dashed line) initially increases dueto recovery by diffusion, and later decreases dueto bleaching. The intensity of immobile fluoro-chromes (dash-dotted line) vanishes exponentially,while the total intensity (solid line) plateaus to avalue where the rate of photobleaching is balancedby diffusive restoration from the surroundingbilayer.

The accuracy of our numerical calculations is ver-ified, in part, by Soumpasis’s analytical solution (seefigure 8 in appendix A) for bilayers without animmobile fraction or acquisition photobleaching.Soumpasis’s solution of the diffusion equation in anunbounded domain gives a FRAP intensity

I ¼ I1 � ðI1 � I0Þ

� 1� e�2tD=t J02tD

t

� �þ J1

2tD

t

� �� �� �; ð3:10Þ

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where I1 is the intensity after full recovery, I0 is theintensity at t ¼ 0, i.e. immediately following FRAPbleaching, and J0 and J1 are the zeroth- and first-order modified Bessel functions of the first kind [42].

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Figure 4. Correcting acquisition photobleaching with partial imaging (r1 , r2): e ¼ 0.05, fm ¼ 1, n0/n1 ¼ 0.2 and r2/r0 ¼ 100.Solid lines are the bleaching curve I/I1 ¼ e2kt. (a) Correction according to equation (4.2): Soumpasis’s analytical solution(dash-dotted line); numerical FRAP time series for r1/r0 ¼ 1, 3, 5 and 10 (dashed lines, top to bottom); corrected FRAPtime series for r1/r0 ¼ 1, 3, 5 and 10 (circles, top to bottom). (b) Phair et al.’s method of collecting reference data: referencetime series with r1/r0 ¼ 2, 5, 10 and 50 (dashed lines, top to bottom). (c) Mueller et al.’s method of collecting reference data:reference time series with r1/r0 ¼ 1, 2, 4 and 8 (dashed lines, top to bottom).

132 Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill

4. RESULTS AND DISCUSSION

4.1. Correcting FRAP acquisitionphotobleaching

Acquisition photobleaching in FRAP is often correctedusing reference data via [46,54]

Ic ¼If

Ir; ð4:1Þ

where If is the observed FRAP data, Ic is the correctedFRAP data (e.g. to be fitted to Soumpasis’s theory)and Ir is the so-called reference data. Recently, it wasrigorously proven [39,47] that acquisition photobleach-ing can be corrected via

Ic ¼ Ifekt ð4:2Þ

when photobleaching kinetics are first order. Thissuggests that the conventional method, as presentedby equation (4.1), is correct if the reference intensitydecays exponentially, i.e. Ir ¼ e2kt.

It is customary to use intensity averaged over theentire FRAP image as reference data [46]. We willrefer to this as Phair et al.’s method. Recently, however,Mueller et al. captured a new time series following con-ventional FRAP, using the same FRAP collectionparameters but without FRAP bleaching. They gener-ated reference data from this new time series at thesame location and in exactly the same way as forFRAP [39]. We will refer to this as Mueller et al.’smethod. Note that the reference time series in Muelleret al.’s method is essentially continuous fluorescencemicro-photolysis (CFM). This is a less popular tech-nique that uses photobleaching to measure self-diffusion coefficients [55]. In CFM, the bilayer is con-tinuously illuminated at medium laser intensity, andthe CFM-spot intensity indicates the rate of photo-bleaching and the diffusive restoration from thesurrounding bilayer.

Using our reaction–diffusion FRAP model, weexamined the utility of equation (4.2) for correcting

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acquisition bleaching. Calculations were performedwith a dimensionless bleaching constant e ¼ ktD ¼

0.05 and mobile fraction fm ¼ 1, so perfectly correcteddata would reproduce Soumpasis’s solution. We testedconditions under which Phair et al.’s and Muelleret al.’s methods for collecting reference data can beused to ascertain the bleaching constant. These testscompare the respective reference time series with thebleaching curve I/I1 ¼ e2kt. To model reference dataaccording to Mueller et al.’s method, the geometry ofour reaction–diffusion FRAP model is modified tosimulate a CFM experiment (see appendix B fordetails).

Simulations with partial imaging of the reservoir, i.e.r1/r0 , r2/r0 ¼ 100, are shown in figure 4. Note thatsolid lines, shown for convenient reference, are thebleaching curve I/I1 ¼ e2kt. Figure 4a shows FRAPtime series for various image sizes r1/r0 ¼ 1, 3, 5 and10 (dashed lines, top to bottom), and the accompanyingcorrected time series (circles, top to bottom) accordingto equation (4.2). The corrections agree with Soumpa-sis’s solution (dash-dotted line) when the image radiusr1/r0�10; otherwise there is ostensible over-correction.Figure 4b shows reference time series collected accord-ing to Phair et al.’s method with r1/r0 ¼ 2, 5, 10 and50 (dashed lines, top to bottom). As expected, the refer-ence time series approach the bleaching curve (solidline) when r1/r0� 1. Figure 4c shows reference datacollected using Mueller et al.’s method with r1/r0 ¼ 1,2, 4 and 8 (dashed lines, top to bottom). Again, thereference time series approach the bleaching curve(solid line) when r1/r0� 1, but the method achievescloser correspondence to the bleaching curve withsmaller values of r1/r0 than with Phair et al.’s method.

Another situation of interest is when the entire reser-voir is imaged, i.e. r1 ¼ r2. This is often the case whenimaging cells [39,46] or finely patterned bilayers [56].Again, we find that acquisition bleaching can be accu-rately corrected using equation (4.2) when r2/r0 ¼ r1/r0 � 10. Moreover, both Phair et al.’s and Muelleret al.’s methods can be adopted to collect reference

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Figure 5. (a) Laser scanning confocal microscopy of FRAP (radius r0 � 5.13 mm, upper right) and CFM (radius r1 � 2.7r0, lowerleft) spots. (b) FRAP time series (circles), reference intensity time series (triangles), and an exponential fit (solid line). (c) Least-squares fit of Soumpasis’s solution to the raw (solid line, circles) and corrected (dashed line, squares) FRAP time series. (d) Least-squares fit of the reaction–diffusion FRAP model (solid line) to the raw FRAP time series (circles) furnishing Ds � 1.05 mm2 s21,k � 0.0013 s21 and fm � 0.98 with x2 � 0.02.

Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill 133

data even when r2/r0 ¼ r1/r0 , 10 (see figure 11 inappendix C).

To accurately obtain the bleaching constant, refer-ence time series must decay exponentially and, thus,overlap the bleaching curve. This requires diffusion tobe absent during the collection of reference data.When the entire reservoir is imaged, Mueller et al.’smethod can accurately reproduce the bleaching con-stant (reference curve overlaps exactly with thebleaching curve) for all image sizes, because bleachingoccurs uniformly over the entire reservoir. In this case,only bleaching dynamics can be resolved in a CFM pro-cess (see figure 10a in appendix B). Furthermore, thereference region does not need to be coincident withthe FRAP imaging. In fact, intensity collected overthe whole or any sub-region of the CFM spot can beused as reference data. Phair et al.’s method stillrequires a finite image size to approximate the bleachingconstant due to the effect of the FRAP-bleaching in theimage centre.

With partial imaging, diffusion is negligible in aCFM process only when e � 1, i.e. tk� tD (seefigure 10b in appendix B). For a CFM time series tobe satisfactorily adopted as reference time series inMueller et al.’s method, the illumination and FRAPimaging intensities must be identical, thereby fixingtk. Thus, we can then only increase the image size r1

to achieve e � 1. By using only the central region ofa CFM spot as the reference region, the image size r1

required to achieve e � 1 is reduced. As shown infigure 2, the central location in a CFM spot bleaches

J. R. Soc. Interface (2011)

faster than in the periphery. On the other hand, Phairet al.’s method requires a larger image size, not onlybecause of the FRAP-bleached spot in the centre, butalso because of diffusion at the image boundary.

Mueller et al. [39] still found their acquisition bleach-ing correction affects apparent protein binding rates.This might be owing to violation of assumptions inthe theoretical model. For example, fluorochromes inthe cell (whole cell imaged) are not uniformly distribu-ted, so there remains diffusion surrounding the referenceregion. We also note that when FRAP image series areused as reference data, a sub-region of the FRAP imageis sometimes used as the reference region [57]. However,the position and size of such a region has to be carefullychosen so that diffusion from the bleached fluoro-chromes in the FRAP spot and the unbleached onesat the image boundary are both negligible.

4.2. Self-diffusion coefficient from FRAP timeseries

To minimize the scan time, we imaged the FRAP spotand a CFM reference spot. Representative snapshots ofa c � 3 mol% DSPE-PEG2k-CF doped DOPC bilayerare shown in figure 5a. The FRAP spot (upper right)has radius r0 � 5.13 mm, and CFM spot (lower left)has radius r1 � 2.7r0. The imaged regions are separatedby a distance L � 25 mm, so the characteristic time forlipopolymers with diffusivity Ds � 1 mm2 s21 to tra-verse the gap is �L2/Ds � 150 s, which is much longerthan the characteristic FRAP recovery time tD � r0

2/

134 Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill

Ds � 7 s. Figure 5b shows the corresponding FRAP timeseries (circles) and reference time series (triangles).Here, the reference data are the average intensitieswithin the central circular region (with radius r0) ofthe CFM spot. The solid line is an exponential fit tothe reference data giving I/I1 � 0.97e20.0039t. Notethat the reference time series does not decay exponen-tially, which suggests that r1 � 2.7r0 is not largeenough for diffusion to be negligible within the centralcircular region of the CFM spot. Figure 5c shows cor-rected FRAP time series (squares) obtained byapplying equation (4.2) with k � 0.0039 s21. Here, thecorrected intensity—normalized with the initially uni-form intensity—is greater than one, indicating thatacquisition bleaching is over-corrected, as expectedfrom figure 4a. Fitting Soumpasis’s equation (3.10)(solid line) to the uncorrected FRAP data (circles)yields Ds � 1.42 mm2 s21 and n1/n1 � 0.92 with thesquared 2-norm of the residuals x2 � 0.01. The least-squares fit (dashed line) of Soumpasis’s solution tothe corrected time series (squares) furnishes Ds �0.71 mm2 s21 and n1/n1 � 1.1 with x2 � 0.08. Here,fitting Soumpasis’s solution to the corrected timeseries is unsatisfactory due to over-correcting of acqui-sition bleaching. Finally, figure 5d shows the least-squares fit (solid line) of our reaction–diffusion FRAPmodel to the uncorrected FRAP data (circles). This fur-nishes Ds � 1.05 mm2 s21, k � 0.0013 s21, and fm � 0.98with x2 � 0.02.

Note that fitting our FRAP model to experimentaldata requires reasonable initial guesses of the fittingparameters Ds, k and fm to converge upon the correctvalues. Using simulated FRAP data, we found that Ds

from fitting Soumpasis’s equation (3.10) to the FRAPtime series, k from fitting an exponential decay to thereference time series, and setting fm ¼ 1 furnished satis-factory starting values. Comparing the least-squares fitsof Ds from the reaction–diffusion FRAP model with thevalues obtained from Soumpasis’s solution reveals thatacquisition bleaching manifests as a faster recovery(see figure 8), thereby overestimating Ds. Note thatSoumpasis’s solution does not yield a bleaching con-stant, and I1/I1 may be reasonably taken to be themobile fraction fm only when the bleaching intensityn0/n1 ¼ I0/I1 ¼ 0.

In principle, fitting the reaction–diffusion CFM modelto experimental CFM time series furnishes Ds, k and fm. Inpractice, however, we found the fitting to be sensitive toinitial estimates of the least-squares fitting parameters.One reason is that the accuracy of CFM depends on theratio tD/tk (see [55]). Our CFM images were obtainedat the same time as FRAP imaging, so the bleaching isnot strictly continuous due to line-scanning limitationsof the confocal microscope. Similarly to Irrechukwu &Levenston [58], the bleaching constant must be treatedas an apparent value, since it depends on instrument-specific settings, such as laser-illumination intensity.

4.3. Self-diffusion coefficient concentrationdependence

We synthesized bilayers containing various concen-trations of DSPE-PEG2k-CF in DOPC. For each

J. R. Soc. Interface (2011)

bilayer synthesis, we performed FRAP replicates in sep-arate regions, each time furnishing Ds, k and fm asdetailed in §4.2. Least-squares fits were rejected if thesquared 2-norm of the residuals x2 . 0.02. Next, theaverage and standard deviation of Ds and fm from thereplicates were computed. Finally, data from severalseparately synthesized bilayers—with the same lipopo-lymer concentration—were combined to obtainweighted average of Ds and fm using the standard devi-ations as weights (see [59, p. 57]). Representative dataare summarized in table 1 for bilayers with c �1 mol% DSPE-PEG2k-CF in DOPC. We found thatPEG-lipids are mobile at concentrations up to c �5 mol%, with the mobile fraction fm � 0.95. As a con-trol, we applied a similar methodology to obtain self-diffusion coefficients and mobile fractions of DOPE-NBD in DOPC.

Self-diffusion coefficients Ds are plotted as a functionof fluorochrome concentration c in figure 6 for DSPE-PEG2k-CF (circles) and DOPE-NBD (squares) inDOPC. In striking contrast to the DOPE-NBD control,the self-diffusion coefficient of the lipopolymerdecreases monotonically with increasing concentration.Interestingly, the DOPE-NBD control has a veryweak, non-monotonic concentration dependence. Wefit the lipopolymer diffusion coefficient data to anempirical interpolation formula

Ds ¼D0

ð1þ acÞ ; ð4:3Þ

where D0 is the diffusion coefficient at infinite dilution,and a is a constant that characterizes the degree towhich diffusion is hindered at finite lipopolymerconcentrations.

Least-squares fitting (solid line) of equation (4.3) tothe data in figure 6 furnishes D0 � 3.36 mm2 s21 anda � 0.56 when c is expressed as a mole percent. Alsoshown in figure 6 are lipopolymer self-diffusion coeffi-cients (triangles) obtained by fitting Soumpasis’ssolution to the uncorrected FRAP time series. Asexpected, Ds from Soumpasis’s solution is larger thanthat obtained from the reaction–diffusion FRAPmodel. Least-squares fitting (dashed line) of equation(4.3) to the data furnishes D0 � 3.81 mm2 s21 and a �0.49. For reasons discussed in §4.2, we consider theresults derived from the reaction–diffusion FRAPmodel to be more accurate, so the values D0 �3.36 mm2 s21 and a � 0.56 will be adopted for thefollowing theoretical interpretation.

Note that the diffusion coefficient of lipopolymerDSPE-PEG2k-CF at infinite dilution, D0 �3.36 mm2 s21, is somewhat higher than the valueinferred for the DOPE–NBD control. While it is diffi-cult to provide a conclusive interpretation of thisdiscrepancy without invoking membrane elasticity andcurvature, for example, the lower mobility of DOPE-NBD might reflect a stronger hydrodynamic couplingof the DOPE-NBD doped bilayer with the glass sup-port. This is not unreasonable given that the bilayerhosting DOPE-NBD can be much closer to the glassthan the lipids in DSPE-PEG2k-CF doped bilayers.

Table 1. Summary of FRAP data for bilayers containing c � 1 mol% DSPE-PEG2k-CF in DOPC. Here, data from fourseparately synthesized bilayers, each with several replicate FRAP experiments with in each bilayer, are combined to furnish theweighted average self-diffusion coefficient Ds and mobile fraction fm.

bilayersynthesis measurement

Ds

measured Ds averageDs weightedaverage

fmmeasured fm average

fm weightedaverage

1 1 2.68 2.27+ 0.51 96.79 94.44+2.942 2.09 95.833 2.17 97.444 2.02 89.285 1.39 96.156 2.63 92.357 2.90 93.22

2 1 2.27 2.09+ 0.24 94.86 93.25+3.342 2.42 98.633 1.93 89.974 2.22 90.055 1.83 94.246 1.88 91.73

3 1 2.15 2.19+ 0.09 96.63 98.08+1.522 2.31 97.113 2.20 98.604 2.09 99.99

4 1 2.10 1.91+ 0.26 98.13 95.90+2.012 1.60 95.333 2.03 94.24

2.16+0.04 96+ 0.51

0 0.01 0.02 0.03 0.04 0.05 0.060.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

concentration, c

self

-dif

fusi

onco

effc

ient

, Ds (

µm2

s−1 )

Figure 6. Self-diffusion coefficients Ds of lipopolymer DSPE-PEG2k-CF in DOPC. Circles are from least-squares fittingof the reaction–diffusion FRAP model to uncorrectedFRAP time series, with least-squares fitting of interpolationformula 4.3 (solid line) furnishing D0 � 3.36 mm2 s21 anda � 0.56. Triangles are from fitting Soumpasis’s solution touncorrected FRAP time series, with interpolation formula4.3 (dashed line) furnishing D0 � 3.81 mm2 s21 and a � 0.49.Squares are self-diffusion coefficients of DOPE-NBD inDOPC.

Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill 135

4.4. Comparison with literature

Literature-reported self-diffusion coefficients for lipopoly-mer-doped bilayers formed by vesicle fusion aresummarized in table 2 with the accompanying lipid

J. R. Soc. Interface (2011)

composition, substrate cleaning procedure, and other par-ameters known to affect bilayer fluidity. For clarity, thedata are plotted in figure 7. Note that we have indicateda vanishing self-diffusion coefficient in figure 7 from thestudy of Kaufmann et al. [57], whereas they actuallyreported that there was no recovery after photobleaching.

Recall, the DSPE-PEG2k-CF lipids in our work aremobile at all concentrations studied (c � 5 mol%)with mobile fractions fm � 95%. [18] also observedmobile DSPE-PEG2k lipids in supported egg-PCbilayers at concentrations up to 5 mol%. However,[57] found that 1 mol% NBD-PC bilayers are ostensiblyinhomogeneous, with no discernable recovery afterbleaching, when the mole fraction of DOPE-PEG2k�5 mol%. Thus, mounting evidence suggests that quali-tative differences in bilayer structure often manifestonly when well into the brush regime, perhaps furtherthan inferred by Albertorio et al. [18].

Now turning to the concentration dependence of lipo-polymer self-diffusion coefficients, Albertorio et al. [18]reported self-diffusion coefficients of Alexa Fluor-594-labelled NH2-PEG2k-DSPE that decrease with increasingconcentration at concentrations c � 1.5 mol%. Accordingto Kaufmann et al. [57], the mole fraction for the PEGmushroom-to-brush transition [60] is

ct ¼Al

R2F; ð4:4Þ

where Al is the cross-sectional area of a lipid, and thePEG-chain Flory radius

RF ¼ amn3=5p ; ð4:5Þ

where am is the monomer size and np the degree of polymer-ization. Literature suggests that am � 0.35–0.43 nm and

Table 2. Summary of literature-reported self-diffusion coefficients in lipopolymer-doped bilayers formed by vesicle fusion.

system I II-1 II-2 III-1 III-2Ds, fm Ds Ds Ds, fm Ds, fm

c � 0 mol% — 4.0+ 0.1 1.2+ 0.4, 95+ 2 —c � 0.5 mol% 2.62+ 0.12,

97.89+ 0.753.8+ 0.2 4.0+0.25 — —

c � 1 mol% 2.16+ 0.04,96.20+ 0.51

— 4.0+0.15 1.5+ 0.1,97+0.4

—

c � 1.5 mol% — 3.9+ 0.2 3.0+0.15 2.2+ 0.4, 97+ 2 —c � 2 mol% 1.61+ 0.08,

96.92+ 0.94— — — 1.1+0.4,

99+1c � 3 mol% 1.23+ 0.03,

94.62+ 0.65— 2.4+0.12 1.9+ 0.2,

98+0.3—

c � 5 mol% 0.90+ 0.03,97.58+ 0.41

3.9+ 0.3 1.4+0.10 no recovery —

bulk lipids DOPC egg-PC egg-PC POPC POPCpolymer DSPE-PEG2k DOPE-PEG2k DSPE-PEG2k DOPE-PEG2k DSPE-PEG2kfluorephore PEG2k-CF 0.5 mol% TR-

PEAlexa Fluor-594-

PEG2k1 mol% NBD-PC PEG2k-CF

substrate cover-slips glass slides cover-slipssubstrate

cleaningPiranha etching baked UVO cleaning

FRAP fitting numerical [27] [42]bleaching fitting neglected correcteda

reference this work [18] [18] [57] [57]

aIntensity in a region of the FRAP image was used as reference data.

−1 0 1 2 3 4 5 60

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

concentration, c (mol%)

self

-dif

fusi

onco

effc

ient

, Ds (

µm2

s−1 )

Figure 7. Summary of measured self-diffusion coefficients withlines to guide the eye. Circles: DSPE-PEG2k-CF from thisstudy at various DSPE-PEG2k-CF concentrations. Squares:DSPE-PEG2k-Alexa Fluor-594 from [18] at various DSPE-PEG2k-Alexa Fluor-594 concentrations. Up-triangle: DSPE-PEG2k-CF from [57] at one DSPE-PEG2k-CF concentration.Diamonds: 0.5 mol% TR-PE from [18] at various DOPE-PEG2k concentrations. Left-triangles: 1 mol% NBD-PCfrom [57] at various DOPE-PEG2k concentrations. Notethat [57] reported an absence of recovery after photobleachingat 5 mol% DOPE-PEG2k, which we depict here as a vanishingdiffusion coefficient. Data are listed in table 2.

136 Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill

Al � 0.60–0.70 nm2 [61,62], so the mushroom-to-brushtransition for PEG2k (np¼ 45) would occur at concen-trations in the range ct � 3.5–6 mol%. For example, with

J. R. Soc. Interface (2011)

am¼ 0.39 nm, Al¼ 0.65 nm2 and RF� 3.83 nm, we findct � 4.4 mol%. On the other hand, if we interpret themushroom-to-brush transition as occurring when ct¼

Al/(pRF2) (full coverage based on the area occupied by cir-

cles with radius RF), then ct� 1.4 mol%. Indeed, thisestimate is consistent with the abrupt transition in theself-diffusion coefficient observed by Albertorio et al. [18].

Interestingly, our data for DSPE-PEG2k-CF inDOPC decrease continuously with increasing lipopoly-mer concentration, similarly to the smooth manner inwhich the spreading pressure varies according to self-consistent mean-field theory [48]. This suggests thatthe mushroom-to-brush transition in our experimentsis considerably more gradual than inferred byAlbertorio et al. [18] Moreover, the PEG2k-CF chainsin our experiments may not adopt an ostensiblybrush-like configuration until concentrations arereached that are closer to or higher than the 4.4 mol%suggested by equation (4.4). If this is indeed the case,then it would be reasonable to consider our experimentsas having been performed with the PEG2k-CF chainspredominantly in mushroom-like configurations.

We emphasize that quantitative comparison of lat-eral diffusion coefficients in SLBs is difficult becausediffusion is easily affected by bilayer composition, sup-port properties, and other parameters [63,64]. Inrecent years, several studies have tried to systematicallyaddress the influences of such parameters on diffusion.For example, Seu et al. [65] found that the diffusion ofNBD-DOPC in egg phosphatidylcholine (egg-PC) (i)decreases significantly upon the addition of eggphosphatidylethanolamine (egg-PE), (ii) increases sig-nificantly with addition of lyso-phosphatidylcholine(LPC), and (iii) decreases slightly with addition oflyso-phosphatidylethanolamine (LPE). They concluded

Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill 137

that both lipid head group and tail structure affect dif-fusion. In another study, Seu et al. [66] found thatpiranha etched glass-slide supported bilayers are three-fold more fluidic than on baked glass. Finally,Scomparin et al. [64] demonstrated that dynamics inSLBs depend on the type of substrate (glass andmica) and the method of bilayer preparation (Lang-muir–Blodgett deposition and vesicle fusion).

The main differences between the three studies listedin table 2 are (i) lipopolymer tail structure; (ii) bulklipids; (iii) substrate cleaning procedures; and (iv)FRAP fitting methodology. In the study of Albertorioet al., DPPE-PEG2k and DSPE-PEG2k have similarinfluences on the diffusion of Texas Red-DHPE in egg-PC membranes [18]. This shows that differences in lipo-polymer tail structure might not have a significantinfluence on lipopolymer mobilities. Bulk lipids used inthe three studies listed in table 2 are DOPC, egg-PC,and POPC. The main component of egg-PC is POPC,so the bulk lipids in system II and III in table 2 are simi-lar. The striking difference in the dynamics of those twosystems indicates that the bulk/supporting lipid plays aminor role. Also, self-diffusion coefficients fitted usingSoumpasis’s solution agree qualitatively with our reac-tion–diffusion FRAP model (figure 6), indicating thatthe qualitative differences in the three studies are morelikely due to the substrate cleaning procedures. However,in contrast to the observations of Seu et al. [66], lipopo-lymer self-diffusion coefficients in our study (system I) onpiranha etched glasses are actually smaller than thevalues from system II using baked glasses. At present,it seems that every parameter plays a role in thedynamics of the three systems, and that the principalcontribution remains elusive.

4.5. Theoretical interpretation of hinderedself-diffusion

Theory is available to interpret the hindered diffusivityof membrane proteins [16], and their dynamics arereasonably well understood. The dynamics of lipopoly-mers in SLBs is more challenging, because eachlipopolymer comprises a small lipid anchor embeddedin a viscous, two-dimensional fluid, with a large graftedpolymer chain immersed in a low-viscosity, three-dimensional half-space.

For discs with excluded-volume interactions in lipidbilayers, the [7] theory predicts a self-diffusioncoefficient at low concentrations

Ds

D0¼ 1� 2f; ð4:6Þ

where f is the disc area fraction. To compare theory andexperiments for proteins, the experimentally measuredprotein concentration must be converted to an area frac-tion. This conversion rests on accurately specifying a discradius [8,10,15]. For lipopolymers, however, the area fac-tion is not easy to specify, because lipopolymers have alarge polymer coil grafted to a hydrodynamically smalllipid anchor. Nevertheless, if we treat the lipopolymeras a disc with an effective radius ae, then the effectivearea fraction is fe ¼ cpae

2/Al, where c is the lipopolymermole fraction, and Al � 0.65 nm2 is the lipid cross-

J. R. Soc. Interface (2011)

sectional area corresponding to a lipid radius al �0.46 nm. Equation (4.6) then becomes

Ds

D0¼ 1� 2pa2

ecAl

; ð4:7Þ

which, when compared with our empirical fitting formulain equation (4.3) as c! 0, furnishes ae ¼ al

ffiffiffiffiffiffiffiffia=2

p. Thus,

with a � 0.56 (figure 6), the effective radius ae �2.41 nm.

By including hydrodynamic interactions, Bussellet al. [14] derived

Ds

D0¼ 1� 2f 1� 1þ ln 29=23

ln l� g

� �� 0:07

lnl� g; ð4:8Þ

where g � 0.577216 is Euler’s constant, and l ¼ hmf/(aml) with a the hard-cylinder radius, h the lipid–bilayer thickness, ml the lipid–bilayer viscosity and mf

the viscosity of the supporting fluid. To furnish an effec-tive cylinder radius for lipopolymers according toBussell et al.’s theory, we specify an effective radius ae

to calculate Ds/D0 according to equation (4.8). Next,comparing the dilute limit of the theory to the exper-imental data, we obtain an improved estimate of ae.Repeating this procedure until successive values of ae

change by less than one percent furnishes ae �2.92 nm when we set h ¼ 3.2 nm [67,68], ml ¼ 1.2 �1021 kg m21 s21 [69] and mf ¼ 8.9 � 1024 kg m21 s21

(water; [6]). Comparing this value with the effectiveradius as � 2.41 nm from the Scalettar–Abney–Owickitheory (equation (4.7)) suggests that hydrodynamicinteractions play a small role.

It is reassuring that the effective radii for PEG2k-CFare much larger than the lipid head radius. This con-firms that thermodynamic interactions between thetethered polymer chains are significant. Moreover, theeffective radii are both smaller than the PEG-chainFlory radius, which is reasonable when acknowledgingthat the effective hard-core radius here is representinga large, soft polymer coil that is attached to a verysmall and compact anchor. The reasonable size of thePEG2k-CF chains also supports our assumption thatthe surface concentration of lipopolymers equals thestoichiometric amount used in the bilayer synthesis.

Note that DSPE-PEG2k-CF bears a charge 2e atthe lipid–PEG junction, and a charge 2e at thePEG2k-CF junction. Thus, DOPE-NBD in DOPCserves as a valuable control because it also bears a nega-tive charge without a grafted PEG chain. Controlexperiments do not exhibit significant changes in theself-diffusion coefficient with concentration, indicatingthat electrostatic interactions between the chargedlipid heads do not significantly influence lipopolymerdynamics. Rather, the polymer–polymer interactionseems to play an important role. However, the hinderedself-diffusion coefficient is not sufficient to distinguishdirect lipopolymer interactions (thermodynamic inter-actions) from lipopolymer-induced perturbations toflow (hydrodynamic interactions). This competition,with other influences, such as substrate–bilayer inter-actions, will hopefully become clearer withcomplementary knowledge of the gradient diffusioncoefficient.

Table 3. Frictional drag characteristics for DSPE-PEG2k-CFin DOPC leaflets according to theoretical interpretations ofmeasured D0 (see text for details) using equations (4.12) and(4.13).

ap (nm) lp (N s m21) ll (N s m21) e bs (N s m23)

0 0 1.21 � 1029 0.68 1.48 � 108

2.41 4.04 � 10211 1.17 � 1029 0.65 1.35 � 108

2.92 4.90 � 10211 1.16 � 1029 0.64 1.33 � 108

3.83 6.43 � 10211 1.15 � 1029 0.63 1.29 � 108

138 Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill

4.6. Frictional drag in polymer-grafted bilayers

From the diffusion coefficient at infinite dilution, theEinstein relation furnishes a lipopolymer drag coeffi-cient

lt ¼ kBT=D0; ð4:9Þ

where kBT is the thermal energy. With D0 �3.36 mm2 s21, we have lt � 1.21 � 1029 N s m21 atT � 295 K. Let us now consider the total drag coeffi-cient on a lipopolymer as the sum

lt ¼ lp þ ll; ð4:10Þ

where lp is the drag coefficient of the polymer chain,and ll the drag coefficient of the anchoring lipid. Fur-thermore, approximating the polymer as a sphere withradius ap gives

lp ¼ 6pmfap; ð4:11Þ

where mf is the viscosity of the supporting fluid (water).As an extension of the [5] theory, the [70] theory

gives

ll ¼ 4phe2

4þ eK1ðeÞ

K0ðeÞ

� �; ð4:12Þ

where h is the membrane viscosity; K0 and K1 arezeroth- and first-order modified Bessel functions of thesecond kind; and

e ; a

ffiffiffiffibs

h

sð4:13Þ

is a dimensionless disc (lipid head) radius (not to beconfused with the dimensionless bleaching constant)with bs the inter-leaflet friction coefficient. Setting a �al � 0.46 nm and h � 0.08 nN s m21 (half the valuefor a free lipid bilayer h � 0.16 nN s m21 [71]), equation(4.13) furnishes the values of e and bs listed in table 3for several estimates of ap.

Evidently, all reasonable values of ap yield lp�ll,which, in turn, furnishes values of e and bs that arepractically independent of the polymer size. Thisdemonstrates that the total frictional drag on lipopoly-mers is overwhelmingly dominated by hydrodynamicfriction within the leaflet that hosts the lipid anchor.A similar conclusion can be drawn from histidine-tagged EGFP anchored to mono-layers [72]. Note alsothat the dimensionless lipopolymer anchor radius e �0.65 corresponds closely with e � 0.73 for NBD-PE in1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine(SOPC; [72]).

5. CONCLUSIONS

We developed a reaction–diffusion FRAP model thatintegrates a finite immobile fraction and first-orderphotobleaching kinetics. We used this model to evalu-ate commonly adopted photobleaching correctionmethodologies for quantitative FRAP in the literature,and applied it to furnish self-diffusion coefficients (andmobile fraction) for a fluorescent lipopolymer DSPE-PEG2k-CF in DOPC supported bilayers over a wide

J. R. Soc. Interface (2011)

range of DSPE-PEG2k-CF concentrations. To quantifythe influence of the grafted PEG chains on thermodyn-amic and hydrodynamic interactions, we performedsimilar experiments furnishing the self-diffusion coeffi-cient of DOPE-NBD in DOPC over a similar range ofDOPE-NBD concentrations.

We conclude that the limiting diffusion coefficient D0

reflects—almost entirely—hydrodynamic frictionwithin the two-dimensional leaflet in which the lipopo-lymer is anchored. This supports the hypothesis thatthere is a negligible difference between the dynamicsof lipids in the top and bottom leaflets of these SLBs,assuming lipopolymers are equally distributed betweenboth leaflets.

Perhaps surprisingly, the lipopolymer self-diffusioncoefficient is significantly hindered by thermodynamicinteractions between the grafted PEG chains. More-over, theoretical interpretation of the data suggeststhat hydrodynamic interactions between the PEGchains are weak compared to thermodynamic(excluded-volume) interactions. Thus, based on the [7]theory of diffusion, we expect an enhancement of thegradient diffusion coefficient of DSPE-PEG2k-CF inDOPC with increasing concentration. This may haveimportant practical implications for interpreting thedynamics of biologically relevant lipopolymers, andtechnological applications involving lipopolymermembranes.

R.J.H. gratefully acknowledges support from the NaturalSciences and Engineering Research Council of Canada andthe Canada Research Chairs program; H.-Y.Z. thanksMcGill University and the Faculty of Engineering, McGillUniversity, for generous financial support through a RichardH. Tomlinson Fellowship and a McGill EngineeringDoctoral Award (Hatch Graduate Fellowships inEngineering), respectively.

APPENDIX A. GENERAL SOLUTION OFTHE REACTION–DIFFUSION FRAPMODEL

General solutions of the reaction–diffusion FRAPmodel are examined in figure 8 for partial imaging(r1 , r2). Note, however, that the results are practicallythe same as when imaging the entire domain, i.e. whenr1 ¼ r2� r0. Figure 8a shows the FRAP spot timeseries for r1/r0 ¼ 1 and several values of r2/r0 ¼ 2, 3,5 and 10 (bottom to top), without acquisition

0 10 20 30 400.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0(a) (b) (c)

scaled time, t/τD

0 10 20 30 40

scaled time, t/τD

0 10 20 30 40

scaled time, t/τD

norm

aliz

ed in

tens

ity, I

/I∞

Figure 8. FRAP time series as predicted by the reaction–diffusion FRAP model (dashed lines) with partial imaging (r1 , r2). Inall examples, the initial bleaching intensity n0/n1 ¼ 0.2. Solid lines are Soumpasis’s analytical solution for FRAP without photo-bleaching (e ¼ 0) and without an immobile fraction ( fm ¼ 1). (a) Varying the reservoir radius r2/r0 ¼ 2, 3, 5 and 10 (bottom totop) with r1/r0 ¼ 1, e ¼ 0 and fm ¼ 1. (b) Varying the bleaching constant e ¼ 0.005, 0.01, 0.02 and 0.05 (top to bottom) withfm ¼ 1, r1/r0 ¼ 10 and r2/r0 ¼ 100. (c) Varying the mobile fraction fm ¼ 0.6, 0.7, 0.8 and 0.9 (bottom to top) with e ¼ 0, r1/r0 ¼ 10 and r2/r0 ¼ 100.

5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0no

rmal

ized

inte

nsity

, I/I

∞

Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill 139

photobleaching (e ¼ 0) and without immobile fluoro-chromes ( fm ¼ 1). Increasing r2/r0 decreases theoverall recovery rate. Model predictions (dashed lines)closely agree with Soumpasis’s theory (solid lines)when r2/r0 � 10. Figure 8b shows model predictionsfor several dimensionless photobleaching rate constantse ¼ 0.005, 0.01, 0.02 and 0.05 (top to bottom), alsowithout immobile fluorochromes ( fm ¼ 1). As expected,the model reproduces Soumpasis’s theory (solid line) ase ! 0 with r2/r0� 1. However, there are significantqualitative differences when t/tD � 1 and e . 0. Notethat even weak photobleaching affects the recoveryrate and plateau. Finally, figure 8c shows model predic-tions for several mobile fractions fm ¼ 0.6, 0.7, 0.8 and0.9 (bottom to top), without acquisition photobleach-ing (e ¼ 0). Again, there are significant deviationsfrom Soumpasis’s theory when fm , 1, with the inten-sity after recovery reflecting the mobile fraction.

scaled time, t/τD

Figure 9. Continuous fluorescence micro-photolysis (CFM)time series as predicted by the reaction–diffusion model:e ¼ 0.5, fm ¼ 0.7, r1/r0 ¼ 1 and r2/r1 ¼ 100. The solid line isthe average intensity of fluorochromes in the CFM spot, andthe dashed and dash-dotted lines are, respectively, the contri-butions from the mobile and immobile fractions.

APPENDIX B. APPLICATION OF THEREACTION–DIFFUSION FRAP MODEL TOCONTINUOUS FLUORESCENCE MICRO-PHOTOLYSIS

Continuous fluorescence micro-photolysis (CFM) isanother methodology for measuring self-diffusion coeffi-cients using photobleaching [55]. The CFM spot isuniformly and continuously illuminated at mediumlaser intensity. Modifying the initial condition of thereaction–diffusion FRAP model captures CFM-spotintensity dynamics. The CFM model has r0 ¼ r1 withinitial conditions for the mobile fluorochromes

nmðr; 0Þ ¼ n1fm; 0 � r � r2 ðB 1Þ

and for the immobile fluorochromes

niðr; 0Þ ¼ n1ð1� fmÞ; 0 � r � r2: ðB 2Þ

A representative CFM time series is presented infigure 9. The intensity of immobile fluorochromes (dash-

J. R. Soc. Interface (2011)

dotted line) decays exponentially with the mobile fluoro-chrome intensity (dashed line) decaying more slowly dueto recovery by diffusion. Note that the total intensity(solid line) approaches a plateau where the rate of photo-bleaching is balanced by diffusive restoration.

The influence of several model parameters on CFMtime series are examined in figure 10. For convenientreference, all panels show bleaching curves e2e t/tD

(solid lines). Figure 10a shows solutions for severalvalues of r2/r1 ¼ 1, 2, 3, 5 and 10 (dashed lines,bottom to top) with e ¼ 0.5 and fm ¼ 1. As expected,when imaging the entire domain (r2/r1 ¼ 1), the CFMtime series overlaps the bleaching curve. Moreover,when r2/r1 � 5, the time series are practically

5 10 15 200

0.10.20.30.40.50.60.70.80.91.0

0 5 10 15 20 0 5 10 15 20

(a) (b) (c)

scaled time, t/τD scaled time, t/τD scaled time, t/τD

norm

aliz

ed in

tens

ity, I

/I∞

Figure 10. CFM time series as predicted by the reaction–diffusion model (dashed lines). Bleaching curves (solid lines) have cor-responding values of e. (a) Varying the reservoir radius r2/r1 ¼ 1, 2, 3, 5 and 10 (bottom to top) with e ¼ 0.5 and fm ¼ 1. Notethat the time series with r2/r1 ¼ 1 overlaps the bleaching curve (solid line), and time series with r2/r1 ¼ 5 and 10 overlap eachother. (b) Varying the bleaching constant e ¼ 0.05, 0.5 and 5 (dashed lines, top to bottom) with fm ¼ 1 and r2/r1 ¼ 100. (c) Vary-ing the mobile fraction fm ¼ 0.5, 0.8 and 1 (bottom to top) with e ¼ 0.5 and r2/r1 ¼ 100.

10 20 30 400

0.10.20.30.40.50.60.70.80.91.0

0 10 20 30 40 0 10 20 30 40

(a) (b) (c)

scaled time, t/τD scaled time, t/τD scaled time, t/τD

norm

aliz

ed in

tens

ity, I

/I∞

Figure 11. Correcting acquisition bleaching when the entire reservoir is imaged (r1 ¼ r2): e ¼ 0.05, fm ¼ 1 and n0/n1 ¼ 0.2. Solidlines are the bleaching curve I/I1 ¼ e2et/tD. (a) Correction according to equation (4.2): Soumpasis’s analytical solution (dash-dotted line); numerical FRAP time series for r2/r0 ¼ r1/r0 ¼ 3, 5 and 10 (dashed lines, top to bottom); corrected FRAP timeseries for r2/r0 ¼ r1/r0 ¼ 3, 5 and 10 (circles, bottom to top). (b) Phair et al.’s method of collecting reference data: referencetime series with r2/r0 ¼ r1/r0 ¼ 3, 5 and 10 (dashed lines, bottom to top). (c) Mueller et al.’s method of collecting referencedata: reference time series with r2/r0 ¼ r1/r0 ¼ 1, 5 and 10.

140 Lipopolymer self-diffusion H.-Y. Zhang and R. J. Hill

independent of the reservoir size. Figure 10b shows sol-utions for several values of e ¼ 0.05, 0.5 and 5 (dashedlines, top to bottom) with fm ¼ 1 and r2/r1 ¼ 100. As eincreases, bleaching dominates and the CFM time seriesdecay to lower plateaus. Figure 10c shows solutions forseveral values of fm ¼ 0.5, 0.8 and 1 (dashed lines,bottom to top) with e ¼ 0.5 and r2/r1 ¼ 100. In theabsence of diffusion ( fm ¼ 0), the CFM time series over-laps the bleaching curve. Moreover, increasing fmfurnishes CFM times series that decay toward increas-ingly higher plateaus.

APPENDIX C. CORRECTINGACQUISITION BLEACHING WHENIMAGING THE ENTIRE RESERVOIR

The results of correcting acquisition bleaching for thecase when whole reservoir is imaged, i.e. r1 ¼ r2, are

J. R. Soc. Interface (2011)

presented in figure 11. Figure 11a shows correctedFRAP curves for r2/r0 ¼ r1/r0 ¼ 3, 5 and 10 (circles,bottom to top). With the given bleaching constante ¼ 0.05, the corrected FRAP curve agrees with Soum-pasis’s solution (dash-dotted line) when r2/r0 ¼ r1/r0 �10. Figure 11b shows the reference curves collectedusing Phair et al.’s method for r2/r0 ¼ r1/r0 ¼ 3, 5and 10 (dashed lines, bottom to top). The referencecurves agree with the actual bleaching curve (solidline) when r2/r0 � 10. Figure 11c shows reference datacollected using Mueller et al.’s method for r2/r0 ¼ r1/r0 ¼ 1, 5 and 10 (dashed lines). All the referencecurves overlap the bleaching curve (solid line)calculated using the known bleaching constante ¼ 0.05.

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