Traffic 2012; 13: 1589–1600 © 2012 John Wiley & Sons A/S

doi:10.1111/tra.12008

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Simplified Equation to Extract Diffusion Coefficientsfrom Confocal FRAP Data

Minchul Kang1,2, Charles A. Day1, Anne K.

Kenworthy1,3,4,∗ and Emmanuele

DiBenedetto1,5,∗

1Department of Molecular Physiology and Biophysics,Vanderbilt University School of Medicine, Nashville,TN 37232, USA2Current address: School of Science, Technology, andEngineering Management, St. Thomas University, 16401NW 37th Avenue, Miami Gardens, FL 33054, USA3Department of Cell and Developmental Biology,Vanderbilt University School of Medicine, Nashville,TN 37232, USA4Chemical and Physical Biology Program, VanderbiltUniversity School of Medicine, Nashville, TN 37232, USA5Department of Mathematics, Vanderbilt University,Nashville, TN 37240, USA*Corresponding authors: Anne K. Kenworthy,anne.kenworthy@Vanderbilt.Edu and EmmanueleDiBenedetto, em.diben@Vanderbilt.Edu

Quantitative measurements of diffusion can provide

important information about how proteins and lipids

interact with their environment within the cell and the

effective size of the diffusing species. Confocal fluores-

cence recovery after photobleaching (FRAP) is one of the

most widely accessible approaches to measure protein

and lipid diffusion in living cells. However, straightfor-

ward approaches to quantify confocal FRAP measure-

ments in terms of absolute diffusion coefficients are cur-

rently lacking. Here, we report a simplified equation that

can be used to extract diffusion coefficients from confocal

FRAP data using the half time of recovery and effective

bleach radius for a circular bleach region, and validate

this equation for a series of fluorescently labeled soluble

and membrane-bound proteins and lipids. We show that

using this approach, diffusion coefficients ranging over

three orders of magnitude can be obtained from confocal

FRAP measurements performed under standard imaging

conditions, highlighting its broad applicability.

Key words: confocal laser scanning microscope, diffusion

coefficient, fluorescence microscopy, FRAP, half time of

recovery

Received 31 May 2012, revised and accepted for

publication 9 September 2012, uncorrected manuscript

published online 17 September 2012, published online

10 October 2012

Diffusion is a fundamental process that is relevant over allscales of biology. How rapidly diffusion occurs is charac-terized by the diffusion coefficient D, a parameter that

provides a measure of the mean squared displacementper unit time of the diffusing species. At the cellular level,measurements of D can provide important insights intohow proteins and lipids interact with their environment,such as the binding of transcription factors with DNA andof proteins and lipids with membrane domains (1,2). Inaddition, especially for the case of soluble proteins, anaccurate measure of D can provide important informationabout the effective size of the diffusing species (3).

Currently, biologists have at their disposal a widerange of fluorescence-based techniques to monitor thediffusion of biomolecules in cells, including single-particletracking (SPT), fluorescence correlation spectroscopy(FCS), photoactivation and fluorescence recovery afterphotobleaching (FRAP) (4). Of these approaches, manyrequire specialized equipment, analytical tools or probes.In contrast, confocal FRAP is one of the most accessiblemethods to measure apparent diffusion (or effectivediffusion). In FRAP, the diffusion of a population offluorescently labeled molecules can be studied byphotobleaching the molecules contained within a user-defined region of interest (ROI), and then monitoringfluorescence recovery due to the exchange of bleachedmolecules within the bleach ROI with the surroundingreservoir of unbleached molecules. Most modern confocallaser scanning microscopes (CLSMs) are equipped toperform FRAP measurements, and FRAP measurementsare straightforward to carry out. In addition, FRAP can beperformed using many fluorescently tagged proteins or ondirectly fluorescently labeled molecules (4,5).

Despite the relative ease of performing confocal FRAPmeasurements, straightforward methods to quantitativelyanalyze the resulting data are currently lacking. Mostcurrent approaches to calculate Ds from confocal FRAPdata require fitting of recovery curves with analyticaldiffusion models using customized programs, a processwhich often demands a significant level of knowledge ofFRAP theory (Table 1). Alternatively, FRAP measurementscan be quantified simply in terms of the half timeof recovery (τ1/2), defined as the time required fora bleach spot to recover half way between initialand steady-state fluorescence intensities (16–19). Anadvantage of this approach is that τ1/2 can be readilyextracted from FRAP recovery curves (16–19). However,τ1/2 is strongly dependent on experimental parameterssuch as the nominal radius of the user-defined bleachspot (rn) and the exact bleaching protocol, as wellas the characteristic dynamics of the molecule underinvestigation. Therefore, τ1/2 is an empirical parameter that

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Table 1: Analytic 2D FRAP models

ReferencesBleachinggeometry

Laserprofiles

Postbleachprofile

Diffusion duringphotobleach

Datatypes Microscopy

Targetproteins

Kang et al. (9) Circle in 2 Gaussian (rn) Gaussian (re) Corrected f (t) CLSM/wide-fieldstatic

Slow/fast

Smisdom et al. (13) Circle in 2 Gaussian (ρ) Uniform withGaussian edge

Not corrected f (t) CLSM Slow

Axelrod et al. (6) Circle in 2 Gaussian(rn)/uniform (rn)

Gaussian(rn)/uniform (rn)

Not corrected f (t)/τ1/2 Wide-field static Slow

Braga et al. (8) Circle in 2 Uniform (rn) Gaussian (re) Corrected f (t) CLSM Slow/fastSoumpasis (7) Circle in 2 Uniform (rn) Uniform (rn) Not corrected f (t)/τ1/2 Wide-field static SlowSprague et al. (14) Circle in 2 Uniform (rn) Uniform (rn) Not corrected f (t) Wide-field static Slow/fastWaharte et al. (15) Circle in 2 not available Gaussian (re) Corrected f (x,t) CLSM Slow/fastCurrent study Circle in 2 Gaussian (rn) Gaussian (re) Corrected f (t)/τ1/2 CLSM/wide-field

staticSlow/fast

Angelides et al. (10) Circle in abox

Gaussian (rn) Gaussian (rn) Not corrected f (t) Wide-field static Slow

Deschount et al.(11)

Box in 2 Gaussian (ρ) Uniform withGaussian edge

Not corrected f (t) CLSM Slow

Seiffert &Oppermann (12)

Circle/linein 2

not available Gaussian (re) Corrected f (x,t)/re(t) CLSM Slow

2, infinite 2 dimensional plane; rn, bleaching spot radius = static laser radius; ρ, scanning laser radius; re, effective radius measured from

a fluorescent intensity profile; f (t), fluorescence intensity from an ROI; f (x,t), time course of spreading radius of postbleach profiles.

cannot be readily compared across studies. In contrast,D provides a quantitative measure of diffusion (6,7).Knowledge of the absolute magnitude of D has manyadvantages over measurements of τ1/2. For example,D is uniquely determined by the size of the diffusingmolecule as well as its local environment (20). Therefore,discrepancies between the theoretically expected D andmeasured apparent (or effective) diffusion coefficientsmay indicate possible molecular interactions or morecomplex situations. For example, in cell membranesthe presence of microdomains prevents proteins fromundergoing free diffusion (2). Accurate estimates of Dare also a necessary starting point for reaction–diffusionanalysis (1,21).

One potential solution to the problem of extractingquantitative D values from confocal FRAP data is to utilizethe explicit forms of the classical two-dimensional (2D)FRAP equations for small circular bleaching spots derivedby Axelrod for a Gaussian laser (6) and by Soumpasisfor a uniform laser (7). Both Axelrod and Soumpasis (6,7)reported equations that relate D, τ1/2 and rn for a pureisotropic diffusion model. For example, the Soumpasisequation is given by

Drn = 0.224r2n

τ1/2(1)

where rn is the radius of the uniform bleach laserand the coefficient 0.224 was numerically determined(7). However, the derivation of both the Axelrod andSoumpasis FRAP equations assumed that diffusion duringphotobleaching is negligible. This is not necessarily thecase for confocal FRAP. Due to the long scanning time ofCLSMs (approximate seconds), significant diffusion during

the photobleaching may occur in confocal FRAP. rn thusmay not provide an accurate description of the initialconditions required for this equation to be valid. Failureto take this into account can lead to underestimationof D when the Soumpasis or Axelrod equations areused to analyze confocal FRAP data, especially for fast-diffusing soluble proteins (8,9,22,23). This discrepancycan be resolved using a modified form of the conventionalFRAP equations by using the postbleach fluorescenceintensity profile to correct for any diffusion that occursduring the photobleaching by incorporating an empiricallydetermined measure, the effective bleach radius re fromthe postbleach profile (9). However, this approach alsorequires full fitting of the FRAP recovery curves tocalculate D. In this study, we sought to further simplifythis approach by deriving an equation to calculate D fromconfocal FRAP data obtained using circular bleach regions.

Theory

All the parameters and their meanings are summarizedin Table 2. For simplicity, we assume that cells are flatenough (as in the case of COS7 cells) so that both theplasma membrane as well as the cytosol can be treatedas 2D objects. We also assume that bleaching spot sizeis small compared with the cell size, so that we can treatthe cell as an infinite plane. Having these, laser intensityprofiles for photobleaching lasers in the infinite plane aredescribed as either a Gaussian laser:

Irn (x, y) = 2I0πr2

nexp

(−2

(x2 + y2

)r2n

)

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Table 2: List of parameters and their definitions

Parameter Meaning Parameter Meaning

D Diffusion coefficient F i Prebleach fluorescence intensityτ1/2 Half time of recovery F0 Initial postbleach fluorescence intensityrn Nominal radius = bleaching spot radius

= sampling region radiusF∞ Postbleach steady-state fluorescence intensity

re Effective radius = spreading radius ofpostbleach profile

F1/2 Fluorescence intensity at the half of recovery = [F0 + F∞]/2

q Quantum yield Mf Mobile fraction = [F∞ − F0]/[F i − F0]∈ Attenuation factor for excitation laser τD Diffusion time = r2

e / (4D)

K Bleaching depth parameter Drn D from Soumpasis equation using rn

(Drn = 0.224 r2n

τ1/2

)

γ The ratio of the nominal and effectiveradius = rn/re

Dre D from Soumpasis equation using re

(Dre = 0.224 r2e

τ1/2

)

Ci Prebleach fluorescent proteinconcentration

DConfocal D from = rn/re and τ1/2

(DConfocal = r2e +r2n

8τ1/2

)

n Number of independent experiments N Total number of cells for FRAP data collection

or a uniform laser,

Irn (x, y) = I0πr2

nH

(r2n − x2 − y2

)where H(·) is the Heaviside function. In both cases, rn isdefined as a nominal radius, and the excitation lasers arerepresented as εIrn (x, y) for attenuation factor ε(�1).

If we assume the concentration of fluorescent proteinsC(x,y,t) satisfies the diffusion equation

Ct = D�C

where D (μm2/s) is a diffusion coefficient and� = (∂2/∂x2) + (∂2/∂y2), then the solution can be com-puted from a convolution of the fundamental solutionof the diffusion equation and an initial condition as

C (x, y, t) =∫ ∫

C(x − x ′, y − y ′, 0

)�Dt

(x ′, y ′) dx ′dy ′

where the fundamental solution of the diffusion equationin the infinite plane is defined as

�Dt (x, y) = 14πDt

exp(

−x2 + y2

4Dt

).

Therefore, the fluorescence intensity from the bleach ROIcan be computed (6,9) from

F (t) = q∫ ∫

εIrn (x, y) C (x, y, t) dx dy

where q is the quantum yield of the fluorophoresand C(x,y,t) describes the concentration of fluorescentproteins within the confocal volume at time t.

It has been empirically shown that a confocal postbleachprofile can be described as a simple Gaussian function(constant minus Gaussian) (21);

C (x, y, 0) = Ci

(1 − K exp

(−2

(x2 + y2

)r2e

)), (2)

where Ci is the prebleach fluorescent protein concentra-tion, and re is the half width at the approximately 14%of bleaching depth from the top. We define re as theeffective radius of a postbleach profile, in contrast to thenominal radius (rn) from a user-defined bleaching spotradius.

Following the computation in Axelrod et al. (6,9), for asolution of the diffusion equation with unknown diffusioncoefficient D and an initial condition given by eqn (2), adiffusion FRAP equation for confocal FRAP can be found as

F (t) = Fi

{1 − K

1 + γ2 + 2t/τD

}Mf + (1 − Mf ) F0 (3)

where τD = r2e / (4D) and γ = rn/re. Mf is defined as

Mf = F∞ − F0

Fi − F0

for prebleach fluorescence intensity (F i ), postbleach initialfluorescence intensity (F0) and postbleach steady-statefluorescence intensity (F∞). K can be computed from eqn(3) at t = 0 (i.e. F(0) = F0),

F0 = Fi

{1 − K

1 + γ2

}

⇒ K = (Fi − F0)

Fi(1 + γ2). (4)

If we let F1/2 = F0+F∞2 , i.e. the fluorescence intensity at

the half time of recovery then, by definition, F(τ1/2) = F1/2.As F1/2 can be related to Mf by

F1/2 = (Fi − F0)

2Mf + F0,

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by setting F(τ1/2) = F1/2:

F1/2 = Fi

{1 − K

1 + γ2 + 2τ1/2/τD

}Mf + (1 − Mf )F0

⇒ 1 + γ2 + 2τ1/2

τD= K

1 − (Fi +F0)2Fi

;

⇒ 1 + γ2 + 2τ1/2

τD=

(Fi −F0)Fi

(1 + γ2

)1 − (Fi+F0)

2Fi

,

where we applied the definition of K in eqn (4). Bymultiplying 2Fi to both the denominator and numerator inthe right hand side, we obtain

1 + γ2 + 2τ1/2

τD= 2 (Fi − F0)

(1 + γ2

)Fi − F0

1 + γ2 + 2τ1/2

τD= 2

(1 + γ2

);

2τ1/2

τD=

(1 + γ2

).

Finally, by solving for D in τD = r2e / (4D), after applying

γ = rn/re, we have

DConfocal = r2e + r2

n

8τ1/2. (5)

When re = rn in eqn (4) (i.e. bleaching is instantaneous),we obtain D = 0.25 r2

nτ1/2

which is essentially identical to theSoumpasis equation (1). The approximately 3% differencein the proportionality constants in eqns (1) and (5) is due tothe different assumptions on either Gaussian or uniformlaser profiles.

Results and Discussion

To validate our new FRAP equations (eqns (3) and (5)), weused them to extract D values from confocal FRAP dataobtained using a CLSM under standard imaging conditionsfor a series of proteins and lipid probes localized either tothe plasma membrane, or within the cytosol (Figure 1).The proteins and lipid probes studied included Alexa488-conjugated cholera toxin B-subunit (Alexa-CTxB), aglycolipid-binding bacterial toxin commonly studied asa marker of lipid rafts; a model GPI-anchored protein(YFP-GL-GPI); a fluorescent lipid analog (DiIC16); andsoluble EGFP (Enhanced Green Fluorescent Protein) inthe cytosol (EGFP). Additionally, the protein Flotillin-1,which is diffusely distributed across the cell surface whenoverexpressed (24) was examined. Previously reporteddiffusion coefficients for these molecules span over threeorders of magnitude (Table 3), covering a range of D valuesthat are physiologically relevant for most studies of proteinand lipid dynamics in cells.

We performed confocal FRAP analysis of each of thesemolecules using a circular bleach region with a nominal

Table 3: Diffusion coefficients in cells previously reported in theliterature for the molecules examined in the current study byconfocal FRAP

Marker D (μm2/second) Methods References

CTxB 0.15–0.4 FRAP, FCS (5,25,26)YFP-GL-GPI 0.2–1.3 FRAP,SPT (5,26,27)DiIC16 0.9–5 FRAP, FCS, SPT (5,26,28,29)EGFP 20–70 FRAP, FCS (21,30,31)

bleach radius (rn) of 1.1 μm by repetitively scanningthe bleach region. We verified that when diffusion wasinhibited by fixing the sample prior to bleaching thatthe radius of the bleach region was similar to theuser-defined nominal bleach radius (Figure 2). Next, wecalculated the effective bleach radius (re) and bleachdepth (K ) from the postbleach fluorescence profile(Figure 3A,B). For slowly diffusing molecules such asCTxB, the effective bleach radius calculated from thepostbleach profile was only slightly larger than the nominalbleach radius (Figure 2). However, for the majority ofmolecules examined exchange of fluorescently taggedproteins outside of bleach region was evident. Thiseffect was especially pronounced for the fastest diffusingmolecules (Figure 3A,B). Importantly, the postbleachprofiles were well described by an exponential functionfor all of the molecules examined. This indicates that thisapproximation provides a reasonable description of thedata even for molecules with widely ranging diffusionalmobilities and thus our FRAP equations are valid underthese conditions.

We next interpolated the recovery curves to obtain τ1/2(Figure 3C, Table 4), and computed D either by fitting theFRAP data with eqn (3) (Figure 3C) or by plugging thesenumbers directly into the simplified DConfocal equation(eqn (5)). As illustrated in Figure 4, essentially identical Dvalues were obtained using eqn (3) (DFitting) and eqn (5)(DConfocal) (Student’s t-test, p > 0.05) showing that the twoapproaches yield similar results as expected. In addition,the D values obtained in this way were in good agreementwith the range of values previous reported from studiesusing more elaborate approaches to quantify D (Table 3).Moreover, the estimated Ds obtained using DConfocal andDFitting are independent of experimental conditions asassessed by comparing measurements of EGFP diffusionin the cytoplasm using different bleaching spot size andphotobleaching times (Figure 5 and Table 5 (9), Student’st-test, p > 0.05).

To illustrate the importance of using the corrected valuesof bleach radius to calculate D values, the same FRAPdata were analyzed by Soumpasis equation (1) usingeither rn (Drn ) or re (Dre ). This approach yielded Ds thatwere significantly different from those measured usingeqn (3) or (5) (Figure 4, Student’s t-test, p < 0.05). Therelation between DConfocal and Drn can be easily seen from

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A B

D E

C

Figure 1: Representative images of the subcellular distribution of the proteins and lipids studied. COS7 cells transfected orexogenously labeled with (A) Alexa-conjugated cholera toxin B-subunit (Alexa-CTxB), (B) a model GPI-anchored protein (YFP-GL-GPI), (C)Flotillin-1-RFP (Flot-1-RFP), (D) a fluorescent lipid analog (DiIC16) or (E) EGFP. Scale bar = 10 μm.

Table 4: FRAP parameters and measured Ds of various markers.

Markersparameter

Alexa-CTxB(n = 5, N = 56)

YFP-GL-GPI(n = 3, N = 36)

RFP-Flot(n = 3, N = 33)

DiIC16

(n = 4, N = 58)EGFP

(n = 3, N = 37)

re (μm) 2.7 ± 0.2 3.2 ± 0.1 2.3 ± 0.1 3.0 ± 0.1 7.1 ± 0.2τ1/2 (second) 4.1 ± 0.7 1.8 ± 0.1 0.97 ± 0.05 0.43 ± 0.28 0.16 ± 0.03K 0.95 ± 0.04 0.97 ± 0.02 0.79 ± 0.06 0.64 ± 0.06 0.61 ± 0.09Mf 0.98 ± 0.04 0.99 ± 0.02 0.95 ± 0.04 0.93 ± 0.02 0.95 ± 0.04DFitting(μm2/second) 0.26 ± 0.03 0.73 ± 0.04 0.72 ± 0.08 2.7 ± 0.7 41.3 ± 8.2

re (μm), effective radius measured from the averaged postbleach profiles; τ1/2 (second), halftime-of-recovery of a FRAP curve; K ,photobleaching depth parameter; Mf , mobile fraction estimated by data fitting; DFitting, diffusion coefficient determined by data fitting;n, number of independent experiments; N, total number of cells. Mean ± SD over n.

0 < γ ≤ 1 and

Drn

DConfocal= 1.8γ2

1 + γ2 ,Dre

DConfocal= 1.8

1 + γ2 (6)

which indicates that

Drn < DConfocal < Dre . (7)

Therefore, Drn and Dre can be understood as the upperand lower bounds for D. Importantly, eqns (6) and (7)provide the analytic relations between the equations thatrelate D and τ1/2 for conventional and confocal FRAP

in terms of γ, the ratio rn/re. This relationship alsoindicates that τ1/2 will not scale proportionally with D underconditions where re > rn, providing further evidence thatτ1/2 is not a good approximator of diffusional mobility.For example, depending on the exact imaging conditionsused, molecules with similar values of D can showdifferent values of τ1/2 (compare results for YFP-GL-GPIand RFP-Flot in Table 4).

The non-fitting method using eqn (5) heavily hinges onobtaining accurate measurements of re and τ1/2, whichcan be achieved by analysis of large amount of FRAP

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A

B C

Figure 2: Diffusion of Alexa-CTxB on live versus fixed cells. A) Representative images of Alexa-CTxB on the plasma membraneduring a FRAP experiment on either live or fixed cells for rn = 1.1 μm. Scale bar = 1 μm. B) Postbleach profiles of Alexa-CTxB on theplasma membranes of live (�, n = 12) and fixed (©, n = 5) cells. (C) FRAP data of Alexa-CTxB on the plasma membranes of live (�,n = 12) and fixed (©, n = 5) cells. As the bleach ROI is slightly off center in our system as seen in the images of (A) at t = 0, a correctionwas made to align the center of the postbleach profile to determine re.

Table 5: Comparison of FRAP parameters for EGFP in the cytosol obtained by confocal FRAP under different experimental conditions

1.1 μm diameter ROI 2.2 μm diameter ROI

Experimental setup parameters 20 iterations (N = 8) 40 iterations (N = 10) 20 iterations (N = 10) 40 iterations (N = 8)

rn (μm) 0.55 0.55 1.1 1.1re(μm) 7.2 7.3 8.6 9.3τ1/2 (second) 0.28 ± 0.10 0.30 ± 0.12 0.22 ± 0.02 0.28 ± 0.02K 0.29 ± 0.01 0.32 ± 0.02 0.38 ± 0.01 0.41 ± 0.01Mf 0.89 ± 0.04 0.89 ± 0.03 0.84 ± 0.01 0.87 ± 0.02

rn (μm) user defined nominal radius; re (μm), effective radius measured from the averaged postbleach profiles; τ1/2 (second), halftime-of-recovery of an individual FRAP curve; K , photobleaching depth parameter; Mf , mobile fraction calculated by eqn (3); iterations, thenumber of cycles in photobleaching laser scans. FRAP curves (n = 1) taken from (9) were re-analyzed without correction for backgroundand photofading. Mf s found in Table 4 were estimated from data fitting and are therefore larger than Mf s in Table 5. N, total number ofcells. Mean ± SE over N.

data (n > 3, and N > 10). Here, the statistics of individualre, τ1/2, D and Mf were carefully compared with thevalues from the averaged FRAP curve for a set ofexperiments. We found that there was no significantdifference between the values found from averaged FRAPdata and mean values of the parameters obtained overmultiple individual FRAP data as long as the FRAP dataset

size is large (Figure 6, Student’s t-test, p > 0.05). Workingwith averaged FRAP data is more advantageous than usingthe mean value arising from individual measurementsfrom a computational cost perspective, especially for noisydata from a small ROI size (rn) or fast diffusion. We alsocarefully compared D and Mf obtained from individualFRAP data with the values from the averaged FRAP curve

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A B C

Figure 3: Confocal FRAP data of Alexa-CTxB, YFP-GL-GPI, Flot-1-RFP, DiIC16 and EGFP. A) Representative pre- and postbleachimages. Dashed circles represent user-defined bleaching spots with diameter 2.2 μm (i.e. rn = 1.1 μm). To speed data acquisition for theFRAP analysis, we cropped the imaging window to make it slightly larger than the bleaching spot size. Scale bar = 1 μm. B) Averagedpostbleach profiles for N = 12∼14 cells along the diagonals of postbleach images (�) and the best fitting Gaussian (eqn (2), – ).Dotted vertical lines show x = ± rn (user-defined bleach ROI). Dash-dot horizontal lines are 14% of bleaching depth from the top. C)Representative FRAP recovery curves (�) and the best fit to eqn (3) ( – ). Insets show full FRAP curves for Alexa-CTxB and YFP-GL-GPI.In (B) and (C), y-axes are normalized fluorescence intensities, i.e. F i = 1. Since the bleach ROI is slightly off center in our system, acorrection was made to determine re as described in the Materials and Methods.

for a set of experiments to check if the normalization ofthe FRAP curves incidently masks biological variability,such as expression level-dependent diffusion coefficients(32). However, for the proteins and lipids examined here,D and Mf obtained from individual FRAP data did notshow any significant differences from those obtained fromthe averaged FRAP curves (Figure 6, Student’s t-test,p > 0.05).

In a previous study, we found that re varies with both rnand bleaching time, and is more sensitive to rn than thebleaching time. For larger rn and longer bleaching time, alarger re is obtained (9,21). In addition, the sensitivity ofre to experimental definition of rn and bleaching time ishigher for rapidly diffusing proteins (9,21). Nevertheless,

variations of re and τ1/2 are constrained by the relationgiven by eqn (5) for large range of K as long as the postbleach profiles can be described by a Gaussian (eqn (2)).

In summary, we derived and tested a new, simplifiedequation for the quantitative analysis of confocal FRAPdata based on the pure isotropic diffusion model. Anoverall workflow to determine the apparent diffusioncoefficient, D from τ1/2, rn and re is summarized inFigure 7. Taking advantage of widely available technology,this new approach should be accessible to manybiologists, since most researchers already have accessto CLSMs. Importantly, this method does not requiregeneration of new constructs or the use of specializedfluorescent dyes or labels, and can be applied to both

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Figure 4: Comparison of diffusion coefficients determined

by various schemes. Comparison of diffusion coefficientsdetermined by FRAP data fitting (DFitting, eqn (3)), versus theDConfocal equation (eqn (5)), or the Soumpasis equation usingeither rn (Drn ) or re (Dre ) in log scale. Ds were found fromaveraged FRAP curves (N ≥ 12 cells per experiment) for three ormore separate experiments (n≥3). Error bars represent standarderrors. Dashed boxes show D’s reported in the literature (Table 3).*p < 0.05 compared to DFitting, Student’s t-test.

soluble and plasma membrane-associated molecules asillustrated here. Finally, it should be possible to readilyincorporate this method into automated approachesto perform and analyze FRAP (33), and thus providequantitative values for use in systems biology.

Materials and Methods

Cell labeling and reagentsCOS-7 cells were acquired from ATCC (Manassas). Cells were maintainedin DMEM containing 10% fetal bovine serum at 37◦C and 5%CO2. Cells were plated on coverslips 2 days prior to experiments.Cells were transfected 1 day prior to imaging with the model GPI-anchored protein (YFP-GL-GPI) (5), Flotillin-1-RFP (Flot-1-RFP, the giftof Dr Ai Yamamoto, Columbia University) (34) or EGFP (Clontech)using FuGENE 6 transfection reagent (Roche Diagnostics). DiIC16(1,1′-dihexadecyl-3,3,3′,3′-tetramethylindocarbocyanine perchlorate) andAlexa488 fluorophore-conjugated cholera toxin B-subunit from Vibriocholerae were obtained from Invitrogen. For exogenous labeling the cellswere rinsed twice with media and then incubated for 5 min at roomtemperature with 100 nM Alexa488-CTxB or 5 μg/mL DiIC16. Cells werethen rinsed twice with media and imaged. Cells were maintained in DMEMsupplemented with 1 mg/mL Bovine Serum Albumin and 25 mM HEPESbuffer during imaging. For FRAP on fixed samples, cell were labeled livewith Alexa488-CTxB (as above). Immediately after labeling, cells were fixedwith 3.4% paraformaldehyde for 15 min at room temperature. Cells werethen rinsed with 1× PBS and mounted in Fluoromount G supplementedwith 25 mg/mL DABCO (1,4 diazabicyclo[2.2.2]octane) (Sigma-Aldrich) andallowed to solidify overnight prior to imaging.

FRAP methodsFRAP experiments were carried out on a Zeiss LSM510 confocalmicroscope (Carl Zeiss MicroImaging) using filter sets provided by themanufacturer. Imaging was performed using a 40 × 1.3 NA Zeiss Plan-Neofluar objective at 4× digital zoom. The confocal pinhole was set

Figure 5: Comparison of diffusion coefficients for EGFP

in the cytosol obtained by confocal FRAP under different

experimental conditions as calculated using the Soumpasis

equation or DConfocal equation. Diffusion coefficients weredetermined by FRAP data fitting (DFitting, eqn (3)), by the DConfocalequation (eqn (5)), by the Soumpasis equation using rn (Drn ), andby the Soumpasis equation using re (Dre ) for individual confocalFRAP curves (N ≥ 6). Dashed box indicates the range of EGFP’sdiffusion coefficients in the cytosol reported in the literature.re was measured from an averaged postbleach profile (n = 1,N = 10 cells) and Ds were obtained from individual FRAP data(n = 1, N = 8, 10, 10 and 8 cells). *p < 0.05 compared to DFitting,Student’s t-test.

between 1.78 and 1.81 Airy units. The imaging window was furthernarrowed to a square observation ROI of 70 × 70 pixel (7.7 × 7.7 μm).The 70 × 70 window contained a circular bleach ROI 20 pixels (2.2 μm)in diameter. FRAP conditions were optimized for each molecule understudy. Samples were imaged and bleached using a 30 mW Argon laser(488 and 514 laser lines) or 1.0 mW HeNe laser (543 nm laser line). Laserpowers for prebleach and postbleach imaging were maintained between6.8 and 17.1 nW (488 or 514 nm) or between 13.8 and 21.0 nW (543 nm).Bleaching of Alexa 488-CTxB was performed using the 488 nm line at 0.99μW. EGFP and Flot-1-RFP were bleached using the 488 nm line at 2.28μW, and YFP-GL-GPI was bleached using the 514 nm line at 1.38 μW. Tobleach DiIC16 we combined the 488 nm line (40 nW), 514 (360 nW) and543 (540 nW). Bleaching regions were scanned 10 (Alexa-CTxB, YFP-GL-GPI) or 20 times (EGFP, Flot-1-RFP or DiIC16). Prebleach and postbleachimages were collected with either no line averaging or with line averagingof 2. Time series images were collect using 2 seconds delay (Alexa-CTxB), 0.25 seconds delay (YFP-GL-GPI) or no delay (Flot-1-RFP, DiIC16,EGFP). This resulted in bleach times of 1.12 seconds (Alexa-CTxB), 0.69seconds (YFP-GL-GPI), 0.56 seconds (Flot-1-RFP), 0.49 seconds (DiIC16)and 0.27 seconds (EGFP). During recovery, images were collected every2.27 seconds (Alexa-CTxB), 0.52 seconds (YFP-GL-GPI), 0.14 seconds(Flot-1-RFP), 0.068 seconds (DiIC16) or 0.035 seconds (EGFP). All FRAPwas performed at 37◦C using a stage heater.

Correction for background and photofadingTo correct for background (Fbk), fluorescence from a cell-free area ofthe coverslip was measured under the same conditions as FRAP wasperformed. To correct for observational photofading (FFading(t)), a timeseries was collected as for the FRAP studies, except no photobleachingstep was performed. The raw FRAP data, FRaw(t), were corrected for

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A B

C D

Figure 6: Comparison of values obtained from averaged FRAP data versus mean of values from individual FRAP data.Comparison of mean ± SE of (A) re, (B) τ1/2, (C) D and (D) Mf determined using averaged FRAP data from more than three independentexperiments with 10 cells (n ≥ 3, N = 10) or means from 10 individual FRAP data in a single experiment (N = 10). Error bars representstandard errors. p > 0.05 cross comparison, Student’s t-test.

background and photofading by

FCorrected (t) = FRaw (t) − FbkFFading (t) − Fbk

.

Finally, the corrected data were normalized by the prebleach intensity,F i

Corrected as

F (t) = FCorrected (t)

FiCorrected

.

τ1/2 measurementsTo measure τ1/2 from the FRAP data a linear interpolation method wasused. For the FRAP data {F(0), F(t1), F(t2), · · ·, F(tn)} such that F(0) = F0and F(tn) = F∞, the fluorescence intensity at half of recovery is defined asF1/2 = (F0 + F∞)/2. If F(tk ) = F1/2 for some tk then we chose τ1/2 = tk . IfF(tk ) < F1/2 < F(tk + 1) then we chose

τ1/2 = tk +[F1/2 − F

(tk

)][F

(tk+1

) − F(tk

)] (tk+1 − tk

).

re measurementsIn order to obtain normalized initial postbleach profiles, fluorescenceintensities were measured along a diagonal of the square observation

ROIs of 70 × 70 pixel (7.7 × 7.7 μm) from the prebleach image obtainedimmediately prior to and after photobleaching. Between the profilesobtained from two diagonals, the dataset with better symmetry waschosen for analysis. The profiles were then normalized by dividing thepostbleach profile by the prebleach profile. The normalized postbleachprofiles were averaged over datasets (N = 12∼14). Since the bleach ROIwas slightly off center in some cases, if necessary, the symmetry axis wasfound and set to be x = 0 from the normalized mean postbleach profile.The mean profiles were then fitted to

fPostbleach (x) = 1 − K exp

(− x2

r2e

)

for K and re using a nonlinear least-squares fitting routine (nlinfit.m)available in MATLAB® (version 7.10, R2010a, The Mathworks Inc.).

Alternatively, re can be determined by direct measurement followingthree steps. First, K can be determined from the bleaching depth inthe normalized postbleach profile as referred as in Figure 8. Then,the half width of cross-sections between the horizontal line at theheight of 0.86 K from the bottom of the postbleach profile (Figure 8)and the postbleach profiles yields re without involving any fitting(Figure 8).

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Figure 7: Workflow of applying the DConfocal equation.

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Figure 8: Determination of re from a postbleach profile.

FRAP data fitting for D and Mf and weighted residualThe FRAP curves were normalized by the mean prebleach fluorescenceintensity (i.e. F i = 1) and the mean of multiple normalized FRAP datasetswas used for data fitting. Data fitting was carried out for D and Mf bya nonlinear least-squares fitting routine (nlinfit.m) available in MATLAB®(version 7.10, R2010a, The Mathworks, Inc.) minimizing a weightedresidual between averaged FRAP data from 10 experiments (FData(t))and a theoretical FRAP curve (F(t), eqn (3)), where the weighted residualis defined by

||FData (t) − F (t) || =s∫

0

∣∣FData (z) − F (z)∣∣

z +s∫

0

FData (w) dw

dz.

where s is the end time of FRAP data. If F(0) computed using K frompostbleach profile fitting showed any discrepancy from FData(0), K wasrecalibrated using K = (1 + γ2)(1 − FData(0)) before data fitting.

Acknowledgments

This work was supported by R01 GM073846, NSF/DMS 0970008 and aVanderbilt Discovery grant. The funding sources had no role in the studydesign, collection, analysis or interpretation of data, writing the report orthe decision to submit the article for publication.

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