Total internal reflection fluorescence (TIRF) microscopy

I. Modelling cell contact region fluorescence

W. M. REICHERT* and G. A. TRUSKEY

Department of Biomedical Engineering, and Center for Emerging Cardiovascular Technologies, Duke University, Durham, NC 27706,USA

* Author for correspondence

Summary

Total Internal Reflection Fluorescence (TIRF) is apowerful technique for visualizing focal and closecontacts between the cell and the surface. Practicalapplication of TIRF has been hampered by the lackof straightforward methods to calculate separationdistances. The characteristic matrix theory of thindielectric films was used to develop simple exponen-tial approximations for the fluorescence excited inthe cell-substratum contact region during a TIRFexperiment. Two types of fluorescence were exam-ined: fluorescent]}' labeled cell membranes, and afluorescent water-soluble dye. By neglecting the re-fractive index of the cell membrane, the fluorescenceexcited in the cell membrane was modelled by asingle exponential function while the fluorescence inthe membrane/substratum water gap followed aweighted sum of two exponentials. The error associ-

ated with neglecting the cell membrane for an inci-dent angle of 70 degrees never exceeded 2.5%,regardless of the cell-substratum separation dis-tance. Comparisons of approximated fluorescenceintensities to more exact solutions of the fluorescenceintegrals for the three-phase model indicated that theapproximations are accurate to about 1% formembrane/substratum gap thicknesses of less than50 run if the cytoplasmic and water gap refractiveindices are known. The intrinsic error of this modelin the determination of membrane/substratum sep-arations was 10 % as long as the uncertainties in thewater gap and cytoplasmic refractive indices wereless than 1 %.

Key words: TIRF, cell contact, cell membrane.

Introduction

Total internal reflection fluorescence (TIRF) has become astandard technique for exciting spectroscopic phenomenaat interfaces (Reichert, 1989). Briefly, the total internalreflection of visible light at a glass/solution interfaceproduces a region of electromagnetic intensity (evanescentwave) that penetrates a few tenths of a micrometer intothe lower refractive index medium and can excite spectralphenomena in a region confined to the solution side of theinterface. The specific application of TIRF to the imagingof contacts of adherent cells to surfaces was developed byAxelrod et al. (1982). The ability to label specific regions ofthe cell has made TIRF an attractive technique forcharacterizing cell—substratum contacts. Recent appli-cations of TIRF microscopy include the visualization ofamoeba (Todd et al. 1988), fibroblast (Lanni et al. 1985),muscle (Gingell et al. 1986), erythrocyte (Axelrod et al.1986), endothelial (Nakache et al. 1986) and leukemia(Weis et al. 1982) cells on glass or quartz substrata.

Cell adhesions are classified in terms of close contacts(membrane within 30 nm of substratum surface) and focalcontacts (substratum within 15 nm of substratum surface)(Burridge et al. 1988). An important factor in the quanti-tation of cell-surface separations is the development of amodel that relates image intensity with the proximity ofthe membrane to the substratum surface. Gingell, Hea-vens and Mellor recently published the general electro-Joumal of Cell Science 96, 219-230 (1990)Printed in Great Britain © The Company of Biologists Limited 1990

magnetic theory for TIRF illumination of adherent cellsfor the case of a water-soluble fluorescent label in whichthe cell-substratum contact region was modeled as adielectric lamellar structure (Gingell et al. 1987). Calcu-lations of expected fluorescence intensity for several inci-dent angles of the totally reflected beam were presented asa function of the water gap distance between the cellmembrane and the substratum surface. TIRF is muchmore sensitive to gap thicknesses of the order of 0-70 nmthan is the more widely used technique of interferencereflection microscopy (Gingell et al. 1987; Bailey andGingell, 1988).

In the Gingell model, cell-substratum contact regionswere treated optically as a four-phase system consisting ofa glass substratum, a water-filled gap, a lipid membraneand the cell cytoplasm. The glass and cell were treated assemi-infinite media while the water and cell membraneswere treated as thin dielectric films. Closed form ex-pressions were derived for the distance dependence of theelectric field amplitude in the watei1 gap regions for thefour conditions under which the electric field in the watergap, cell membrane and cell cytoplasm was either continu-ous (propagating) or evanescent (decaying). Although theclosed form expressions are exact solutions to the fieldequations, the expressions presented are difficult to applyin the determination of separation distances.

An alternate approach, adopted in this paper, is todevelop simpler expressions for the fluorescence that are

219

valid for the particular conditions of cell-glass contacts inan aqueous medium. Characteristic matrix theory (Han-sen, 1968), in which the same matrix form is used whetherthe field in a given medium is either evanescent orcontinuous, was used to determine the expected TIRFemission excited using transverse electric (TE)-polarizedincident radiation totally reflected at the glass-cell inter-face. Fluorescence excited from fluorescently labelled cellmembranes and cells with cell-substratum water gapsinfused with fluorescent solutions are considered. Thesecalculations show that the lipid membrane may be ignoredwith an error of no greater than 2.5% and the cell-sub-stratum contact may be modelled as a comparatively moresimple three-phase system. Approximate analytical ex-pressions are developed for these two TIRF excitationconfigurations that permit straightforward calculation ofthe cell-substratum water gap thickness from exper-iment.

Theory

Fluorescent emission excited during total internalreflectionConsider a beam of light containing both transverseelectric (TE, perpendicular, or s) and transverse magnetic(TM, parallel, or p) polarized electric vectors incident upona dielectric interface at the angle 6x (Fig. 1). The fluor-escence emission excited by the transmitted component ofthe incident beam between the limits Zi and z2 relative tothe reflecting interface is given by the general expression(Suci and Reichert, 1988):

F = (AjcoBd) I * <p(z)a(z)c(z)T{z) dz, (1)

where <p(z) is the quantum efficiency of the fluorophore,a(z) and c(z) are the molar extinction coefficient andconcentration of the fluorophores, Aj/cos&x is the interfa-cial area of illumination written in terms of the cross-sectional area of the incident beam Au and T\z) is the zdependence of the magnitude squared of the transmittedelectric field amplitude given by:

homogeneous (c(z)=c), then for a given angle of incidenceequation (1) becomes:

= \E(z) (2)

where * designates the complex conjugate and • designatesthe dot product. If one assumes that (1) the quantumefficiency and extinction coefficient of the fluorophore areindependent of the proximity to the totally reflectinginterface and spatial orientation (cp(z)a{z)=(pa), and (2) thefluorophore distribution between the limits Z\ and z2 is

Medium 2

Fig. 1. Schematic illustration of beam incident at the angle #iupon a dielectric interface between medium 1 and medium 2.Vectors on incident beam indicate the orientation of the electricvector for transverse electric (TE) and transverse magnetic(TM) polarizations.

- ' J * T(z)6z, (3)

where K^iAi/cosd^Qae. The above assumptions consti-tute an ideal case and actual experimental conditions maybe subject to spatial variations in fluorescence emissionresulting from fluorophore orientation with respect to theincident beam, fluorophore proximity with respect to thesubstratum interface, and molecular aggregation of flu-orophore molecules. The importance of these phenomenato the quantitative interpretation of TIRF images isconsidered in the Discussion.

For the ideal case, TIRF will occur when the incidentangle exceeds the critical angle for total internal reflection(0c) defined by the higher refractive index of the incidentmedium (rei) and the lower refractive index of a semi-infinite homogeneous solution of fluorophores (n2):

ec = sin-\n2/ni), (4)

in which case the term T\z) reduces to the simple exponen-tial:

= T12(0)exp(-z/d12), (5)

where 7^(0) is the transmitted amplitude squared at then\jni interface and d12 is the depth of penetration of theevanescent wave. Substitution of equations (4) and (5) intoequation (3) followed by integration yields the expressionfor fluorescence excited by TIRF in homogeneous aqueoussolutions between the limits zx and z2:

F = #r12(O)d12[exp(-2l/d12)-exp(-z2/d12)]. (6)

For solutions considered to behave optically as trans-parent media, the terms 7\2(0) and d12 for TE polarizedincident light are given by the expressions:

r12(0) =

dl2 =

1/[l-(rc2/n1)2] (7)

n201-(n2/Ai1)2]1. (8)

Analogous expressions for TM polarized light are alsoavailable (Hansen, 1968).t In the ideal case, values for therefractive indices and the angle of total reflection may besubstituted directly into equation (6) to yield exact sol-utions for the fluorescence excited during a TIRF exper-iment.

Fluorescent emission from thin dielectric filmsWhen the fluorophores reside in a thin dielectric filmadjacent to the incident phase, the expected fluorescentemission is still represented by equation (1), except thatthe term 7\z), the magnitude of the total electric fieldamplitude squared, now consists of two traveling waves: aforeword traveling wave propagating down from the topsurface of the film and a reverse traveling wave propagat-ing up from the bottom surface of the film. Expressions forthis field are derived from two optical principles: (1) thewave equation must be satisfied in each medium, and (2)the tangential components of the electric and magneticfields must be continuous across each interface.

tThe expression for TM polarized transmitted intensity duringtotal internal reflection was reported incorrectly by Reichert(1989). The correct expression (using current notation) shouldhave read:

T rau - T row 2sin2e1-(n2/n1)2

220 W. M. Reichert and G. A. Truskey

The characteristic matrix theory provides a convenientmethod of calculating the tangential boundary conditionsat one interface of a thin dielectric film in terms of theadjacent interface. Consider a film (medium 2) sandwichedbetween two semi-infinite media (medium 1 and medium3). The upper and lower interfaces of the film are definedby the planes 2=0 and z= A2, respectively, where A2 is thefilm thickness. Following the treatment of Hansen (1968),if Q2(0) and Q2(A2) are 1x2 matrices that represent thetangential components of the electric and magnetic fieldamplitudes at the upper film and lower film boundaries,respectively, then Q2(A2) may be calculated from Q2(0) asfollows:

Q2(A2) = N2(A2)Q2(0), (9)

where iV2(A2) is the 2x2 inverse characteristic matrix ofthe film that accounts for reflection and transmission atthe upper and lower film boundaries along with the phasechange of light that traverses the film. Similarly, one cancalculate the tangential components of the electric andmagnetic field amplitudes at any location z in the film,(Q2(z)) using the analogous matrix expression:

Q2(z) = N2(z)Q2(0), (10)

where Q2(z) and N2{z) are identical in form to Q2(^2) and^ ( A ^ , except the location z is substituted for the filmthickness A2. Exact expressions for the matrices inequations (9) and (10) are available elsewhere (Hansen,1968).

The characteristic matrix technique will give the tan-gential (i.e. x and y) components of the electric andmagnetic field amplitudes at any location 2 in a laminateof films as long as the thicknesses and refractive indices ofeach film and the boundary conditions at the first interfaceare known. For TE polarized light, Ey{z) is element 11 ofthe matrix Q2(2), allowing one to calculate directly the 2-dependent distribution of the TE-polarized electric fieldintensity from the expression:

J\2>JE = |.ZSy(.2,)| . (.J--U

The characteristic matrix software used in this paper wasgenerously provided to these authors by W. H. Hansen ofUtah State University, and subsequently modified locallyfor use with Microsoft Fortran in a Macintosh computer. Amore detailed explanation (but less rigorous than Han-sen's derivation) of this technique is available from Rei-chert (1989).

Results

The characteristic matrix theory permits calculation ofthe electric field intensity T[z) in a laminate of thin films.For TE-polarized incident radiation, this technique wasused to calculate values of the evanescent field intensityrelevant to TIRF microscopy of cell-substratum contacts,from which the excited fluorescence values are calculatedfrom equation (3). Although the calculations Eire straight-forward, major simplifications in the analysis can be madeby neglecting the membrane thickness and refractiveindex, which reduces the analysis to a three-phase system(glass/water gap/cytoplasm). This simplification leads tothe derivation of experimentally useful expressions for thefluorescence excited in the cell-substratum contact re-gion.

Evanescent field intensity in cell-substratum contactsConsider a region of a cell adherent to a glass substratum

Glass substratum /i, = 1.52

. Water gap2 /;-,= 1.333

MembraneAm=4nm»m=1.45

Fig. 2. Four-phase optical model of a cell-substratum contactregion. Refractive indices were derived from Gingell et al.(1987).

(Fig. 2). The region directly below the incident glassmedium (/ii = 1.52) is occupied by a water gap (/i2=1.333)with a thickness A2 varying from close to zero to approxi-mately 100 nm, a 4 run thick cell membrane (Am=4nm,^=1.45) , and the cell cytoplasm (713= 1.37), which wastreated as a semi-infinite medium. Referring to equation(1), the fluorescence observed in the cell-substratumcontact region is excited by the electromagnetic intensitytransmitted across the interface. However, the form of thetransmitted intensity is a function of the refractive indi-ces, the incident angle, the water gap thickness and themembrane thickness.

Fig. 3A contains matrix calculations of 7\0), plotted as afunction of incident angle, for a four-phase model consist-ing of glass/water/membrane/cytoplasm for differentvalues of water gap thicknesses ranging from 0^A2^1.5X.Performance of the same calculations, but with a three-phase glass/water/cytoplasm system that neglects thepresence of the cell membrane, yields a plot that appearsto be identical to Fig. 3A (not shown). The percentagedifference between the interfacial intensities calculatedwith and without the cell membrane is shown in Fig. 3Bfor water gap thicknesses of A2^0.5A. In the blue-greenregion of the spectrum theses values correspond to watergap thicknesses of the order of 100 nm or less.

The first conclusion to be drawn from Fig. 3 is that thepresence of the cell membrane has little effect on theinterfacial intensity (i.e. the error of ignoring the cellmembrane reaches a maximum at a zero water gapthickness, but never exceeds an asymptote of 2.5 %). Asdiscussed in the next section, this result affords us theluxury of neglecting the cell membrane. Second, increas-ing the water gap thickness causes the intensity peak nearthe glass/cytoplasm critical angle (0C=64.33) to diminishand be replaced by a growing intensity peak near theglass/water critical angle (0C=61.23). Since the interfacialintensity reaches a maximum at the critical angle, thenthis transformation suggests an effective refractive indexfor the cell-substratum contact that shifts to lower angleswith increasing water gap thickness. Fig. 4 plots theeffective critical angle (incident angle of maximum inter-facial intensity) for a four-phase system (cell membraneincluded in calculation) plotted as a function of the watergap thickness. Here one sees that the critical angleremains essentially constant up to a water gap thicknessof approximately 0.75A before it begins to drop toward aglass/water critical angle (this transformation is not fullycompleted until A2=5A). Also note that the cell membraneappears to increase slightly the effective refractive indexof the water gap/cytoplasm region as evidenced by acritical angle at small water gap thicknesses (0C=64.44),which is slightly greater than the glass/cytoplasm criticalangle (i.e. for 0C=64.44 we obtain an effective refractiveindex of 1.52sin(64.44)=1.3712). Third, the angle of inci-

TIRF microscopy 221

4.0-

-2 .0 -

A , = I.5A

A-.=0.0

60 62 64 66Incident angle (degrees)

68 70

Fig. 3. A. Magnitude squared of transmitted interfacialamplitude (interfacial intensity 7X0)) plotted as a function ofincident angle for water gap thicknesses ranging from0^A2^1.5A; B, percentage error in the calculated interfacialintensity upon neglecting the cell membrane for OsAj^O.SA.Although values were calculated for 0.1A increments in watergap thickness, only selected gap thicknesses are displayed forclarity: A2/A=0.0 (D), 0.1 (•), 0.2 (O), 0.4 (•) , 0.5 (A), 0.8 (A),1.1 (<0>) and 1.5 ( • ) . All values calculated for TE-polarized lightat 0.2 angular increments.

dence for a TLRF experiment must be greater than thecritical angles for both the cytoplasm/glass interface andthe water/glass interface (e.g. 0C=7O degrees) to ensurethat an evanescent field exists throughout the regionoccupied by the cell-substratum contact.

Fig. 5 presents calculations of the evanescent intensityin a cell-substratum contact region, for an incident angleof 70 degrees, plotted against water gap thicknesses from0<A2^A. The filled and open circles again represent theevanescent intensity decay for simple glass/cytoplasm andglass/water interfaces, respectively. Similarly, we notethat the evanescent intensity undergoes a transition froma glass/cytoplasm to a glass/water system with increasingwater gap thickness. However, in all cases the evanescentintensity decays essentially to zero within one wave-length's distance from the interface and virtually nofluorescence would be excited beyond this distance. FromFig. 5, the important parameter that determines theintensity during a TIRF experiment (if one maintains thesame assumptions that went into development of equation

ang

ical

cnt

ive

Eff

ect

i

64-

63-

62-

61-

h-l t 1glass/cytoplasm

\

\

I\

glass/water

0.0 0.5 1.0 15Water gap thickness (A:/lambda)

Fig. 4. Effective critical angle plotted as a function of watergap thickness. Effective critical angle determined from incidentangle of maximum interfacial intensity vis-a-vis Fig. 3A for

A including all values calculated at 0.1A increments.

2.0-

1.5-

1.0-

0.5-

0.0-

VX}A2=A

A2=0

/

1 - , -

0.0 0.5 1.0 1.5Distance from interface (;/lambda)

Fig. 5. Calculations of the decay of the evanescent electric fieldamplitude squared (T(z)) plotted as a function of distance fromthe glass surface for water gap thicknesses ranging from 0 to A.Filled and open circles represent values of 1Xz) calculated forsimple glass/cytoplasm and glass/water interfaces, respectively.Values calculated using an incident angle of 70 degrees and TE-polarized light.

(3)) is the overlap integral between the decaying evan-escent intensity and the fluorophore distribution near theinterface. Greatly simplified expressions for the fluor-escence overlap integral, and thus fluorescence excited ina cell-substratum contact region, can now be derived byneglecting the presence of the cell membrane.

Simplified expressions for fluorescence excited in labeledmedia of cell membraneFor a fluorescent solution in the water gap the emission

222 W. M. Reichert and G. A. Truskey

intensity is proportional to the integral of the evanescentfield intensity over the thickness of the water gap:

Fg = Kg fAaT(2)d2, (12)J o

while the fluorescence intensity in the membrane isproportional to the integral over the membrane.

0.02

J A2

dz, (13)

where Kg and Km are the constant terms defined inequation (3) specific to fluorophores distributed in thewater gap and in the cell membrane, respectively.

However, the membrane is very thin relative to thewavelength of visible light (e.g. 4nm/500nm=0.008) andthe electric field amplitude squared is approximatelyconstant throughout the membrane. Consequently, ifAm/A approaches zero, then 7XA2+Am) approaches 7\A2),the transmittivity expression for a single thin film diele-tric at a distance A2 from the interface (see Appendix). Theintegral in equation (13) is now simply equal to the valueof 7\A2) multiplied by the membrane thickness:

Fm = (14)

Since Kg and K, are assumed to be constant for a givenincident angle, dx, the following primed expressions areused for further analysis:

= fAlJ o

z (15)

(16)

Equations (15) and (16) may be interpreted as excitedfluorescence normalized to the incident intensity and theproduct of the fluorophore concentration and quantumefficiency. In the following calculations, Km is assumed tobe equal to Kg, although in practice they may be quitedifferent.

Numerical integration of the data in Fig. 5 according toequation (15) (using Simpson's rule), and a direct matrixcalculation of equation (16) yields values of normalizedfluorescence excited from solutions in the water gap andfrom fluorescently labeled membranes. The result of thesecalculations are presented in Fig. 6 in which fluorescenceintensities for both labeling techniques are plotted as afunction of water gap thickness for an incident angle of 70degrees and TE polarized light. The excitation of fluoro-phores in the water gap increases with water gap thick-ness, while the fluorescence from a dye-labeled membranedecreases with water gap thickness as the membranemoves farther from the glass surface. Both techniquesreach an asymptotic intensity for a water gap thickness ofthe order of the wavelength of light, while possessing asimilar sensitivity to changes in the distance of the cellmembrane to the glass surface.

Effective interfacial amplitude and depth of penetrationValues for the decaying evanescent field (Fig. 5) andfluorescence intensities (Fig. 6) appear to fit the generalshape of simple exponentially decaying functions forA2<0.5A. Consequently, equation (5) was rewritten in thegeneral form:

TIz) = Tefi{0) exp(-z/deff), (17)

where TefKO) and <iefr are effective values for the transmit-ted interfacial amplitude squared and depth of pen-etration, respectively. If T^O) and defr are assumed to be

O

0.01-

X>

E

0.00

•0.30

•0.20

0.10

v-0.00

a.aon

0.0 0.5 1.0 1.5 2.0Water gap thickness (Ai/lambda)

Fig. 6. Normalized values of fluorescence excited in the cellmembrane (equation (23)) and in the water gap (equation (22))plotted as a function of water gap thickness. Values calculatedusing a three-phase glass/water/cytoplasm model for anincident angle of 70 degrees and TE-polarized light.

independent of water gap thickness (which they are not),then substitution of equation (17) into equations (15) and(16) yields the following approximate expressions:

(18)

(19)= ren<0) exp(- A2/defr)A

According to equations (7) and (8), the effective termsrefl{0) and deff may be expressed as a function of aneffective refractive index neff that encompasses the com-bined influence of the water gap and cytoplasm on theevanescent wave:

Ten<0) =

defr =

(20)

(21)

where n^ and n'^ are effective refractive indices associ-ated with T"?n(0) and de{f, respectively.

For each value of A2/A in Fig. 5, T^O) was determinedfrom the evanescent intensity at z=0 and nen- was calcu-lated from equation (20). The effective depth of pen-etration was determined from the slope of a semi-log plotof the data in Fig. 5 and n'^ calculated from equation (21).(Semi-log plots of 7X2) versus z showed distinctly differentslopes across the water gap and cell cytoplasm, and couldnot be considered linear for intermediate values of A2/A).Both Tefl<0) and den undergo a transition from the valuesfor a simple glass/cytoplasm interface at small values ofA2/A, to that of a simple glass/water interface at largervalues of A2/A (Fig. 7 A). However, the transition of thesetwo terms between the limiting interfacial conditionsoccurs at two distinctly different rates. Examination of thewater/cytoplasm effective refractive indices calculatedfrom equations (20) and (21) (Fig. 7B) indicates that by thetime the effective depth of penetration has begun torespond to the presence of the water gap (i.e. A2/A=0.5)the transmitted interfacial amplitude squared has essen-tially completed the transition to a glass/water interface.This difference in the behavior of Ten(0) and den is clearlydue to the fact that cfeff is determined essentially over adistance equal to the wavelength of the incident radiation,while Teff(0) is established by conditions immediatelyadjacent to the interface.

For an incident angle of 70 degrees, Table 1 contains

TIRF microscopy 223

A. 7"eff(0) and dM versus A2

0.20

0.19

0.18

0.17

0.16

0.151.37

1 36

1.35

u•o

c>t>

Si•o

jo3CJ

cdtJ

•o

cTO

C

_oCBu ,<U

<L>D.

"ox:Q.

•u

>

o

c<u

03

tcUCS

EU

E

•1.34

1.33 1.330 1 2 3Water gap thickness (Ai/lambda)

Fig. 7. Fit of effective transmitted interfacial amplitude, depthof penetration (A) and refractive index (B) to evanescentlydecaying intensity profile in Fig. 5 plotted as a function ofwater gap thickness. Note that the shapes of the refractiveindex curves follow very closely to the parameters from whichthey are calculated.

Table 1. Fits of effective transmitted interfacialamplitude, depth of penetration, and refractive index to

calculated normalized fluorescence

Glass/waterGlass/cytoplasm

MembraneWater gap

T2(0)

2.0262.494

TWO)

2.4812.657

d12jx

0.15510.1970

<W>0.15520.1264

1.521.52

n«fr

1.3691.380

n2

1.3331.37

n'«(r

1.3331.261

values of Ti2(0) and d^/h calculated from equations (7)and (8) for simple glass/water and glass/cytoplasm inter-faces (designated as glass/water and glass/cytoplasm for«2=l-333 and 1.37, respectively), and values of Teff{0),defl/A, neff and n'eff derived from fits to the calculatedmembrane (equation (19)) and water gap (equation (18))fluorescence data in Fig. 6 (designated as membrane andwater gap, respectively). Note that fluorescence excited inthe cell membrane yielded fits of Te{!(0), den/X, nef{ and n'enconsistent with those obtained for simple interfaces,whereas physically unreasonable values were obtained forfluorescence excited in the water gap. These results indi-cate that: (1) the fluorescence excited in the cell membranecan be approximated by equation (19) using values ofTen<0) and deff for simple glass/cytoplasm and glass/waterinterfaces, respectively; while (2) the fluorescence excitedin the water gap cannot be fit accurately to equation (18).The poor fit of water gap fluorescence to equation (18)results from the term 71

e(j{0)exp(—z/d^, which, because!Tefl(0) and den vary with water gap thickness (Fig- 7A),cannot be integrated analytically to obtain Tefl{0)<iefl{l —exp(-A2/defr)].

Fit of membrane fluorescence to a single exponentialOn the basis of the results presented in Table 1, the

Incident angle

0

«

«

•

D

1

A

68°

70°

72°

74°

76°

78°

80°

-4O'.O 0.2 0.4 o'.6 0.8 1.0

Water gap thickness (A^lambda)

Fig. 8. Semi-log plot of membrane fluorescence normalized tothe membrane thickness Qog^m/A^) plotted as a function ofwater gap thickness for several incident angles of totalreflection and TE-polarized light. Symbols represent valuescalculated using a three-phase glass/water/cytoplasm model(equation (23)), continuous lines represent exponentialapproximations (equation 29)).

membrane-excited fluorescence may be approximated byequation (22) because:

; r13(0) (22a)

de f f=d1 2 , (22b)

where T13(0) refers to the transmitted interfacial ampli-tude for a simple glass/cytoplasm interface, given byequation (7), where 712= 1.37 is substituted for n2, and d12is the depth of penetration for a simple glass/waterinterface, given by equation (8), where n2= 1.333. Thesuitability of this approximation is confirmed by com-paring calculated values of membrane fluorescence usingequations (16) and (19). For a constant membrane thick-ness (Am=4nm), comparison of equations (16) and (19)reduces to a comparison of matrix calculated values of7\A2) to its approximation T13(0)exp(-A2/d12). Fig. 8contains semi-log plots of matrix calculated (continuouslines) and approximated (symbols) membrane fluor-escence normalized to the membrane thickness. Thesevalues are plotted as a function of water gap thickness forseveral angles of incidence beyond the glass/cytoplasmcritical angle for total reflection. Equation (19) clearlyrepresents a good approximation of membrane-excitedfluorescence that, plotted in a semi-log format, fits astraight line with a slope equal to -l/d12 and ay interceptequal to 7/13(0).

Fit of water gap fluorescence to weighted exponentialsSince fluorescence excited in the water gap is poorly fit bya single exponential function, obtaining a simple ex-pression for water gap fluorescence is not as straightfor-ward as it was for membrane-excited fluorescence. Fig. 9compares matrix calculated values of water gap fluor-escence (filled circles) with those values calculated for asimple glass/water interface and a simple glass/cytop-

224 W. M. Reichert and G. A. Truskey

glass/cytoplasm

*-*-*-* •-glass/water

0.0 0.5 10 1.5 2.0Water gap thickness (A2/lambda)

Fig. 9. Water gap fluorescence plotted as a function of watergap thickness for an incident angle of 70 degrees and TE-polarized light. Filled circles represent values calculated viaequation (16) for a three-phase glass/water/cytoplasm model,upper and lower continuous lines represent values calculatedfor simple glass/cytoplasm and glass/water interfaces.

lasm interface (continuous lines). Owing to the stronginfluence of the cytoplasm on the evanescent decay atsmall water gap thicknesses, the calculated water gapfluorescence is slightly greater than the glass/water ap-proximation, for gap thicknesses less than A/2. At largerwater gap thicknesses the fluorescence intensity is domi-nated by the glass/water interface.

One can conclude from Fig. 9 that the actual distance ofthe cell membrane from the glass surface is intermediatebetween those values predicted by the simple interfacesalone. A model that conforms to this observation is aweighted expression:

FJA2) = [(1- (23)

where w is a weighting factor and F' 12^2) and i<"13(A2) arethe normalized fluorescence intensities excited to a depthof A2 from a simple glass/water and glass/cytoplasminterfaces, respectively. Evaluation of equation (15) usingequations (7) and (8) yields the following expression for

F' 12(Aa) = T12(0)d12[l - exp( - A2/d12)] (24)

F 13( A2) = rls(0)d13[l - exp(- A2/d13)]. (25)

The weighting factor co in equation (23) should rangefrom unity for small gap thicknesses (F'g(A2)=ir'i3(A2)) tozero for large gap thicknesses (F'g(A2)»F'i2(A2)). The formof the weighting factor is therefore obtained from the ratio:

co = (Fg(A2) - - F'12(A2)), (26)

which may be plotted against water gap thickness. A semi-log plot of this ratio (Fig. 10) reveals that the weightingfactor is well represented by an exponential function:

co = exp(-A2/y), (27)

where y=0.1786A.Greater insight into the form of the weighting factor

decay rate y is obtained by noting that the terms 1 — co andco in equation (23) are solutions to the following integrals:

J Ajexp(-z/y)dz = 1 - exp(-A2/y) (28a)

0

CO= 1/y IJ A,

exp(-z/y)dz = exp(-A2/y), (28b)

which are proportional to the evanescent intensity in thewater gap and in the cytoplasm, respectively, for a fielddecaying with a depth of penetration equal to y. For thedata in Fig. 10, the value of y=0.1786A is intermediatebetween the corresponding depths of penetration for sim-ple glass/water and glass/cytoplasm interfaces (Table 1),which is close to the average value (1/2(0.1551A+0.1970A)=0.1761A).

The simple replacement of y with the average depth ofpenetration would be generally applicable only if itresulted in good estimates of water gap fluorescence atseveral angles of total reflection. Fig. 11 compares 'actual'water gap fluorescence values (symbols) calculated usingmatrix theory to 'approximated' water gap fluorescencevalues (continuous lines) plotted as a function of water gapthickness for several angles of total reflection. The approx-imated values were calculated from equations (23)-(27)and weighting factor decay of:

Y + d13), (29)

where d12 and d13 are given in equation (8) with appropri-ate substitutions. The values contained in Fig. 9 for anincident angle of 70 degrees can be analyzed in terms of aneffective water/cytoplasm refractive index. Usingequation (20), the slope of the data in Fig. 10 (y=0.1786)yields an effective refractive index of 1.357 while theaverage depth of penetration (dave=0.1761) yields thenearly identical value of 1.355, which differs by less than0.15%. The apparent ability to simply use the averagedepth of penetration and still obtain such good fits tomatrix theory is somewhat fortuitous in that the refrac-tive indices of water and cytoplasm differ by less than 3 %and neither phase totally dominates the form of theevanescent decay (Fig. 5).

Discussion

The fluorescence excited at regions of close contact be-tween the cell and surface must be quantitated if TIRF isto be used to determine cell-substratum separation dis-tances. Direct application of thin film optics yields cum-bersome expressions in which fluorescence intensity is a

-50.0 0.5 1.0 1.5 2.0Water gap thickness (A^lambda)

Fig. 10. Semi-log plot of weighting factor plotted against watergap thickness showing that the weighting factor fits anexponential with a constant rate of decay. Data for this plotwere derived from individual data points (•) in Fig. 8.

TIRF microscopy 225

0.5-r

0.00.0 0.1 0.2 0.3 0.4 0.5Water gap thickness (A^lambda)

Fig. 11. Water gap fluorescence plotted as a function of watergap thickness for several angles of total reflection and TE-polarized light. Continuous lines represent approximated watergap fluorescence calculated from matrix theory, symbolsrepresent water gap fluorescence calculated from equations(23)-(27) using a three-phase glass/water/cytoplasm model.

nonlinear function of the separation distance (A2). Sincethe refractive indices of water and cytoplasm are similarand the cell membrane is very thin, simplified expressionsfor fluorescence excitation can be developed. These re-lations for fluorescence excitation may provide a straight-forward estimation of the separation distance.

The simplest method of modelling the optics of thin filmsis through the use of an effective refractive index. In thecase of a TLRF experiment of adherent cells, such anapproach is valid only for fluorescence excited in a dye-labeled membrane (Table 1) for which the interfacialevanescent intensity depended upon the refractive indexof the cytoplasm (equation (22a)) and the depth of pen-etration depended upon the refractive index of water(equation (22b)). The use of effective refractive indices tomodel the fluorescence excited in the water gap proved tobe unreasonable, and the calculated fluorescence insteadwas well represented by a weighted sum of the fluor-escence excited at a glass/water and a glass/cytoplasminterface (equation (23)).

Equation (19) for membrane-excited fluorescence, sub-stituted with equations (22a) and (22b), can also be derivedfrom the exact analytical expressions pertinent to thin-film optical theory and is valid as long as the refractiveindices in the water gap and cytoplasm are similar (seeAppendix). However, equation (23) cannot be derived inthe same manner, but the weighting factors in thisexpression were shown to be proportional to the evan-escent intensity in the water gap and in the cell cytoplasm,which provided a physical explanation for the observed fit.The observed fit of both of these approximate expressionswas very good compared to more exact matrix calculatedvalues (Figs 8 and 11).

The sources of error in the development of the aboveapproximate expressions for fluorescence excited in thecell membrane (equations (19), (22a) and (22b) and in thewater gap (equation (23)) of a cell-substratum contactarise from two assumptions. (1) The presence of the cellmembrane can be neglected, thus reducing a four-phase

glass/water/membrane/cytoplasm model to a more trac-table three-phase glass/water/cytoplasm model. (2) Asimple combination of two-phase glass/water and glass/cytoplasm interfaces can be used to approximate closelythe fluorescence intensity excited in a three-phase glass/water/cytoplasm system.

The argument for neglecting the cell membrane wasthat the cell membrane was much thinner than thewavelength of light and therefore did not appreciablyaffect the evanescent intensity calculations (see Appen-dix). The effect was demonstrated in Fig. 3 over a broadrange of incident angles by noting that the calculatedinterfacial intensity appeared to be unaltered if oneneglected the cell membrane in the calculation scheme.However, the exponentially decaying evanescent intensitydoes not exist just at the interface, but extends in the zdirection to a depth of one wavelength from the sub-stratum surface into the liquid phase (Fig. 5). The percent-age error associated with neglecting the cell membranewas determined more accurately by comparing the evan-escent field intensity distribution in a cell contact regionwith and without the cell membrane (not shown). For thecase of TE-polarized light and an incident angle of 70degrees, the percentage difference between the two valuesnever exceeded 2.5% for water gap thicknesses in therange 0=£z/A=Sl, which was the same level of error ob-served when only the interfactial intensity was considered(Fig. 3B).

Figs 8 and 11 clearly demonstrate that the error associ-ated with the derived approximate expressions (assump-tion (2), above) are negligible for water gap thicknesses inthe range of 0^z/A=£l for several angles of total reflection.However, the approximation error in the crucial regionbetween z/X=0 and 2/A=0.1 is not clearly delineated inthese data. In the blue-green portion of the visible spec-trum (A=500nm), these narrow gap thicknesses corre-spond to cell separations of 0-50 run (i.e. focal and closecontacts). Further error calculations (not shown) revealedthat the approximation error starts at zero and increaseswith water gap thickness. For an incident angle of 70degrees with TE-polarized light, the error for membrane-excited fluorescence was roughly linear reaching a valueof 1.2 % for a gap thickness of A2/A=0.1, while the error inwater gap fluorescence reaches a plateau of 0.5% atA2/A=0.1. Although the error for membrane fluorescencereaches a level of 10 % for a water gap thickness of A2/A= 1(not shown), the excited fluorescence has decayed essen-tially to zero by then (Fig. 6) and the net effect remainsminimal.

Other sources of error that may complicate applicationof these approximations (equations (22) and (23)) arechanges in the fluorophore quantum efficiency as a func-tion of distance from and orientation to the substratumsurface, fluorescence excited by scattered light, and inho-mogeneous fluorophore distribution. For protein adsorp-tion, the excitation of fluorescence by light scattered fromthe solid/liquid interface is significant (Reichert, 1989).Comparison of IRM and TLKF images of the same cell haveshown good correspondence between the appearance offocal contacts and fluorescence intensity, indicating thatexcitation appears to be confined to the evanescent volumeat the substratum surface (Lanni et al. 1985). The possibleimportance of scatter has not, however, been experimen-tally investigated for adherent cells until recently (Grappaet al. 1990).

The proximity and orientation of the fluorophore withrespect to the substratum surface may affect emission

226 W. M. Reichert and G. A. Truskey

efficiency (Lanni etal. 1985). Recent studies using cya-nine-impregnated 50 nm thick Langmuir-Blodgett (L-B)films of fatty acids showed that the emission efficiencydepended slightly upon the distance of the dye layer fromthe substratum surface (Suci and Reichert, 1988). How-ever, rigorous calculations of the total power dissipated bya point source dipole (the harmonic oscillator analog to anemitting fluorophore) in the same dielectric environmentshowed an emission decrease of approximately 1 % overthe thickness of the L-B film (Suci, 1988). Since Dil(Molecular Probes)-labeled membranes of adherent cellsrepresent a similar system, one would anticipate a smallattentuation of emitted fluorescence as the membraneapproaches the substratum surface. Such attenuationscould lead to a slight overestimation of cell-substratumseparations.

Membranes are also birefringent, as are L-B films, withan optical axis normal to the membrane surface(nx=ny^nT) (Suci and Reichert, 1990). Therefore, if one isto minimize birefringent effects the excitation radiationshould be restricted to TE polarization that is parallel tothe net membrane molecular orientation at the sub-stratum surface. The electric vector of TE polarized light isalso parallel to the transition moment of a cyanine labelwhen the membrane is parallel to the substratum surface,thus maximizing fluorescence excitation.

Finally, local aggregations of fluorophores can alsoproduce spurious effects. Dye aggregations may producehigh levels of excited fluorescence, which may be misinter-preted as close membrane contacts. On the other hand, ifintermolecular quenching is significant, or dye is excludedfrom regions of high protein concentration, then lowerthan expected emission levels may result. Although wecurrently have no evidence to support or refute either ofthese aggregation effects, one would speculate that dyeaggregation would be minimal in homogeneous solutionsof fluorophores (e.g. fluorescein-labeled dextrans), butcould be significant in fluorescently labeled lipid struc-tures where dye aggregation is known to exist (Vaidya-nathan et al. 1985).

Owing to the potential for spatial variations in theconcentration of membrane fluorescence, estimates of cellseparations based upon measurements at a single anglemay be subject to significant errors. Taking measurementsof membrane fluorescence at one location on the mem-brane for several angles above the critical angle can yielddirect estimates of A2. Taking the natural logarithm ofequation (19), which upon rearrangement, yields thefollowing linear function with respect to l/dW> with aslope of — A2 andy intercept of ln[Tefl{0)Am/cos $i]:

Incos20i

ft(0)Aml _ Aa_O 8 2 e i J drt'

(30)

where TefT<0) and deff are given by equations (20) and (21),respectively. Assuming Am, 6U nx and A are known, thevalue of A2 is sensitive to the choice of n'eff.

The refractive indices used in this paper (Fig. 2) wereselected as representative values for calculation purposes.In order to estimate the error associated with an inaccur-ate estimate of refractive index /i'eff, values oflntF'm/cos2^] were calculated with constant effective re-fractive index values of rcefr=1.37 and n'eB=1.36 over anincident angle range of 68-80 degrees for three differentvalues of A2, with Am=4nm, A=514.5nm and «i=1.52.For each angle, l/defr was calculated using seven differentvalues of n'effthat covered a reasonable range of refractiveindices (re'efr= 1.333, 1.34, 1.35, 1.36, 1.37, 1.38 and 1.39).

4.50

4.25-

3.25

3.00-

2.75

A2=20nm

0.004 0.008 0.012l/deff(nm~')

0.016

Fig. 12. \n[Fm(&2)/cos28i] versus 1/d^ plot for values ofmembrane-excited fluorescence and depth of penetration(equations (19)-(21)) calculated for several angles of incidence(0!=68-8O degrees), three values of water gap thickness(A2=20, 50 and 100 nm), and seven values of water gap effectiverefractive index (n'eH=1.333 (•), 1.34 (D), 1.35 (A), 1.36 (A),1.37 (•), 1.38 (O) and 1.39 (• ) , where Am=4nm, A=514.5nm,n,fr=1.37 and n ^ l . 5 2 . Note that the water gap thickness andtransmitted interfacial amplitude can be determined directlyfrom the slope and y intercept of a straight line fit to each dataset.

Values of ln[F'm/cos20] were plotted against l/den-for thevarious assumed values of n'efr, producing a family ofcurves (Fig. 12). For each estimated value of /i'effand defT,the data were well fit by a straight line (correlationcoefficient 0.994—1.000) and the values of water gapthickness (A2) were obtained directly from the slope usingequation (23) (Table 2).

From the data in Table 2 one can construct percentageerror plots (not shown) of water gap thickness error andcytoplasmic refractive index error against water gapeffective refractive index error. The error in water gapthickness was linear with water gap effective refractiveindex error and independent of water gap thickness (i.e. a% error in ra'eff results in a 10 % error in A^, while thecytoplasmic effective refractive index increased nonli-

Table 2. Water gap thickness and effective cytoplasmicrefractive index calculated from linear regression

ln[F'm/cos2d1] versus / d lAg (nni)

2050

100

2050

100

1.333

Water gap effective refractive index guess (n'o(r)

1.340 1.350 1.360 1.370 1.380

Water gap thickness (Aj) from slope (nm)

1.390

25.16 22.31 21.20 20.00 18.68 17.16 15.3062.89 55.76 53.00 50.00 46.69 42.90 38.24

125.78 111.53 106.00 100.00 93.37 85.80 76.48

Cytoplasmic refractive index (n.(r) from y intercept

1.3841.4031.428

1.3781.3891.406

1.374 1.370 13661.380 1.370 1.3591.390 1.370 1.346

1.3611.3461.317

1.3551.3301.278

TIRF microscopy 227

nearly with water gap effective refractive index error andincreased linearly with water gap thickness (i.e. a 1 %error in n'en results in approximately 0.5, 1.0 and 2.5%errors in nan for A2=20, 50 and 100 nm, respectively). A1 % error in effective refractive index centered about avalue of 1.37 (Table 2) corresponds to a refractive indexrange of 1.356-1.384, which encompasses the cytoplasmicrefractive index range reported elsewhere (Lanni etal.1985). However, it should be noted that the error in thecytoplasmic refractive index would not affect the determi-nation of water gap thickness from membrane-excitedfluorescence data, unless the incident angle was notsufficiently far from the critical angle to ensure totalinternal reflection.

Variations in the water gap and cytoplasmic refractiveindex are important because the region between a focalcontact and the surface, and the adjacent cytoplasm, areknown to contain receptors for the adhesion proteins suchas fibronectin and vitronectin. The high density of proteinsin a small volume of fluid could produce a local increase inthe refractive index. In fact, refractive indices as high as1.36 to 1.40 have been assigned to focal contacts (Lannietal. 1985; Bereiter-Hahn etal. 1979). Consequently, agiven location in a cell-substratum contact could have acytoplasmic or water gap refractive index of as low as1.333 or as high as 1.4, which could produce a localvariation in the critical angle from 61 to 67 degrees. Anincident angle in this range may result in locations in thecell image that are not being excited by evanescentradiation. One way to avoid this problem is to collect dataover the incident angle range of 70-80 degrees.

Alternatively, cell separation distances can be obtainedusing a water-soluble label such as a low molecular weightdextran. In this paper we have shown that the water-soluble dye and membranes labeled by TIRF techniquesyield, in theory, equivalent information on cell-substratum contacts. However, the determination of cell-substratum gap thickness from water-soluble dyes (e.g.fluorescein-labeled dextran markers) is less direct.Equations 23-25 and 27, which describe fluorescence inthe water gap, are implicit in A2 and require estimates ofboth neft and n'efj. An iterative scheme, such as Newton'smethod, can be used to determine the value A2 by mini-mizing the deviation between the measured and calcu-lated fluorescence for several angles of total reflection. Thewater-soluble marker TIRF technique is discussed indetail elsewhere (Gingell etal. 1987) and the reader isreferred to the literature for further details.

Conclusions

Here we have presented simple exponential approxi-mations for fluorescence excited in the membrane and inthe membrane/substratum water gap of a cell adherent toa glass substratum. Expressions were derived in which allterms, except the water gap thickness A2, can be deter-mined from expressions for the transmission of light atsimple dielectric interfaces by knowing the refractiveindices of the glass, water and cytoplasm, and the angle oftotal reflection. Representative calculations were per-formed for TE-polarized light for incident angles rangingfrom 68—80 degrees, which corresponds to the typicalrange used in a TIRF experiment at the glass/waterinterface. The error associated with these approximationsdid not exceed 2.5% for a membrane separation typical offocal and close membrane/substratum contacts. By fitting

the derived approximations to fluorescence collected fromdye-labeled cell membranes (equation (29)) or to fluor-escence from solutions in the membrane/substratumwater gap (equation (30)) one can, in principle, calculatethe distance from the cell membrane to the substratumsurface. Currently we are examining membrane-labeledfibroblasts and endothelial cells under TIRF illuminationusing digital image analysis, which allows us to quanti-tate the image brightness from specific segments of the cellmembrane. By collecting TIRF images at several angles oftotal reflection, we plan to use equation (33) to transformthe change in local image brightness into local values ofcell-substratum separation (Grapa et al. 1990).

Appendix

Consider two dielectric films with refractive indices n2 andrtm and thicknesses A2 and Am sandwiched between twosemi-infinite media with refractive indices n^ and n3. Andlet 81, 62, 6m and 63 be the angles of light propagation ineach medium with respect to the interfacial normals. Thesquare of the electric field amplitude transmitted throughthe two thin films {T(z)=T(&2+&m)) is given by themagnitude squared of the transmission coefficient of thetwo-film laminate:

= |T (A.I)

where r is given by Heavens (1960):

r12r2m

r12r2mrm3 exp(-2i<5m)],where t=(- l ) 1 / 2 , b2=(.2nA2/\)n2co8d2, <5m=(2^Am/A)«mcosem; and r12,r2m,rm3 and Ji2,(2m,im3 are the Fresnelreflection and transmission coefficients for the first, secondand third interfaces of the laminate. For TE-polarizedlight, the Fresnel reflection coefficients for the first inter-face between the incident medium (ni) and the first film(n2) are given by:

t12 = n2cos62)

with identical expressions for the second and third inter-faces appropriately substituted with n2, n^, n3, 02, Sm

an^03-

Now let rijn be a 4 nm cell thick membrane adjacent to aglass substratum (see Fig. 2 in main text of paper). In thisconfiguration, the membrane is much thinner than thewavelength of the incident light (i.e. 4nm/500nm=0.008). Taking the limit of Am/A->0 we observe that <5m-*0,which implies that the membrane vanishes (n^—*n3)resulting in the convergence of rm3—>0 and tms-*!. Substi-tuting these limits into the above expression for T yields:

lim T = (A.2)

where the thickness and refractive index of the cellmembrane are neglected. Substitution of equation (A.2)into equation (A.I) and evaluating yields the well-knowntransmittivity expression for a single thin-film dielectricduring normal reflection and refraction of light (Born andWolf, 1986):

A^-228 W. M. Reichert and G. A. Truskey

However, when the incident angle is beyond the criticalangle for total reflection for both a glass/water and aglass/cytoplasm interface, then the field is evanescent inboth the water gap and the cell cytoplasm, in which casecos#2 and cos^a becomes negative and imaginary.

cos02 = -

cos03 = -

- 1]' (A.3a)

- 1]*. (A.3b)

Substitution of equations (A.2), (A.3a) and (A.3b) intoequation (A.I), and again evaluating yields the followingexpression for 7XA2) during total internal reflection:

H2f23 exp(-A2/d12),(A.4)

where <i12 is the depth of penetration for total internalreflection at the i i /n 2 (i.e. glass/water) interface (seeequation (8) of main text) and where for TE-polarized lightwe have r?2=l and:

l -Wrt ! ) 2

^23 - [ •2[n2sin201 -

[n?sin201-/i§]1

r?. =- raj]' - [rfoin'ft -- n§]' + [n

foin'ft - n2f~Vfsin2^ - ntf]

(A.5)

Figure Al shows values of the bracketed exponentialprefactor in equation (A.4) plotted as a function of watergap thickness for TE-polarized light at several angles oftotal reflection, where rai=1.52, «2=1.333 and n3=1.37.Note that, except for angles close to the glass/cytoplasmcritical angle, the exponential prefactor term is essentiallyconstant and equal to the value at the glass/cytoplasminterface. This result is due to the fact that the refractive

8.K

u

0

I—I—I—I—•—t—I—I—1

-t—0—0-

•—*—«—»—*—I—t—•-

I I I I I I I I I

"4—*—»—4—4—4—i—4—1

Incident angle

0 68°

• 70°

« 72°

• 74°

D 76°

• 78°

4 80°

0.0 0.2 0.4 0.6 0.8 1.0Water gap thickness (A2/lambda)

Fig. Al. Exponential prefactor in equation (A.4) plotted aa afunction of water gap thickness for several angles of totalreflection and TE-polarized light. Note that except for thevalues at 68 degrees, this prefactor remains essentially equal tothe value for a glass/cytoplasm interface (A2/A=0).

indices of water and cytoplasm differ only by about 3 %,and during total internal reflection we see from the aboveexpressions that ^==0 and t^^l. Therefore, the exponen-tial prefactor in equation (A.4), subject to the conditionthat n3 approximates «2, is very nearly equal to thetransmission term for a simple glass/cytoplasm interface.Finally, subjecting 7TA2+Am) to the limit of Am/A->0 andrecognizing that 713=712 yields:

lim 7XA2 + Am) = T13(0) exp(-A2/d12), (A.6)

where Ti3(0) and di2 are the square of the transmittedinterfacial amplitude and the depth of penetration termsfor a simple glass/cytoplasm and glass/water interfaces(see equations (7) and (8) of main text of paper).

This work was supported by research grants from the WhitakerFoundation, NSF grants CBT-8796331 and CBT-8746074, andNIH grant HL32132. The authors thank W. N. Hansen of UtahState University for providing a copy of the characteristic matrixsoftware.

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TIRF microscopy 229

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230 W. M. Reichert and G. A. Truskey