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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 a
powerful technique for visualizing focal and close
contacts between the cell and the surface. Practical
application of TIRF has been hampered by the lack
of straightforward methods to calculate separation
distances. The characteristic matrix theory of thin
dielectric films was used to develop simple exponen-
tial approximations for the fluorescence excited in
the cell-substratum contact region during a TIRF
experiment. Two types of fluorescence were exam-
ined: fluorescent]}' labeled cell membranes, and a
fluorescent water-soluble dye. By neglecting the re-
fractive index of the cell membrane, the fluorescence
excited in the cell membrane was modelled by a
single exponential function while the fluorescence in
the membrane/substratum water gap followed a
weighted 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 fluorescence
intensities to more exact solutions of the fluorescence
integrals for the three-phase model indicated that the
approximations are accurate to about 1% for
membrane/substratum gap thicknesses of less than
50 run
if the cytoplasmic and water gap refractive
indices are known. The intrinsic error of this model
in the determination of membrane/substratum sep-
arations was
10
%
as long as the uncertainties in the
water gap and cytoplasmic refractive indices were
less than
1
%.
Key words: TIRF, cell contact, cell membrane.
Introduction
Total internal reflection fluorescence (TIRF) has become
a
standard technique
for
exciting spectroscopic phenomena
at interfaces (Reichert, 1989). Briefly,
the
total internal
reflection
of
visible light
at a
glass/solution interface
produces
a
region of electromagnetic intensity (evanescent
wave) that penetrates
a few
tenths
of a
micrometer into
the lower refractive index medium
and can
excite spectral
phenomena
in a
region confined
to the
solution side
of
the
interface.
The
specific application
of
TIRF
to the
imaging
of contacts
of
adherent cells
to
surfaces
was
developed
by
Axelrod et
al.
(1982). The ability to label specific regions of
the cell
has
made TIRF
an
attractive technique
for
characterizing cell—substratum contacts. Recent appli-
cations
of
TIRF microscopy include
the
visualization
of
amoeba (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
focal
contacts (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 a
model that relates image intensity with
the
proximity
of
the 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 cells
for
the
case
of a
water-soluble fluorescent label
in
which
the cell-substratum contact region
was
modeled
as a
dielectric lamellar structure (Gingell
et al.
1987). Calcu-
lations
of
expected fluorescence intensity
for
several inci-
dent angles of the totally reflected beam were presented
as
a function
of the
water
gap
distance between
the
cell
membrane
and the
substratum surface. TIRF
is
much
more sensitive
to gap
thicknesses
of
the order
of
0-70 nm
than
is the
more widely used technique
of
interference
reflection microscopy (Gingell
et al. 1987;
Bailey
and
Gingell, 1988).
In
the
Gingell model, cell-substratum contact regions
were treated optically
as a
four-phase system consisting of
a glass substratum,
a
water-filled gap,
a
lipid membrane
and
the
cell cytoplasm.
The
glass
and
cell were treated
as
semi-infinite media while
the
water
and
cell membranes
were treated
as
thin dielectric films. Closed form
ex-
pressions were derived
for the
distance dependence
of
the
electric field amplitude
in the
watei1
gap
regions
for the
four conditions under which
the
electric field
in the
water
gap,
cell membrane and cell cytoplasm was either continu-
ous (propagating)
or
evanescent (decaying). Although
the
closed form expressions
are
exact solutions
to the
field
equations,
the
expressions presented
are
difficult
to
apply
in
the
determination
of
separation distances.
An alternate approach, adopted
in
this paper,
is to
develop simpler expressions
for the
fluorescence that
are
219
valid for the particular conditions of cell-glass contacts in
an aqueous medium. Characteristic matrix theory (Han-
sen, 1968), in which the same matrix form is used whether
the field in a given medium is either evanescent or
continuous, was used to determine the expected TIRF
emission excited using transverse electric (TE)-polarized
incident radiation totally reflected at the glass-cell inter-
face.
Fluorescence excited from fluorescently labelled cell
membranes and cells with cell-substratum water gaps
infused with fluorescent solutions are considered. These
calculations show that the lipid membrane may be ignored
with an error of no greater than 2.5% and the cell-sub-
stratum contact may be modelled as a comparatively more
simple three-phase system. Approximate analytical ex-
pressions are developed for these two TIRF excitation
configurations that permit straightforward calculation of
the cell-substratum water gap thickness from exper-
iment.
Theory
Fluorescent emission excited during total internal
reflection
Consider a beam of light containing both transverse
electric (TE, perpendicular, or s) and transverse magnetic
(TM, parallel, or p) polarized electric vectors incident upon
a dielectric interface at the angle 6x (Fig. 1). The fluor-
escence emission excited by the transmitted component of
the incident beam between the limits Zi and z2 relative to
the 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 and
concentration 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 z
dependence of the magnitude squared of the transmitted
electric field amplitude given by:
homogeneous (c(z)=c), then for a given angle of incidence
equation (1) becomes:
= \E(z) (2)
where
*
designates the complex conjugate and
•
designates
the dot product. If one assumes that (1) the quantum
efficiency and extinction coefficient of the fluorophore are
independent of the proximity to the totally reflecting
interface and spatial orientation
(cp(z)a{z)=(pa),
and (2) the
fluorophore distribution between the limits Z\ and z2 is
Medium 2
Fig. 1. Schematic illustration of beam incident at the angle #i
upon a dielectric interface between medium 1 and medium 2.
Vectors on incident beam indicate the orientation of the electric
vector 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 may
be subject to spatial variations in fluorescence emission
resulting from fluorophore orientation with respect to the
incident beam, fluorophore proximity with respect to the
substratum interface, and molecular aggregation of flu-
orophore molecules. The importance of these phenomena
to the quantitative interpretation of TIRF images is
considered in the Discussion.
For the ideal case, TIRF will occur when the incident
angle exceeds the critical angle for total internal reflection
(0c) defined by the higher refractive index of the incident
medium (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 the
n\jni interface and d12 is the depth of penetration of the
evanescent wave. Substitution of equations (4) and (5) into
equation (3) followed by integration yields the expression
for fluorescence excited by TIRF in homogeneous aqueous
solutions 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 polarized
incident 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 also
available (Hansen, 1968).t In the ideal case, values for the
refractive indices and the angle of total reflection may be
substituted directly into equation (6) to yield exact sol-
utions for the fluorescence excited during a TIRF exper-
iment.
Fluorescent emission from thin dielectric films
When the fluorophores reside in a thin dielectric film
adjacent to the incident phase, the expected fluorescent
emission is still represented by equation (1), except that
the term 7\z), the magnitude of the total electric field
amplitude squared, now consists of two traveling waves: a
foreword traveling wave propagating down from the top
surface of the film and a reverse traveling wave propagat-
ing up from the bottom surface of the film. Expressions for
this field are derived from two optical principles: (1) the
wave equation must be satisfied in each medium, and (2)
the tangential components of the electric and magnetic
fields must be continuous across each interface.
tThe expression for TM polarized transmitted intensity during
total internal reflection was reported incorrectly by Reichert
(1989).
The correct expression (using current notation) should
have read:
T rau - T row 2sin2e1-(n2/n1)2
220 W. M. Reichert and G. A. Truskey
The characteristic matrix theory provides a convenient
method of calculating the tangential boundary conditions
at one interface of a thin dielectric film in terms of the
adjacent interface. Consider a film (medium 2) sandwiched
between two semi-infinite media (medium 1 and medium
3).
The upper and lower interfaces of the film are defined
by the planes 2=0 and
z=
A2,
respectively, where A2 is the
film thickness. Following the treatment of Hansen (1968),
if Q2(0) and Q2(A2) are 1x2 matrices that represent the
tangential components of the electric and magnetic field
amplitudes at the upper film and lower film boundaries,
respectively, then Q2(A2) may be calculated from Q2(0) as
follows:
Q2(A2) = N2(A2)Q2(0), (9)
where iV2(A2) is the 2x2 inverse characteristic matrix of
the film that accounts for reflection and transmission at
the upper and lower film boundaries along with the phase
change of light that traverses the film. Similarly, one can
calculate the tangential components of the electric and
magnetic 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 film
thickness A2. Exact expressions for the matrices in
equations (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 and
magnetic field amplitudes at any location 2 in a laminate
of films as long as the thicknesses and refractive indices of
each film and the boundary conditions at the first interface
are known. For TE polarized light, Ey{z) is element 11 of
the matrix Q2(2), allowing one to calculate directly the 2-
dependent distribution of the TE-polarized electric field
intensity from the expression:
J\2>JE
=
|.ZSy(.2,)|
. (.J--U
The characteristic matrix software used in this paper was
generously provided to these authors by W. H. Hansen of
Utah State University, and subsequently modified locally
for use with Microsoft Fortran in a Macintosh computer. A
more 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 of
the electric field intensity T[z) in a laminate of thin films.
For TE-polarized incident radiation, this technique was
used to calculate values of the evanescent field intensity
relevant to TIRF microscopy of cell-substratum contacts,
from which the excited fluorescence values are calculated
from equation (3). Although the calculations Eire straight-
forward, major simplifications in the analysis can be made
by neglecting the membrane thickness and refractive
index, which reduces the analysis to a three-phase system
(glass/water gap/cytoplasm). This simplification leads to
the derivation of experimentally useful expressions for the
fluorescence excited in the cell-substratum contact re-
gion.
Evanescent field intensity in cell-substratum contacts
Consider a region of a cell adherent to a glass substratum
Glass substratum
/i,
= 1.52
. Water gap
2 /;-,= 1.333
Membrane
Am=4nm
»m=1.45
Fig. 2. Four-phase optical model of a cell-substratum contact
region. Refractive indices were derived from Gingell et al.
(1987).
(Fig. 2). The region directly below the incident glass
medium (/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 was
treated as a semi-infinite medium. Referring to equation
(1),
the fluorescence observed in the cell-substratum
contact region is excited by the electromagnetic intensity
transmitted across the interface. However, the form of the
transmitted intensity is a function of the refractive indi-
ces,
the incident angle, the water gap thickness and the
membrane thickness.
Fig. 3A contains matrix calculations of 7\0), plotted as a
function of incident angle, for a four-phase model consist-
ing of glass/water/membrane/cytoplasm for different
values 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 the
presence of the cell membrane, yields a plot that appears
to be identical to Fig. 3A (not shown). The percentage
difference between the interfacial intensities calculated
with and without the cell membrane is shown in Fig. 3B
for water gap thicknesses of A2^0.5A. In the blue-green
region of the spectrum theses values correspond to water
gap thicknesses of the order of
100
nm or less.
The first conclusion to be drawn from Fig. 3 is that the
presence of the cell membrane has little effect on the
interfacial intensity (i.e. the error of ignoring the cell
membrane reaches a maximum at a zero water gap
thickness, but never exceeds an asymptote of 2.5
%).
As
discussed in the next section, this result affords us the
luxury of neglecting the cell membrane. Second, increas-
ing the water gap thickness causes the intensity peak near
the glass/cytoplasm critical angle (0C=64.33) to diminish
and be replaced by a growing intensity peak near the
glass/water critical angle (0C=61.23). Since the interfacial
intensity reaches a maximum at the critical angle, then
this transformation suggests an effective refractive index
for the cell-substratum contact that shifts to lower angles
with increasing water gap thickness. Fig. 4 plots the
effective critical angle (incident angle of maximum inter-
facial intensity) for a four-phase system (cell membrane
included in calculation) plotted as a function of the water
gap thickness. Here one sees that the critical angle
remains essentially constant up to a water gap thickness
of approximately 0.75A before it begins to drop toward a
glass/water critical angle (this transformation is not fully
completed until A2=5A). Also note that the cell membrane
appears to increase slightly the effective refractive index
of the water gap/cytoplasm region as evidenced by a
critical angle at small water gap thicknesses (0C=64.44),
which is slightly greater than the glass/cytoplasm critical
angle (i.e. for 0C=64.44 we obtain an effective refractive
index 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 66
Incident angle (degrees)
6870
Fig. 3. A. Magnitude squared of transmitted interfacial
amplitude (interfacial intensity 7X0)) plotted as a function of
incident angle for water gap thicknesses ranging from
0^A2^1.5A; B, percentage error in the calculated interfacial
intensity upon neglecting the cell membrane for OsAj^O.SA.
Although values were calculated for 0.1A increments in water
gap thickness, only selected gap thicknesses are displayed for
clarity: 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 light
at 0.2 angular increments.
dence
for a
TLRF experiment must
be
greater than
the
critical angles
for
both
the
cytoplasm/glass interface
and
the water/glass interface
(e.g.
0C=7O degrees)
to
ensure
that
an
evanescent field exists throughout
the
region
occupied
by the
cell-substratum contact.
Fig.
5
presents calculations
of the
evanescent intensity
in
a
cell-substratum contact region,
for an
incident angle
of 70 degrees, plotted against water
gap
thicknesses from
0<A2^A.
The
filled
and
open circles again represent
the
evanescent intensity decay
for
simple glass/cytoplasm
and
glass/water interfaces, respectively. Similarly,
we
note
that
the
evanescent intensity undergoes
a
transition from
a glass/cytoplasm
to a
glass/water system with increasing
water
gap
thickness. However,
in all
cases
the
evanescent
intensity decays essentially
to
zero within
one
wave-
length's distance from
the
interface
and
virtually
no
fluorescence would
be
excited beyond this distance. From
Fig.
5, the
important parameter that determines
the
intensity during
a
TIRF experiment
(if
one maintains
the
same assumptions that went into development
of
equation
ang
ical
cnt
ive
Effect
i
64-
63-
62-
61-
h-l t 1
glass/cytoplasm
\
\
I
\
glass/water
0.0
0.5 1.0 15
Water
gap
thickness (A:/lambda)
Fig.
4.
Effective critical angle plotted
as a
function
of
water
gap thickness. Effective critical angle determined from incident
angle
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-
V
X
}
A2=A
A2=0
/
1
-,
-
0.0
0.5 1.0 1.5
Distance from interface (;/lambda)
Fig.
5.
Calculations
of
the decay
of
the evanescent electric field
amplitude squared (T(z)) plotted
as a
function
of
distance from
the glass surface
for
water
gap
thicknesses ranging from
0 to
A.
Filled
and
open circles represent values
of
1Xz)
calculated
for
simple 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
the
interface. Greatly simplified expressions
for the
fluor-
escence overlap integral,
and
thus fluorescence excited
in
a cell-substratum contact region,
can now be
derived
by
neglecting
the
presence
of the
cell membrane.
Simplified expressions
for
fluorescence
excited
in
labeled
media of cell membrane
For
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 evanescent
field intensity over the thickness of the water gap:
Fg = Kg fAaT(2)d2, (12)
J
o
while the fluorescence intensity in the membrane is
proportional to the integral over the membrane.
0.02
J A2
dz,
(13)
where Kg and Km are the constant terms defined in
equation (3) specific to fluorophores distributed in the
water gap and in the cell membrane, respectively.
However, the membrane is very thin relative to the
wavelength of visible light (e.g. 4nm/500nm=0.008) and
the electric field amplitude squared is approximately
constant throughout the membrane. Consequently, if
Am/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). The
integral in equation (13) is now simply equal to the value
of 7\A2) multiplied by the membrane thickness:
Fm = (14)
Since Kg and K, are assumed to be constant for a given
incident angle, dx, the following primed expressions are
used for further analysis:
=
fAl
J
o
z (15)
(16)
Equations (15) and (16) may be interpreted as excited
fluorescence normalized to the incident intensity and the
product of the fluorophore concentration and quantum
efficiency. In the following calculations, Km is assumed to
be equal to Kg, although in practice they may be quite
different.
Numerical integration of the data in Fig. 5 according to
equation (15) (using Simpson's rule), and a direct matrix
calculation of equation (16) yields values of normalized
fluorescence excited from solutions in the water gap and
from fluorescently labeled membranes. The result of these
calculations are presented in Fig. 6 in which fluorescence
intensities for both labeling techniques are plotted as a
function of water gap thickness for an incident angle of 70
degrees 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 membrane
decreases with water gap thickness as the membrane
moves farther from the glass surface. Both techniques
reach an asymptotic intensity for a water gap thickness of
the order of the wavelength of light, while possessing a
similar sensitivity to changes in the distance of the cell
membrane to the glass surface.
Effective interfacial amplitude and depth of penetration
Values for the decaying evanescent field (Fig. 5) and
fluorescence intensities (Fig. 6) appear to fit the general
shape of simple exponentially decaying functions for
A2<0.5A. Consequently, equation (5) was rewritten in the
general 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.
a
on
0.0 0.5 1.0 1.5 2.0
Water gap thickness (Ai/lambda)
Fig. 6. Normalized values of
fluorescence
excited in the cell
membrane (equation (23)) and in the water gap (equation (22))
plotted as a function of water gap thickness. Values calculated
using a three-phase glass/water/cytoplasm model for an
incident 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 terms
refl{0) and
deff
may be expressed as a function of an
effective refractive index neff that encompasses the com-
bined influence of the water gap and cytoplasm on the
evanescent 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 determined
from 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 plot
of the data in Fig. 5 and n'^ calculated from equation (21).
(Semi-log plots of
7X2)
versus z showed distinctly different
slopes across the water gap and cell cytoplasm, and could
not be considered linear for intermediate values of A2/A).
Both Tefl<0) and den undergo a transition from the values
for a simple glass/cytoplasm interface at small values of
A2/A, to that of a simple glass/water interface at larger
values of A2/A (Fig. 7
A).
However, the transition of these
two terms between the limiting interfacial conditions
occurs at two distinctly different rates. Examination of the
water/cytoplasm effective refractive indices calculated
from equations (20) and (21) (Fig. 7B) indicates that by the
time the effective depth of penetration has begun to
respond 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 clearly
due to the fact that
cfeff
is determined essentially over a
distance equal to the wavelength of the incident radiation,
while Teff(0) is established by conditions immediately
adjacent 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
0.15
1.37
1
36
1.35
u
•o
c
>
t>
Si
•o
jo
3
CJ
cd
tJ
•o
c
TO
C
_o
CB
u,
<U
<L>
D.
"o
x:
Q.
•u
>
o
c
<u
0
3
tc
U
C
S
E
U
E
•1.34
1.331.33
0
12 3
Water
gap
thickness (Ai/lambda)
Fig.
7. Fit of
effective transmitted interfacial amplitude, depth
of penetration (A)
and
refractive index (B)
to
evanescently
decaying intensity profile
in
Fig.
5
plotted
as a
function
of
water gap thickness. Note that
the
shapes
of
the refractive
index curves follow very closely
to the
parameters from which
they
are
calculated.
Table
1.
Fits
of
effective
transmitted interfacial
amplitude, depth of penetration,
and
refractive index
to
calculated normalized fluorescence
Glass/water
Glass/cytoplasm
Membrane
Water gap
T2(0)
2.026
2.494
TWO)
2.481
2.657
d12jx
0.1551
0.1970
<W>
0.1552
0.1264
1.52
1.52
n«fr
1.369
1.380
n2
1.333
1.37
n'«(r
1.333
1.261
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
calculated
membrane (equation
(19)) and
water
gap
(equation
(18))
fluorescence data
in
Fig.
6
(designated
as
membrane
and
water gap, respectively). Note that fluorescence excited
in
the cell membrane yielded fits
of
Te{!(0),
den/X,
nef{
and
n'en
consistent with those obtained
for
simple interfaces,
whereas physically unreasonable values were obtained
for
fluorescence excited
in the
water gap. These results indi-
cate that: (1)
the
fluorescence excited
in
the cell membrane
can
be
approximated
by
equation
(19)
using values
of
Ten<0)
and
deff
for simple glass/cytoplasm
and
glass/water
interfaces, respectively; while
(2) the
fluorescence excited
in
the
water gap cannot
be fit
accurately
to
equation (18).
The poor
fit of
water
gap
fluorescence
to
equation
(18)
results from
the
term 71e(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 exponential
On
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
to
the membrane thickness Qog^m/A^) plotted
as a
function
of
water gap thickness
for
several incident angles
of
total
reflection
and
TE-polarized light. Symbols represent values
calculated using
a
three-phase glass/water/cytoplasm model
(equation (23)), continuous lines represent exponential
approximations (equation
29)).
membrane-excited fluorescence
may be
approximated
by
equation (22) because:
; r13(0)
(22a)
deff=d12,
(22b)
where T13(0) refers
to the
transmitted interfacial ampli-
tude
for a
simple glass/cytoplasm interface, given
by
equation (7), where
712=
1.37
is
substituted
for n2, and d12
is
the
depth
of
penetration
for a
simple glass/water
interface, given
by
equation
(8),
where n2= 1.333.
The
suitability
of
this approximation
is
confirmed
by com-
paring calculated values
of
membrane fluorescence using
equations
(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
of
7\A2)
to its
approximation T13(0)exp(-A2/d12). Fig.
8
contains semi-log plots
of
matrix calculated (continuous
lines)
and
approximated (symbols) membrane fluor-
escence normalized
to the
membrane thickness. These
values
are
plotted
as a
function
of
water gap thickness
for
several angles
of
incidence beyond
the
glass/cytoplasm
critical angle
for
total reflection. Equation
(19)
clearly
represents
a
good approximation
of
membrane-excited
fluorescence that, plotted
in a
semi-log format, fits
a
straight line with
a
slope equal
to -l/d12 and ay
intercept
equal
to
7/13(0).
Fit
of
water gap fluorescence
to
weighted exponentials
Since fluorescence excited
in the
water gap
is
poorly
fit by
a 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.
9
compares matrix calculated values
of
water
gap
fluor-
escence (filled circles) with those values calculated
for a
simple 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 2.0
Water gap thickness (A2/lambda)
Fig. 9. Water gap
fluorescence
plotted as a function of water
gap thickness for an incident angle of
70
degrees and TE-
polarized light. Filled circles represent values calculated
via
equation (16) for a three-phase glass/water/cytoplasm model,
upper and lower continuous lines represent values calculated
for simple glass/cytoplasm and glass/water interfaces.
lasm interface (continuous lines). Owing to the strong
influence of the cytoplasm on the evanescent decay at
small water gap thicknesses, the calculated water gap
fluorescence is slightly greater than the glass/water ap-
proximation, for gap thicknesses less than A/2. At larger
water gap thicknesses the fluorescence intensity is domi-
nated by the glass/water interface.
One can conclude from Fig. 9 that the actual distance of
the cell membrane from the glass surface is intermediate
between those values predicted by the simple interfaces
alone. A model that conforms to this observation is a
weighted expression:
FJA2)
= [(1-(23)
where
w
is a weighting factor and F' 12^2) and i<"13(A2) are
the normalized fluorescence intensities excited to a depth
of A2 from a simple glass/water and glass/cytoplasm
interfaces, respectively. Evaluation of equation (15) using
equations (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 range
from unity for small gap thicknesses (F'g(A2)=ir'i3(A2)) to
zero for large gap thicknesses (F'g(A2)»F'i2(A2)). The form
of 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 weighting
factor 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
and
co
in equation (23) are solutions to the following integrals:
J
Ajexp(-z/y)dz = 1 - exp(-A2/y) (28a)
0
CO= 1/y I
J A,exp(-z/y)dz = exp(-A2/y), (28b)
which are proportional to the evanescent intensity in the
water gap and in the cytoplasm, respectively, for a field
decaying with a depth of penetration equal to y. For the
data in Fig. 10, the value of y=0.1786A is intermediate
between 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 of
penetration would be generally applicable only if it
resulted in good estimates of water gap fluorescence at
several angles of total reflection. Fig. 11 compares 'actual'
water gap fluorescence values (symbols) calculated using
matrix theory to 'approximated' water gap fluorescence
values (continuous lines) plotted as a function of water gap
thickness 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 an
incident angle of
70
degrees can be analyzed in terms of an
effective water/cytoplasm refractive index. Using
equation (20), the slope of the data in Fig. 10 (y=0.1786)
yields an effective refractive index of 1.357 while the
average depth of penetration (dave=0.1761) yields the
nearly identical value of
1.355,
which differs by less than
0.15%.
The apparent ability to simply use the average
depth of penetration and still obtain such good fits to
matrix 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 the
evanescent decay (Fig. 5).
Discussion
The fluorescence excited at regions of close contact be-
tween the cell and surface must be quantitated if TIRF is
to 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 2.0
Water
gap
thickness (A^lambda)
Fig. 10. Semi-log plot of weighting factor plotted against water
gap thickness showing that the weighting factor fits an
exponential with a constant rate of
decay.
Data for this plot
were derived from individual data points (•) in Fig. 8.
TIRF microscopy 225
0.5-r
0.0
0.0 0.1 0.2 0.3 0.4 0.5
Water gap thickness (A^lambda)
Fig. 11. Water gap fluorescence plotted as a function of water
gap thickness for several angles of total reflection and TE-
polarized light. Continuous lines represent approximated water
gap fluorescence calculated from matrix theory, symbols
represent water gap fluorescence calculated from equations
(23)-(27) using a three-phase glass/water/cytoplasm model.
nonlinear function of the separation distance (A2). Since
the refractive indices of water and cytoplasm are similar
and the cell membrane is very thin, simplified expressions
for 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 films
is through the use of an effective refractive index. In the
case of a TLRF experiment of adherent cells, such an
approach is valid only for fluorescence excited in a dye-
labeled membrane (Table 1) for which the interfacial
evanescent intensity depended upon the refractive index
of 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 to
model the fluorescence excited in the water gap proved to
be unreasonable, and the calculated fluorescence instead
was well represented by a weighted sum of the fluor-
escence excited at a glass/water and a glass/cytoplasm
interface (equation (23)).
Equation (19) for membrane-excited fluorescence, sub-
stituted with equations (22a) and (22b), can also be derived
from the exact analytical expressions pertinent to thin-
film optical theory and is valid as long as the refractive
indices in the water gap and cytoplasm are similar (see
Appendix). However, equation (23) cannot be derived in
the same manner, but the weighting factors in this
expression 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 expressions
was very good compared to more exact matrix calculated
values (Figs 8 and 11).
The sources of error in the development of the above
approximate expressions for fluorescence excited in the
cell membrane (equations (19), (22a) and (22b) and in the
water gap (equation (23)) of a cell-substratum contact
arise from two assumptions. (1) The presence of the cell
membrane 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) A
simple combination of two-phase glass/water and glass/
cytoplasm interfaces can be used to approximate closely
the fluorescence intensity excited in a three-phase glass/
water/cytoplasm system.
The argument for neglecting the cell membrane was
that the cell membrane was much thinner than the
wavelength of light and therefore did not appreciably
affect the evanescent intensity calculations (see Appen-
dix).
The effect was demonstrated in Fig. 3 over a broad
range of incident angles by noting that the calculated
interfacial intensity appeared to be unaltered if one
neglected the cell membrane in the calculation scheme.
However, the exponentially decaying evanescent intensity
does not exist just at the interface, but extends in the z
direction 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 membrane
was determined more accurately by comparing the evan-
escent field intensity distribution in a cell contact region
with and without the cell membrane (not shown). For the
case of TE-polarized light and an incident angle of 70
degrees, the percentage difference between the two values
never exceeded 2.5% for water gap thicknesses in the
range 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 in
the range of 0^z/A=£l for several angles of total reflection.
However, the approximation error in the crucial region
between z/X=0 and 2/A=0.1 is not clearly delineated in
these 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 close
contacts). Further error calculations (not shown) revealed
that the approximation error starts at zero and increases
with water gap thickness. For an incident angle of 70
degrees with TE-polarized light, the error for membrane-
excited fluorescence was roughly linear reaching a value
of 1.2
%
for a gap thickness of
A2/A=0.1,
while the error in
water gap fluorescence reaches a plateau of 0.5% at
A2/A=0.1.
Although the error for membrane fluorescence
reaches 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 remains
minimal.
Other sources of error that may complicate application
of these approximations (equations (22) and (23)) are
changes in the fluorophore quantum efficiency as a func-
tion of distance from and orientation to the substratum
surface, fluorescence excited by scattered light, and inho-
mogeneous fluorophore distribution. For protein adsorp-
tion, the excitation of fluorescence by light scattered from
the solid/liquid interface is significant (Reichert, 1989).
Comparison of IRM and
TLKF
images of the same cell have
shown good correspondence between the appearance of
focal contacts and fluorescence intensity, indicating that
excitation appears to be confined to the evanescent volume
at the substratum surface (Lanni et al. 1985). The possible
importance of scatter has not, however, been experimen-
tally investigated for adherent cells until recently (Grappa
et al. 1990).
The proximity and orientation of the fluorophore with
respect 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 efficiency
depended slightly upon the distance of the dye layer from
the substratum surface (Suci and Reichert, 1988). How-
ever, rigorous calculations of the total power dissipated by
a point source dipole (the harmonic oscillator analog to an
emitting fluorophore) in the same dielectric environment
showed an emission decrease of approximately
1 %
over
the thickness of the L-B film (Suci, 1988). Since Dil
(Molecular Probes)-labeled membranes of adherent cells
represent a similar system, one would anticipate a small
attentuation of emitted fluorescence as the membrane
approaches the substratum surface. Such attenuations
could lead to a slight overestimation of cell-substratum
separations.
Membranes are also birefringent, as are L-B films, with
an optical axis normal to the membrane surface
(nx=ny^nT) (Suci and Reichert, 1990). Therefore, if one is
to minimize birefringent effects the excitation radiation
should be restricted to TE polarization that is parallel to
the net membrane molecular orientation at the sub-
stratum surface. The electric vector of TE polarized light is
also parallel to the transition moment of a cyanine label
when the membrane is parallel to the substratum surface,
thus maximizing fluorescence excitation.
Finally, local aggregations of fluorophores can also
produce spurious effects. Dye aggregations may produce
high levels of excited fluorescence, which may be misinter-
preted as close membrane contacts. On the other hand, if
intermolecular quenching is significant, or dye is excluded
from regions of high protein concentration, then lower
than expected emission levels may result. Although we
currently have no evidence to support or refute either of
these aggregation effects, one would speculate that dye
aggregation would be minimal in homogeneous solutions
of fluorophores (e.g. fluorescein-labeled dextrans), but
could 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 the
concentration of membrane fluorescence, estimates of cell
separations based upon measurements at a single angle
may be subject to significant errors. Taking measurements
of membrane fluorescence at one location on the mem-
brane for several angles above the critical angle can yield
direct estimates of A2. Taking the natural logarithm of
equation (19), which upon rearrangement, yields the
following linear function with respect to l/dW> with a
slope of — A2 andy intercept of ln[Tefl{0)Am/cos $i]:
Incos
20i
ft(0)Aml _ Aa_
O82ei J drt' (30)
where TefT<0) and
deff
are given by equations (20) and (21),
respectively. Assuming Am, 6U nx and
A
are known, the
value of A2 is sensitive to the choice of
n'eff.
The refractive indices used in this paper (Fig. 2) were
selected as representative values for calculation purposes.
In order to estimate the error associated with an inaccur-
ate estimate of refractive index
/i'eff,
values of
lntF'm/cos2^] were calculated with constant effective re-
fractive index values of rcefr=1.37 and n'eB=1.36 over an
incident angle range of 68-80 degrees for three different
values of A2, with Am=4nm, A=514.5nm and «i=1.52.
For each angle, l/defr was calculated using seven different
values of n'effthat covered a reasonable range of refractive
indices (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
0.0040.008 0.012
l/deff(nm~')0.016
Fig. 12. \n[Fm(&2)/cos28i] versus 1/d^ plot for values of
membrane-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 effective
refractive 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.52. Note that the water gap thickness and
transmitted interfacial amplitude can be determined directly
from the slope and y intercept of a straight line fit to each data
set.
Values of ln[F'm/cos20] were plotted against l/den-for the
various assumed values of n'efr, producing a family of
curves (Fig. 12). For each estimated value of /i'effand
defT,
the data were well fit by a straight line (correlation
coefficient 0.994—1.000) and the values of water gap
thickness (A2) were obtained directly from the slope using
equation (23) (Table 2).
From the data in Table 2 one can construct percentage
error plots (not shown) of water gap thickness error and
cytoplasmic refractive index error against water gap
effective refractive index error. The error in water gap
thickness was linear with water gap effective refractive
index error and independent of water gap thickness (i.e. a
% error in ra'eff results in a
10 %
error in A^, while the
cytoplasmic effective refractive index increased nonli-
Table 2.
Water
gap thickness and effective cytoplasmic
refractive index calculated from linear regression
ln[F'm/cos2d1] versus /dl
Ag (nni)
20
50
100
20
50
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 15.30
62.89 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
1.384
1.403
1.428
1.378
1.389
1.406
1.374 1.370 1366
1.380 1.370 1.359
1.390 1.370 1.346
1.361
1.346
1.317
1.355
1.330
1.278
TIRF microscopy 227
nearly with water gap effective refractive index error and
increased 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). A
1
%
error in effective refractive index centered about a
value of 1.37 (Table 2) corresponds to a refractive index
range of 1.356-1.384, which encompasses the cytoplasmic
refractive index range reported elsewhere (Lanni etal.
1985).
However, it should be noted that the error in the
cytoplasmic refractive index would not affect the determi-
nation of water gap thickness from membrane-excited
fluorescence data, unless the incident angle was not
sufficiently far from the critical angle to ensure total
internal reflection.
Variations in the water gap and cytoplasmic refractive
index are important because the region between a focal
contact and the surface, and the adjacent cytoplasm, are
known to contain receptors for the adhesion proteins such
as fibronectin and vitronectin. The high density of proteins
in a small volume of fluid could produce a local increase in
the refractive index. In fact, refractive indices as high as
1.36 to 1.40 have been assigned to focal contacts (Lanni
etal. 1985; Bereiter-Hahn etal. 1979). Consequently, a
given location in a cell-substratum contact could have a
cytoplasmic or water gap refractive index of as low as
1.333 or as high as 1.4, which could produce a local
variation in the critical angle from 61 to 67 degrees. An
incident angle in this range may result in locations in the
cell image that are not being excited by evanescent
radiation. One way to avoid this problem is to collect data
over the incident angle range of 70-80 degrees.
Alternatively, cell separation distances can be obtained
using a water-soluble label such as a low molecular weight
dextran. In this paper we have shown that the water-
soluble dye and membranes labeled by TIRF techniques
yield, 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 in
the water gap, are implicit in A2 and require estimates of
both neft and
n'efj.
An iterative scheme, such as Newton's
method, 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. The
water-soluble marker TIRF technique is discussed in
detail elsewhere (Gingell etal. 1987) and the reader is
referred to the literature for further details.
Conclusions
Here we have presented simple exponential approxi-
mations for fluorescence excited in the membrane and in
the membrane/substratum water gap of a cell adherent to
a glass substratum. Expressions were derived in which all
terms,
except the water gap thickness A2, can be deter-
mined from expressions for the transmission of light at
simple dielectric interfaces by knowing the refractive
indices of the glass, water and cytoplasm, and the angle of
total reflection. Representative calculations were per-
formed for TE-polarized light for incident angles ranging
from 68—80 degrees, which corresponds to the typical
range used in a TIRF experiment at the glass/water
interface. The error associated with these approximations
did not exceed 2.5% for a membrane separation typical of
focal and close membrane/substratum contacts. By fitting
the derived approximations to fluorescence collected from
dye-labeled cell membranes (equation (29)) or to fluor-
escence from solutions in the membrane/substratum
water gap (equation (30)) one can, in principle, calculate
the distance from the cell membrane to the substratum
surface. Currently we are examining membrane-labeled
fibroblasts and endothelial cells under TIRF illumination
using digital image analysis, which allows us to quanti-
tate the image brightness from specific segments of the cell
membrane. By collecting TIRF images at several angles of
total reflection, we plan to use equation (33) to transform
the change in local image brightness into local values of
cell-substratum separation (Grapa et al. 1990).
Appendix
Consider two dielectric films with refractive indices n2 and
rtm and thicknesses A2 and Am sandwiched between two
semi-infinite media with refractive indices n^ and n3. And
let 81, 62, 6m and 63 be the angles of light propagation in
each medium with respect to the interfacial normals. The
square of the electric field amplitude transmitted through
the two thin films {T(z)=T(&2+&m)) is given by the
magnitude squared of the transmission coefficient of the
two-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 Fresnel
reflection and transmission coefficients for the first, second
and third interfaces of the laminate. For TE-polarized
light, 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 a
glass substratum (see Fig. 2 in main text of
paper).
In this
configuration, the membrane is much thinner than the
wavelength 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 cell
membrane are neglected. Substitution of equation (A.2)
into equation (A.I) and evaluating yields the well-known
transmittivity expression for a single thin-film dielectric
during normal reflection and refraction of light (Born and
Wolf,
1986):
A^-
228 W. M. Reichert and G. A. Truskey
However, when the incident angle is beyond the critical
angle for total reflection for both a glass/water and a
glass/cytoplasm interface, then the field is evanescent in
both the water gap and the cell cytoplasm, in which case
cos#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) into
equation (A.I), and again evaluating yields the following
expression for 7XA2) during total internal reflection:
H2f23
exp(-A2/d12),(A.4)
where <i12 is the depth of penetration for total internal
reflection at the ii/n2 (i.e. glass/water) interface (see
equation (8) of main text) and where for TE-polarized light
we have r?2=l and:
l-Wrt!)2
^23
-[•
2[n2sin201 -
[n?sin201-/i§]1
r?.
=
-
raj]'
-
[rfoin'ft
-
-
n§]'
+ [n
foin'ft -
n2f~V
fsin2^
-
ntf]
(A.5)
Figure Al shows values of the bracketed exponential
prefactor in equation (A.4) plotted as a function of water
gap thickness for TE-polarized light at several angles of
total reflection, where rai=1.52, «2=1.333 and n3=1.37.
Note that, except for angles close to the glass/cytoplasm
critical angle, the exponential prefactor term is essentially
constant and equal to the value at the glass/cytoplasm
interface. 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 1.0
Water gap thickness (A2/lambda)
Fig.
Al.
Exponential prefactor
in
equation (A.4) plotted
aa a
function
of
water gap thickness
for
several angles
of
total
reflection
and
TE-polarized light. Note that except
for the
values
at
68 degrees, this prefactor remains essentially equal
to
the 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 above
expressions that ^==0 and t^^l. Therefore, the exponen-
tial prefactor in equation (A.4), subject to the condition
that n3 approximates «2, is very nearly equal to the
transmission term for a simple glass/cytoplasm interface.
Finally, subjecting 7TA2+Am) to the limit of Am/A->0 and
recognizing that 713=712 yields:
lim
7XA2
+ Am)= T13(0) exp(-A2/d12), (A.6)
where Ti3(0) and di2 are the square of the transmitted
interfacial amplitude and the depth of penetration terms
for 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 Whitaker
Foundation,
NSF
grants CBT-8796331
and
CBT-8746074,
and
NIH grant HL32132.
The
authors thank
W. N.
Hansen
of
Utah
State University
for
providing
a
copy of the characteristic matrix
software.
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(Received
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230 W. M. Reichert and G. A. Truskey
