Download General electromagnetic theory of total internal reflection fluorescence

General electromagnetic theory of total internal reflection fluorescence:
the quantitative basis for mapping cell-substratum topography
D. GINGELL 1 , 0. S. HEAVENS2 and J. S. MELLOR'
1
2
Department of Anatomy and Biology as applied to Medicine, The Middlesex Hospital Medical School, Cleveland Street, London 117, UK
Department of Physics, University of York, York, UK
Summary
Total internal reflection fluorescence (TlKF) has
recently been used to look at the contacts made
between cells and a glass surface on -which they
are spread. Our method utilizes the fluorescence
of a water-soluble dye that acts as an extracellular aqueous volume marker. Fluorescence is
stimulated by the short-range electric field near
the glass surface that exists under conditions of
total internal reflection. Since fluorescence is normally generated beneath a spread cell and not
beyond it, the fluorescence of the image is related
to the size of the cell-glass water gap. The
images obtained are remarkable for their detail,
contrast and the absence of confusing granularity
due to cytoplasmic heterogeneity, which is commonly seen in interference reflection (IRM)
images.
We here develop a rigorous electromagnetic
theory of total internal reflection in layered structures appropriate for cell contacts and apply it
to quantitative TIRF. We show that: (1) TIRF,
unlike IRM, can report cell-glass gaps in a way
that is practically independent of the detailed
physical properties of the cell; (2) TIRF is also
far more sensitive than IRM for measuring cellglass water gaps up to =100 nm. These striking
results explain the image quality seen by TIRF.
As the initial step towards verifying our theory
we show that measurement of the fluorescence
stimulated by total internal reflection at a simple
glass-water interface matches theoretical predictions.
Introduction
(Axelrod, 1981; Weiss et al. 1982). Here we present a
quantitative basis for its use and show that it is far
superior to IRM for studying cell contacts. We realize
that not all who are interested in the application of
TIRF will wish to do battle with the mathematics, so
our paper is organized so that the results are fully
understandable without reference to the theoretical
section (pp. 2-8).
The basis of TIRF is as follows. Consider a collimated beam of light in glass (refractive index H\),
incident at an angle <p on a planar boundary with a
transparent medium of lower index n2, where (p
exceeds the critical angle </>c(«2, «i). In this case total
internal reflection occurs; all the energy is reflected
back into the glass and no transmitted wave propagates
through «2- F ° r the interface between a microscope
coverslip and water (for example), Snell's Law gives:
Knowledge of the pattern of contacts between a cell
and the surface on which it is spread is basic to
understanding cell attachment and movement on solid
substrata. A decade ago the re-introduction of interference reflection microscopy (IRM) heralded a major
advance in the understanding of cell contacts and their
relationship to the cytoskeleton. Nevertheless IRM has
very serious drawbacks and although image interpretation has been placed on a quantitative basis (Gingell &
Todd, 1979; Gingell et al. 1982) insufficient knowledge
of dimensions and refractile properties of the cell
periphery is still a substantial practical limitation. Even
qualitative misinterpretations of IRM images are easily
made (Gingell, 1981).
An alternative procedure to IRM employs total
internal reflection fluorescence (TIRF). This powerful
light-microscopic tool was first used in cell biology only
recently to study the close approach of cells to glass
Journal of Cell Science 87, 677-693 (1987)
Printed in Great Britain © The Company of Biologists Limited 1987
Key words: electromagnetic theory, total internal reflection
fluorescence, evanescent wave, cell-substratum contact.
= arcsin(n z /«i).
677
Although there is no net energy flow into medium n-i
there is a very short-range electromagnetic disturbance
in the second medium near the interface, called an
evanescent wave. This will often have a complicated
waveform but its amplitude falls exponentially into
medium n-i normal to the interface, and dies out in less
than a wavelength. This phenomenon has a long
history and was anticipated by Newton. It forms the
basis of a type of beam-splitting prism and, most
importantly, occurs in fibre optic transmission lines.
The relatively recent interest shown by chemists in this
seemingly rather arcane phenomenon stems from the
fact that if a fluid medium contains dissolved fluorescent molecules, emission can be stimulated by the
evanescent wave in a very restricted zone within one
wavelength of the glass surface. This has been used
for several decades to study the adsorption of fluorescent macromolecules at optical interfaces (see Hlady
et al. 1985) but it is only in the past few years that cell
biologists have become aware of the subject and have
used it to look at cells spread on glass. Axelrod (1981)
reported the use of the fluorescent lipid analogue Dil to
label cell surfaces and he was the first to demonstrate
strictly localized fluorescence where cells come close to
glass.
Gingell et al. (1985) demonstrated that striking
images showing the topography of the cell-glass apposition zone can be obtained using a fluorescent extracellular water-soluble dye acting as an aqueous volume
marker. This produces dark contacts against a bright
background by virtue of exclusion of the extracellular
aqueous volume at contacts. The essential point of this
technique is that no fluorescence is normally stimulated
from the aqueous medium beyond the cell, since the
evanescent wave does not penetrate the several micrometres necessary to cross the cytoplasm. It is for
precisely this reason that the volume marker technique
is able to provide a unique map of the cell-glass contact
zone by reporting variations in the thickness of the
aqueous region between a living cell and its transparent
substratum.
In the following analysis we develop rigorous expressions for the electrical energy in the cell-glass gap
under conditions of TIR illumination, by solving
Maxwell's equations for all the conditions likely to arise
at cell contacts. From these equations we predict the
variation in the stimulated fluorescence with cell-glass
separation and discuss several factors that may influence the results. A critical comparison of T I R F and
IRM is made. Finally, we compare the experimentally
measured fluorescence at a simple glass-water interface
with our theoretical predictions.
(b)
(a)
\E
\E
\
V
E
\
V
D
7fc
n,
*C(»4.«|) < 4» 0C
(d)
(c)
\E
A
v
/i i.}-,.£,.,
Fig. 1. Layered dielectric model of cell-glass apposition.
Light is incident at an angle (f> (which exceeds the critical
angle) on the interface between glass (>i\) and a film of
aqueous medium (« 2 ) of thickness t\. The cell membrane
is represented by an isotropic dielectric film of thickness
tz~h and refractive index n3. The average cytoplasmic
index is W4. Complex waves (£1 ...E4) have complex
amplitudes in the positive .v direction (A...Z5) and negative
.v direction (A' ... C'). Quantities /3 r , yr are defined in the
text.
678
D. Gingell et al.
Fig. 2. The four situations that can arise when an
evanescent wave E exists in medium n2 are: in case (a) the
waves in media 3 and 4 are also evanescent; in (b) a
continuous wave C exists in medium 3; whereas in (c) it
exists in medium 4. In case (d) the media 3 and 4 support
continuous waves. The relationships between the incident
angle <p and critical angles <pc(nr,)i\) are explained in the
text.
Theoretical
We confine the analysis to the case of a plane s-polarized wave with electric vector perpendicular to the plane of
incidence (i.e. parallel to the reflecting interface), incident at angle <p>(f>c(n2,nl) on a glass-liquid interface
(x = 0). The bilayer membrane and the aqueous gap beneath it are simply represented as two thinfilmsof thickness
t2 — t\ and *i, respectively. The symbol" represents a complex quantity. A diagram of the layered system is shown
in Fig. 1 in which the y direction is perpendicular to the plane of the paper. The amplitudes EVT will in general be
complex. The incident wave has components Ey in the y direction but Ex = E~ = 0. Amplitudes Evr and refractive
indices nr are referred to media r = 1... 4. The quantities yr, /Jr are defined below. A ...D are complex amplitude
coefficients (phasors) of waves in the ±x direction. In all that follows, ti\ > n2, but distinct and interesting cases
for the different relative magnitudes of n\, «3 and na, will emerge. In region r, the appropriate form of Maxwell's
equations for plane waves of amplitude Ey are:
-nrkl)Ey
dxl
(1)
(2)
o • ax •
where
,
„ /.
When the waves in media 2, 3 and 4 are evanescent, the solutions for the second-order differential equation take the
forms:
Y
< 0
tz>x>tx
F
i = A e~'Y
~
Ey3 = C e~p'x + €'
ftA
"
'
x>tz
where
The relationship (eqn (2)) between electric and magnetic components gives:
Hai = {Ayt e~'YlX -A'YI
e'Y'x)/ck0^
(5)
The Maxwell boundary conditions for the parallel components of the electric and magnetic fields are:
Eyl(0) = Ej2(0)
Ey2(ti) =Ey3(tl)
Ey3(t2) = Ey4t2)
^
(6)
Hzl(0) = Ha2(0)
Mapping cell-substratum topography 679
Setting the incident amplitude A = 1 we obtain a set of linear simultaneous equations:
A1
-A'y,
-B
+iB~p2
-B'
0
0
0
+ 1=0
-iB'Pz
0
0
0
+ y. = o
0
-C'e
-C'p3
0
eft''
0
0
+ C ' e ft'2
0
0
+ C'p3 eft'2
0
0=0
0
0=0
-Df
+ Dp4 e -ft'z
(7)
0=0
0 =0.
These are solved using determinants to give the complex coefficients A', B, B', C, C', D. Substitution into
equations (3) and (5) gives the electric and magnetic fields in each medium as a function of x. Squared amplitudes,
which are real quantities, are then obtained by multiplying each complex amplitude by its complex conjugate
\EV\ = EV-E*.
(AS an alternative, it is possible to derive a general recurrence relationship for fields in adjacent
layers, but quite a lot of algebra is needed to obtain the explicit solutions considered in this paper.) In a two-film
system, there are four different situations, which can arise according to whether the fields in media 3 and 4 are
.continuous (transmitted, homogeneous) or evanescent (inhomogeneous). Diagrams of these cases are shown in
Fig. 2A-D. Consider the behaviour of the function fiT-»z as the angle of incidence <p> <pc(ti2,>i\) varies:
(8)
when ti\ sin <p>nr the root is positive and /3r is real. However, when n\ sin <p<.nr,
(9)
where j3r is purely imaginary and yr is real. Consequently, when this occurs the wave in medium r becomes
continuous. The significance of this switch is easily understood since Snell's Law requires that the angle of
refraction in a medium adjacent to a dielectric film is independent of the refractive index of the film and depends
only on the indices of the two bounding media. Therefore, we can define a critical angle between ti\ and « r :
n\ sin0! = « r
«? sin 2 0, - n 2 = 0.
The relations:
c
<t> {nr,nx)><p><pc(«2,«i)
imply /3r imaginary and Ev continuous in medium r. Alternatively:
implies /3r real and Ev evanescent in medium r.
When «3 > « 2 < W4 and n^ > « 4 and (p is increased the sequence in Fig. 2 will be d—* b—> a but if 114 > n^ the
sequence will be d—*c—* a. The phenomenon of a continuous wave generated beyond a gap containing an
evanescent wave is called frustrated total internal reflection (FTIR). While the relationships between nT and <p that
we have described do indeed solely determine whether FTIR can occur, the power of the transmitted wave will fall
exponentially with the width of the gap containing the evanescent wave. It falls to around zero for a gap in the order
of A. We shall return later to the subject of FTIR in relation to cell contacts. For a cell surface, medium 3 represents
lipid ( t t 3 ~ l - 4 ; Ninham & Parsegian (1970)) and medium 4 represents cytoplasm (1-36 Sjn 4 ^ 1-37 from
refractometry; Bailey & Gingell (unpublished); Izzard & Lochner (1976)). Since W3 will exceed ;?4 situation c
should not arise in observation on cells.
Having calculated the squared amplitude |i?v2(.v)|2, which is proportional to electrical energy at a particular
depth (JC) in the water gap, we make use of the fact that stimulated fluorescence is proportional to the local electric
field energy (see Appendix 1). The coefficient of proportionality linking emission with stimulation will include
quantum efficiency (Q) and fluorochrome concentration (M). The proportion of the emitted photons detected by a
680
D. Gingell et al.
counting instrument will depend on several factors that are a constant (S) for a given system. Thus detected
fluorescence (Fc) for an area beneath a cell where the water gap is t\ will be:
Fc = QMS
Jo
\EvZ{x) | 2 dx =
(10)
QMSI(tx).
The background fluorescence (F^) at a nearby area without a cell will be:
F b = QMS f" \Ev2(x) \2dx = QMSI(»).
Jo
(11)
Therefore, relative fluorescence is given by:
)•
(12)
The coefficient QMS drops out and the ratio F, obtained from experimental measurements of fluorescence beneath
the cell and in the background, gives the cell—glass separation.
Case (a)
For the situation illustrated in Fig. 2(a) we obtain for the field in the second (aqueous) medium:
(13)
where
a, = (Pi + 0 2
a2 = - ( $ a s
3 /M/^4si
) sinh(5i + fizifiz + p*4) cosh(5i
34) sinhSi — ^ 3 ()3 4 — Pi) coshS]
(14a)
(14b)
2
^zh + P sinh/?2/]) sinh<5i
(14c)
coshd] + (y32p4 sinhp 2 t) + p3 cosh/32J 1) sinhS]
(14d)
and
Whence:
|2 —
(15)
and
(16)
(17)
When « 3 = w4 the expression for \Ey\2 reduces to that for a single film with an evanescent wave in the third
medium:
12
and
2
2yf i [ - ^ ^
+ /3i(/3
/3i 2 sinhj82/,
s m h 2 p V , + p i 3 ( c o s h 2 p ^ 1 - l ) + <1(p-2--l
(19)
sinh/32/, ) 2
For the case £)—»°° (or ^ ^ 0 ; W3—»n2) we obtain the expression for I^Vl2 for an evanescent wave at a single
interface between two bulk media:
1
4n 1 cos (p
f
exp
4^x
(20)
Setting the exponent equal to unity we obtain x, the characteristic wave penetration depth, namely the distance
into medium 2 at which the squared amplitude has fallen to l/e of its value at x = 0. Thus:
vac
x =
(21)
sin2<p Mapping cell-substratum topography
681
A compact expression for / is obtained as usual by integrating the squared amplitude from .v = 0 to infinity.
(22)
jMwy
The form of the evanescent wave Ev{x,z)
shown to be:
Ev(x,z)
at an unbounded interface is relatively straightforward and it can be
*nx sin0 - 8, | e~ f c v ,
= 2«i cos(p • cos( cot \
where the phase angle
^vac
(23a)
/
/
8
\
Pz
a.^tan-'/,
).
\k0nicos<pj
(23b)
The final consideration of case (a) involves solving for the field in the fourth medium:
where a3 and a4 are given in equation (14c,d). Integration between x = t and infinity gives:
2
lit t \tffePi
'' -/B 4 (y?.§+ /&*)•
7(11 2)
If the cytoplasm contained a fluorescent volume marker this expression would give fluorescence versus
cytoplasm-glass distance t2. We shall use it in a simplified model where the cell membrane is omitted and medium
«4 represents the aqueou9 medium on the far side of a thin sheet of cytoplasm (thickness t2~t\) separated from the
glass by an aqueous gap t\.
Case (b)
If there is a continuous wave in the third medium (Fig. 2(b)) we replace P\ by — y 3 and obtain:
|£2l
22 2 fl2,2 , 2
b + p 22 bb4
= - ( y ! - j82j84) sind, + y3(/32 + j84) cosd
where
=
(26)
>
(27a)
(27b)
b 3 = y3 (/34 sinh/32f, + /32 coshj32r,) cosd, + (^264 cosh^2«, - yf si
sinSi
(27c)
b 4 s y3 (/34 cosh/32/, + B2 sinh/32^ 1) cos61 + (/32/34
sinS)
(27d)
s
i
^
When the fourth medium supports a continuous wave /3 4 becomes — y\ leading to:
|2
4y?[{cicosh/32(<1 -.v) + c 2 sinh^ 2 (t 1 - ,v)}2 + {c4 co8h/32(ti - JC) + c3
- P2) sinhj62t\ 'cosh^i + PZ(Y\ y4 - Pl)coshB2t\ •'.3 inh6,} 2
2
1 + 74) COShB2t\ cosh6i + (Yip '3 • PzY*) sinnp2^i •sinh6,}
where
CI = /3 2 y 4 sinh6]
(30a)
C2 = /33y4COSh5,
(30b)
c3=#sinh6,
(30c)
c 4 = /3 2 /3 3 cosh<5 1
(30d)
and 8\ = Pi{tz—1\) as defined earlier. Setting the denominator of equation (29) to^i and integrating with respect
to x:
|(C'
682
+
°2
D. Cn/^e// e< a/.
° 3 + ° 4 ) sinh2j3t, + 2(cf - c2 - c2 + c2)^, + j - (c,c 2 -fc 3 c 4 )(cosh2)3 2 / l - 1 ) | . (31)
P22
J
Case (d)
In this situation continuous waves exist in both the third and fourth media (Fig. 2(d)). We set f}\ = — y\; /J2 = — y2.
and defining &z = Y'i{ti~t\) obtain:
- .,
=
ty2\
4rf[73(73 sin6 2 • sinh/32(<i - x) - fi2 cos<52 • cosh02(*, - x)}1
+y5{y 3 cos<52-sinhj82(f, -x)+/3 2 sin<5 2 • cosh/32(*i ~ x)}2]
.
(32)
2
[73(71X4 — y3l)sinhyS2fi •cos6 2 + /3 2 (yiy 4 + yf)cosh/32ti -sin<52]
[(7i73 - £ 2 y 4 ) sinhj82t, • sin<52 - y 3 £ 2 (yi + y4) cosh£2«i • cos<52]2
d, = Y$ sin2<52 + 74-73 cos 2 6 2
2
z
(33a)
2
d 2 s yf/31 cos 6 2 + yip 2 sin <52
(33b)
d3 = 73^2(74 - 7!) sin<52 cosd2 .
(33c)
Setting the denominator in equation (32) to J2 we have:
}
(34)
When «3 = « 4 we have a single film with a frustrated wave in the third medium:
- '
and
{(7i73-/3l)sinhyS 2 f 1 } 2 +{y3 2 (y 1 + y 3 ) c o s h e r , } 2
2y
,
XHZW1
, ,3,^o..p2H/
^
Absorption
We finally consider the case of absorption in the second medium for case (a) only. If the aqueous medium absorbs
the evanescent wave it is convenient to define a complex index of refraction:
n2 = n2— iK2 ,
(37)
where K2 is the absorption coefficient and n2 is the real part of the refractive index. We replace n2 by iT2 in all field
equations and obtain a complex fi2 (eqn (4)):
]}2 = ko{a-ib),
where
(38a)
2 • 2_L
2
a = n\ sin d> - n 2 •
(38b)
Real and imaginary parts of fi2 are obtained using de Moivre's theorem:
Pi — ^o( a + b ) < coslx] —t sin 1x1 > ,
(39a)
where
W e define
^{^2} - s,
so that
^
^2 = 5 ] +
J52
.
(39c)
Mapping cell—substratum topography
683
This expression for /S2 is now substituted in equation (13) for the field Ey2. Using \Ev2\2 = Ey2 ' E*2
obtain:
\E 12 =
1 y2]
where
jV
jV,
tiftf
finally
+ Np
(NY+XS+NSrf+iNSAUYN^)2'
'<''= ess'<''-*>fe,
sinS2(t, - *
i (
v)
+ e- ' "- te3sinS2(<1-x)+^cos52(i1-x)}.
s
we
V
'
(41a)
x)
N2 = t '^~ {gl cosS2(t, -x) -g2smS2(tl -x)}
+ e - s . d , - * ) ^ ^ CosS2(*, _ x ) + f t s i n 5 2 ( f ] _ x)y .
(41b)
JV 3 = y33(/34cosa2 • sinhCT! + Si coscr2 • cosher — 5 2 sina 2 • sinhaj) cosh^i
• cosher — 5 2 ^4 sin0 2 • sinhCTi + /3f cosa 2 • sinhCTi) sinhdj .
oshai + 5 j sina 2 • sinha]
+5 2 COSCT 2
• cosha^coshSi
+ (5i/3 4 sin0 2 • sinh<7i + 5 2 /3 4 cosa 2 • coshai + /3f sina 2 • coshai) sinh6) .
Ns = /33()34 cosa 2 • coshCT] + Si
COSCT2 •
(41c)
(41d)
sinhcrj - S 2 sina 2 • cosha^ coshdi
+ (S1P4 COSCT2 • sinh<7i — S2^34 sina 2 • coshai + /33 COSCT2 • cosh<7i) sinh^i .
Nf, = j33(j34 sina 2 • sinhCTi + Si sina 2 • coshCT] + S 2 cosa 2 • sinhai) coshoj
+ (Si/3 4 sina 2 • cosher + S2y34 COSCT2 • sinhCT] +^33 sina 2 • sinhCTi) sinhoj .
(41e)
(41 f)
In equation (41) we have used:
<5i = & ( ' 2 ~t\),
Oi=Siti,
oz=S2ti
= -{ft + S,j84) sinho, - (jr33yS4 + S,j83) cosho,
(41g)
^3 = - ( $ - S,j84) sinho, - (fofa - 5,/33) cosh6, .
Integration gives:
+
fe.ft -«i) s.n2S2r, + feg2 + ^ 3 )(cos2S,., - 1) y _
(42)
where J 3 is the denominator of equation (36).
The integral:
I(K2)=r\Ev\zdx,
(43)
Jo
which gives the fluorescence at an unbounded interface in the presence of absorption, can be obtained from our
results by reducing equation (40) for the case t\ —* °o; tz^* t\. After much algebra all terms in /33 and /34 disappear
and we obtain:
giving the characteristic decay depth x = 1/2S. Integration gives the background electric energy in the absorbing
medium:
I(K2) =
\Ev\*dx = —
^—-2-r ,
(45)
which reduces as required to the expression (eqn (22)) for a non-absorbing medium at a simple interface when
K2 = 0 (implying S 2 = 0, whereupon Si = fi2).
684
D. Gingell et al.
Numerical results
The objective of this work is to show how the water gap
between a cell and its substratum can be measured by
the TI R-induced fluorescence of a water-soluble dye in
the gap. Thus we compute curves of fluorescence
versus gap width. Since the electric field of the
evanescent wave in the gap stimulates fluorescence, the
curves are strongly dependent on factors that affect
field energy, namely the angle of incidence <p of the
laser beam on the glass-water interface, and the
refractive index ti\ of the glass. These two factors also
govern how far the evanescent wave penetrates not only
into the gap but also into the adjacent cell. If the cell is
very thin the wave may even penetrate right through it
and stimulate fluorescent molecules on the far side. We
shall present the results of calculations, using the
theory developed above, which show to what extent
measurement of the water gap by T I R F depends on <p
and 77] as well as on the physical properties of the cell
and its shape. We shall also discuss other factors that
may influence the results and give a numerical comparison of the sensitivities of T I R F and IRM.
Dependence of fluorescence on the width of the water
gap for various incident angles (p and refractive
indices of the glass n/
Fig. 3 shows fluorescence (in arbitrary units) stimulated by light (A = 488 nm) incident at angles between
70° and 85°. The curves show how fluorescence
depends on the thickness (t\) of the water layer in
which the evanescent wave develops. Fig. 4 shows
curves of relative fluorescence (F) (fluorescence expressed as a fraction of the bright background value, as
defined in equation (12)) versus t\ for <p values between
65° and 85° for both normal glass (wj = 1-539) and
high-index glass (ii\ = 1-85). The optical properties of
the cell are as shown in Fig. 3. The significant feature
of these curves is the short range of T I R F : in all these
cases 90% of background fluorescence is reached
before the water gap reaches 160 nm. Penetration of the
evanescent wave falls as <p increases. On high-index
glass at high incident angle 90 % of background fluorescence is reached before the water gap reaches
80 nm, giving an appropriate range for measuring
cell-substratum gaps without the evanescent wave
passing right through the cytoplasm of thinly spread
cells (see below).
Relationship between fluorescence and gap width for
various assumptions about the nature of the cell
surface
An important result is shown in Fig. 5. Whereas on
normal glass ((p = 75°; n\ = 1-539) setting the cytoplas*
mic refractive index to either 1-37 or 1-40 makes a
significant difference even at small distances (=17%
error at 20 nm) we find that on high refractive index
glass at the same incident angle the worst error is less
than 2-5nm, right through the useful range from 0 to
70 nm or more. At t\ = 10nm the error is 10%, falling
to 5 % at 30 nm. Furthermore, the presence or absence
of a membrane in the model makes even less difference
than the cytoplasmic variation discussed. We conclude
that gap measurement on a high-index substratum is
virtually independent of assumptions about the refractive index of the cytoplasm and the ce.ll membrane.
Comparison of TIRF with IRM for measuring cellsubstratum topography
Figs 6 and 7 show comparisons of quantitative interference reflection theory (Gingell & Todd, 1979) with
the present TIRF theory. In comparing the merits of
these two techniques we define sensitivity as the change
in signal/background with water gap, dR/dti. In the
case of TIRF, R = F (eqn (12)). The IRM theory has
been experimentally validated and is capable of high
accuracy (Gingell et al. 1982) provided the optical
properties of the components are adequately known.
This, however, can be a serious practical limitation for
measuring cell contacts. IRM is best on low-index
40
30-
20-
10-
(iY)
50
100
Water gap /| (nm)
150
Fig. 3. Calculated fluorescence Fc (arbitrary units)
generated by the evanescent wave in the aqueous gap
beneath a model cell (see Fig. 1) plotted as a function of
gap thickness t\. Curves are shown for several angles of
incidence <p= (i) 70°, (ii) 75°, (iii) 80°, (iv) 85°. Refractive
indices are as follows: glass n\ = 1-539; aqueous medium
/?2 = 1-337; cell membrane n^ = 1-45 (thickness
t2~t\ = 4nm); cytoplasm «.» = 1-37. Wavelength of incident
light A = 488 nm.
Mapping cell-substratum
topography
685
glass, where the differences in amplitudes reflected
from glass and the cell surface are minimized. In this
situation (Fig. 6) it is clear that the IRM image only
begins to give useful information about the cell-glass
gap when it exceeds 30nm or so; at smaller distances
sensitivity is very poor. Furthermore, the result is
dependent on the cytoplasmic refractive index because
IRM image formation depends on phase retardation
due to the refraction of light through the cytoplasm as
well as reflection at the cell periphery. The reflected
amplitude is a function of the refractive indices of the
various layers of the cell surface and the interpretation
of IRM images depends on the details of the assumed
properties of this region. Even under optimal conditions of high illuminating numerical aperture where
the contribution of cell thickness to image formation is
minimized, this contribution still depends critically on
the cytoplasmic refractive index (Bailey & Gingell,
unpublished), and it is for this reason that cytoplasmic
inhomogeneities give IRM images a characteristic
granularity.
80
100
Water gap (,
Fig. 4. Calculated relative fluorescence F developed in a
variable aqueous gap /|. Upper three curves, refractive
index of glass n\ = 1-85; lower three curves, n\ = 1-539.
Incident angles (<p) 65°, 75°, 85° as shown. Other
parameters as in Fig. 3.
IRM
TIRF
10
10
20
30 40 50
Water gap f, (nm)
60
70
Fig. 5. Calculated relative fluorescence F developed in a
variable aqueous gap t\ showing the limited dependence of
fluorescence on cell properties. Upper three curves;
ti\ = 1-85; lower two curves rt\ = 1-539. In curves (i) and
(iv) cytoplasmic refractive index n+ = 1-40; in (iii) and (v)
tit, = 1-37. In (ii) setting nj = 114 gives a situation where the
membrane is (optically) absent.
686
D. Gingell et al.
20
30
40
50
Water gap I, (nm)
60
70
F i g . 6. Comparison of (i), (ii) I R M with (iii), (iv) T I R F .
Relative signal (i.e. signal/background) appropriate to each
method is plotted against the aqueous gap t\. For IRM the
parameters used are glass n\ = 1-539; medium n2 = 1 3 3 7 ,
t\ variable; lower lipid membrane iij = 1 4 5 , 4 n m ;
cytoplasm « 4 = 1-40 (i) or 1-37 (ii), 1000 n m ; upper
membrane (as lower); upper medium (as lower). For
T I R F , w, = 1-539; « 2 = 1-337, t, variable; #i3 = 1-45, 4 n m ;
«4 = 1-40 (iii) or 1 3 7 (iv). A = 4 8 8 n m ; angle of incidence
tf> = 75°only.
Although TIRF developed on low-index glass
depends to a limited extent on the cytoplasmic refractive index (lower curves, Fig. 6) the sensitivity of the
method is very high. On high-index glass (Fig. 7) IRM
is useless below 50 nm but on the other hand TIRF is
seen to be a very sensitive function of the width of the
water gap, rising to almost half the background value
in 30 nm. Even 2nrn is probably measurable (see
Appendix 2).
Dependence of fluorescence on gap width for the
special case where the cytoplasm is very thin
Evidence from quantitative IRM and supported by
electron microscopy shows that cells can make ultrathin
peripheral cytoplasmic lamellae, of ~100nm (Gingell
& Vince, 1982; Mellor & Gingell, unpublished). These
appear as dark areas in IRM, similar to intimate
cell-glass contacts. When such lamellae have formed it
is difficult if not impossible to measure their separation
from the substratum by IRM when their thickness is
unknown. If lamellae have a refractive index different
from bulk cytoplasm the matter is even more uncertain.
In contrast, TIRF offers a way of measuring cell-glass
gaps beneath thin lamellae, utilizing the low wave
penetration available with- glass of high refractive
index. In addition there is the advantage that uncertainty in assigning the cytoplasmic refractive index
becomes unimportant when high-index glass is used.
But first, consider the situation of T I R F on normal
glass shown in Fig. 8. Fluorescence beneath a lamella,
as a function of the water gap, is shown in the
ascending curves. Fluorescence stimulated beyond the
lamella, due to partial penetration of the thin cytoplasmic layer by the evanescent wave, is shown in the
descending curves. For a given lamella thickness,
increasing the water gap (i.e. moving the lamella
further away from the glass) reduces the fluorescence
originating beyond it while increasing that beneath it,
up to the bright background value where relative
fluorescence approaches unity. Results are given for
lamellae of 100 and 200nm. Obviously, much less
fluorescence is generated beyond a thick lamella than a
thin one.
When the same modelling is done with glass of high
refractive index (Fig. 9) it is immediately clear that
very little fluorescence originates beyond the cell, even
with no water gap. When (p = 78° a 100 nm lamella is
effectively infinitely thick. This offers an unambiguous
way of analysing the contacts of cells where such
lamellae form, and this will be expanded in a subsequent paper.
IRM
1-On
0-8-
0-6-
0-4-
50
100
Water gap r, (nm)
0-2-
10
20
30
40
50
Water gap (| (nm)
60
F i g . 7. Curves (i)-(iv) as Fig. 6 except substratum
modelled as glass of high refractive index ii\ = 1 8 5 . Curve
(v) as for (ii) except n^ = n+, i.e. cell membrane optically
deleted.
150
F i g . 8. Evanescent wave can penetrate a thin cytoplasmic
lamella and stimulate fluorescence beyond it. Inset sketch
shows details. Ascending curves: relative fluorescence
generated beneath cell approaches unity as aqueous gap t\
increases. Descending curves: relative fluorescence
generated beyond lamella falls to zero as l\ increases.
Continuous curves, 100nm lamella; broken curves, 200 nm
lamella. Angle of incidence either 76° or 64° as shown.
A = 488nm.
Mapping cell—substratum topography
687
Under what conditions does frustrated total internal
reflection fluorescence (FTIRF) occur and what can
be learned from this effect?
In the results given so far, no reference has been made
to the different possible electromagnetic waveforms in
the cell periphery (Fig. 2). In most situations case (a)
is appropriate, but at sufficiently small (p values, not far
above the critical angle <pc{nz,n\), case (b) and ultimately at <f) values just above </>c(«2.«i) case (d) will
occur. As <p is reduced that part of the cell periphery
with the highest refractive index will be the first to
transmit a continuous wave. We would expect this to be
the plasma membrane bilayer and the conditions for a
continuous wave in n^ but not « 4 are:
When these criteria are satisfied, the plasma membrane
will behave like a parallel wafer guide, and a wave will
be propagated along it, totally internally reflected from
the lipid-water and lipid-cytoplasm boundaries. At
a slightly smaller incident angle a continuous wave
will be transmitted through the cytoplasm as well
(case (a)). This will happen when:
0 c («2i«i)
<(
P <arcsin(«4/«i).
We have in fact seen fluorescence stimulated by laser
light transmitted by FTIR from cells that have regions
very close to glass, when illuminated at incident angles
well above 0 c ( n 2 , n i ) . These fluorescent beams look
Fluorescence
beneath cell
like comet tails, originating at cells and pointing away
from the light source. This phenomenon will be
discussed further in a later paper. The fact that FTIRF
first appears near 69° in the case of chick heart
fibroblasts shows that the part of the cell periphery
with the highest refractive index has an index of 1-40.
This is presumably the mean refractive index of the
lipid bilayer.
To what extent is gap measurement influenced by
absorption of the stimulating evanescent wave by
aqueous fluorochrome molecules?
Fig. 10 illustrates the degree to which perturbation of
the evanescent wave, due to its absorption by the
fluorochrome, affects fluorescence. It can be seen that
relative fluorescence decays more rapidly as the absorption coefficient Kz increases, so that increasing Kz is
similar to increasing the refractive index of glass. This
incidentally suggests that a suitable absorber in the
medium could attenuate the evanescent wave, without
recourse to a substratum of high refractive index.
However, our reason for modelling absorption was to
see whether it was likely to 'bleed off sufficient
evanescent wave energy to cause a significant error
in quantitative work. Since our maximum measured
extinction coefficient for fluorescein isothiocyanate
(6850OM~'cm~') gives an evanescent wave absorption
coefficient of only 5X 10~5 there should be no measurable error from this source (see next section).
1-25
1-20-
115-
110-
105-
50
100
Water gap (| (nm)
Fluorescence
beyond lamella
150
F i g . 9. As Fig. 8 except glass nx = 1-85. Results for
100nm lamella only given. Curves (i), (iii), <p = 78";
curves (ii), (iv), $ = 64°.
688
D. Gingell et al.
1-00
F i g . 10. Ratio of fluorescence in the absence Ft, and
presence F{, of absorption of the stimulating wave, at an
unbounded interface, plotted versus angle of incidence <p.
Coefficient of absorption: (i) A'2 = 0-05; (ii) K2 = 0-005.
Measurement of TIRF at a simple interface
We measured the fluorescence developed by solutions
of fluoresceinated dextran (FD-4; mean M r = 4100;
Sigma, Southampton, UK) at different concentrations, as a function of the angle of incidence exceeding the critical angle, at an unbounded glass-water
interface. The solutions were made by serial dilution of
a stock containing 5mgml~ FD-4 plus 5mgml~
unlabelled dextran (Mr = 6000; Fluka, Buchs, Switzerland) and 25 mM-Hepes buffer, pH 7-2. Dilution with a
similarly buffered l O m g m P 1 solution of unlabelled
6000 Mr dextran obviated small variations in refractive
index, which otherwise arise due to variable dextran
content. All solutions were passed through a sterile
0-2 nm pore size 'Acrodisc' filter (Gelman Sciences
Ltd, Northampton, UK) before use. The refractive
index of the solutions (w2= 1-339) was determined
with an Abbe refractometer (model 60/ED Mkl;
Bellingham & Stanley Ltd, Tunbridge Wells, UK)
illuminated with a low-power beam from an argon ion
laser at A = 488 nm.
The equipment used for photon counting has been
described in detail elsewhere (Gingell et al. 1985). For
fluorescence measurements, several drops of the dextran solutions were placed on the upper surface of a
glass prism (n = 1525) and illuminated by means of an
argon ion laser (A = 488 nm) at angles above the critical
angle (pc(ri2, «i) = arcsin (1-339/1-525). The beam was
spatially filtered and refocused onto the interface with a
half-angle convergence of 0-5°. The error in incident
angle at the centre of the optical field was found to be
less than ±0-05°. The laser used in light control mode
gave a constant power output of 100 raW, and the beam
was attenuated with a neutral density filter (O.D. = 1;
Melles Griot, Aldershot, UK) to prevent significant
photobleaching. Rotation of the laser beam to give
transverse polarized light (i.e. s-polarization, perpendicular to the plane of incidence, to within ±1°) was
obtained with a double Fresnel rhomb (Lexel, Lamda
Photometries, Batford Mill, UK), which screwed to
the front of the laser. Emitted fluorescence was collected using a 63 X water immersion objective (Zeiss,
Oberkochen, Welwyn Garden City, UK) focused on
the interface, using a photomultiplier aperture of
2-5 mm. The laser beam was centred to the optic axis of
the objective by small adjustments to the angle of
incidence and a lateral micrometer movement, to peak
the photon count rate over 100 s. This was repeated
for each solution and for every angle of incidence.
Although there was no evidence of dextran adsorption
to glass (a 10-20 s rinse in deionized water returned the
photon content to background; see also Hlady et al.
1985) the upper surface of the block was rinsed
between dextran solutions with dilute Teepol, then
water and finally acetone. This had no effect on
measured fluorescence but gave very reproducible
values of laser light scattered by imperfections in the
interface. Scatter was measured at each angle of
incidence using lOmgml" 1 unfluoresceinated dextran
in place of FD-4, after removing the Schott OG515
barrier filter. The extinction coefficient e of each FD-4
solution was measured using an LKB Ultrospec 4050
spectrophotometer and the absorption coefficient Ki
(eqn (38)) was calculated according to the relationship
K2 = 2-303ehM/4jl, where M is the fluorescein (fluorochrome) concentration. All optical measurements
were made at 22°C.
Fig. 11 A,B shows that, over the entire range of FD-4
solutions, fluorescence is linearly proportional to fluorochrome concentration. Thus quenching is absent,
even at the highest concentration used. Comparison of
theory with measured fluorescence over a range of
incident angles was performed in two ways. In the first,
experimental values of fluorescence versus (f> are normalized to the fluorescence measured at 62° and it can
be seen that they follow closely the corresponding
theoretical curve, shown as a continuous line in
Fig. 12. The second method involves linear transformation of the curves and permits calculation of the
constant QS defined in equation (11). This equation
gives the background fluorescence Fb at an unbounded
interface for the case of unit incident amplitude. We
introduce a factor of cos<p to take account of the fact
that the proportion of the incident flux falling within
the fixed measuring area, viewed by the photomultiplier, falls with increasing <p. Furthermore, although
the total laser beam power is constant during measurements, the incident amplitude depends on previous
reflections, which change with <p. Setting the angleindependent factors QMS = C and recalling the angular
dependence of / we write:
A((p) is the amplitude of the wave in the glass, which
depends on previous reflections. Substituting for /
from equation (22) and expressing Y\ and fiz from
equations (8) and (9) gives:
C = Fb(<f>)/G(<P),
(47)
where
G(<P) =
Thus a plot of the experimental values Fb((p) versus
the denominator of equation (47), [D(<j>)], should give
a straight line of slope C for a given concentration M.
Fig. 13 shows that a most satisfactory linear relationship is indeed found between Fb((p) and D((f)) for all
concentrations used. From C we obtain the system
constant QS, which relates measured photon count rate
Mapping cell-substratum
topography
689
2-2x10*-
1-5X10 3 -
64-2°
lxH) 5
66-4°
4
5XK)
0
1-0 2-0 3-0 4-0 5-0
0 0-5 1-0
2-0
Concentration of fluoresceinated dextran (mgnil"1)
3-0
4-0
Fig. 11. A,B- Measured fluorescence as a function of the concentration of FD-4 for a range of incident angles. Curves for
the three low incident angles shown in A are partially included as broken lines in B. The latter has an expanded scale
suitable for larger values of (p. The slopes of the regression lines shown (expressed as the ratio of the slope for 62°) are
given with the corresponding theoretical values ( ): 1, (1); 0-39, (0-40); 0-25, (0-24); 0-14, (0-15); 0-09, (0-10); 0-06,
(0-06); 0-04, (0-03); 0-02, (0-02); 0-01, (0-01). Repeat readings (X) taken 9 h later, after the full range of measurements
was completed.
to the integral I(4>, °°) for a given value of M. The
strictly linear curves that pass through the origin
indicate: (1) the correctness of the physical theory for a
simple interface; (2) the measuring system is highly
accurate; (3) excitation of fluorescence by scattered
laser light in our system is negligible; (4) adsorption of
FD-4 onto glass is negligible; (5) photobleaching is
negligible under our experimental conditions; (6) absorption of laser light by the fluorochrome is negligible,
as deduced in the previous section. The last point
means that the energy absorbed from the evanescent
electromagnetic wave, which drives fluorescence, is a
negligible fraction of the electric field energy. The fact
that scatter cannot be a major determinant of fluorescence was shown by direct measurement at 488 nm
without FD-4. Whereas the photon count from fluorescence changes 100-fold from lowest to highest incident angle, the count from scatter changed only
10-fold. It is therefore related to <p by a completely
different functional relationship from that linking fluorescence with (p. Thus any significant contribution to
690
D. Gingell et al.
the total fluorescence stemming from scattered light
would bend the straight lines in Fig. 12.
We should stress that the relatively simple situation
of an evanescent wave at an unbounded dielectric
interface, which is a limiting case of our general theory,
is well known (Born & Wolf, 1975) and has been
investigated using fluorescent monolayers (Carniglia et
al. 1972). However, we are not aware of any previous
comparison of fluorescence generated over the entire
evanescent wave with theoretical values of !{<p, °°) as a
function of incident angle. The results show that our
experimental system is eminently suited to the task of
making reliable accurate quantitative measurements of
cell to glass contacts using the multilayer theory.
Conclusions
The theory of total internal reflection in multilayers
that we have presented is complete insofar as Maxwell's
3X105
2-25X10 5
l-5xlO s -
7-5X104-
62
64
66
68
70
72
74
76
78
80
82
Fig. 12. Fluorescence versus (p normalized to that at
<(> = 62°. Experimental points and theoretical curve shown.
equations provide a complete description of the properties of light. The equations give closed form solutions
for a plane parallel wavefront incident on plane parallel
layers. This is a reasonable description of a collimated
laser beam 'illuminating' cell-glass contacts by TIR.
Provided refractive indices at the appropriate wavelength are known, the problem is solved. However, a
theory of TIRF, relating fluorescence to the electromagnetic stimulating energy, is less straightforward.
Since the probability of photon emission by a fluorescent molecule is proportional to the squared amplitude of the local electromagnetic field (Appendix 1),
the total emission in the evanescent field is proportional
to the integral of the squared amplitude from the
glass—water interface to 'infinity', effectively a few
hundred nanometres. When emission is expressed as
relative fluorescence (eqn (12)) the proportionality
factors QMS cancel out. In a brief discussion of
quantitative T I R F based on cytoplasmic fluorescence,
Lanni et al. (1985) used the theory developed by
Lukosz & Kunz (1977a,b) to compensate for the fact
that the proximity of a dielectric interface perturbs the
emission of fluorescence. However, they conclude that
the effect is likely to be small. Since we get precise
correspondence between the theory for a simple interface and experiment without taking this into account,
either the correction is negligible or other errors
compensate for it.
1000
2000
3000
4000
5000
D(<p)
Fig. 13. Measured fluorescence (photon count rate)
plotted against [D((p)], the denominator of equation (47).
The slope C at each concentration is the coefficient of
proportionality between the measured photon count rate
and the stimulating evanescent wave energy. The slopes of
the regression lines expressed as a ratio of that for
5mgml"' are: 1-0, (1-0); 0-80, (0-80); 0-61, (0-60); 0-38,
(0-40); 0-20, (0-20); 0-09, (0-10); 0-05, (0-05), where the
ratios of the concentrations used are shown in parenthesis.
There remains a problem that is less easy to decide.
Our quantitative volume-marker TIRF method assumes that the entire volume between the substratum
and the plasma membrane is equally accessible to the
fluorescent tracer molecules. However, the presence of
glycoproteins external to the bilayer may result in a
certain volume fraction being unavailable to tracer
molecules. The error will be minimal with a small
tracer such as unconjugated fluorescein or rhodamine.
Gingell et al. (1985) showed that 168 000MT dextran is
excluded from certain zones of cell-glass contact, but
that 4000Mr fluoresceinated dextran and free fluorescein are apparently not. Where a probe is excluded,
an increasing error would occur as the separation
approaches and then falls below the thickness of the
glycoprotein region. By further measurements on the
relative fluorescence at cell contacts using fluorescent
probes of different sizes it should be possible to know
when volume exclusion is influencing the results.
Mapping cell-substratum
topography
691
Summary of results
h is Planck's constant and v is frequency. E is the
amplitude of the electric field. Hence:
(1) T1RF provides an exquisitely sensitive way of
measuring cell-glass apposition distances up to 100 nm
or so.
(2) The method is incomparably more sensitive
than quantitative interference reflection microscopy
(IRM) to small changes in distance, up to 100 nm or
more. At larger separations IRM comes into its own
and TIRF becomes insensitive.
(3) T I R F is relatively insensitive to the optical
properties of the cell periphery and cortical cytoplasm.
On glass of high refractive index this insensitivity is
remarkable. This provides TIRF with a second major
advantage over IRM that is sensitive to the refractive
heterogeneity of cytoplasm (even under optimal conditions of high illuminating numerical aperture) and
the assumed refractive index of the bilayer and peripheral glycoprotein region.
(4) The possibilities of varying evanescent wave
penetration by use of substrata of a range of refractive
indices, and by varying the angle of incidence of the
stimulating beam provide a powerful tool for analysing
cell contacts. With highly attenuated waves it should be
possible to resolve the otherwise difficult question of
the size of the cell-glass gap beneath very thin cytoplasmic extensions.
Despite the drawback of relatively expensive and
complex equipment, including that for recording cell
images at relatively low levels of light (Gingell et al.
1985), we feel that TIRF, even in a qualitative form,
will soon be considered an indispensable tool for
studying cell contacts. In following papers we shall
present an experimental investigation of TIRF using
thin films and a quantitative analysis of the contacts of
spread cells.
D.G. thanks the Science and Engineering Research
Council for supporting this work, and his family for supporting him while deriving equation (40). We are grateful to
Dr P. Tatham for his kind assistance with optical density
measurements.
Appendix 1
The Poynting vector (5) is defined as the energy flux
directed normal to unit area per second. In MKS units
01,= i)
(Ai)
= Nhv,
where iV = number of quanta passing unit cross-section per second, e and e r are the absolute and relative
permittivities of material r and £o, f.io are the electric
and magnetic permittivities of free space, respectively,
692
D. Gingell et al.
(A2)
Suppose photons flow axially along a cylinder of unit
cross-sectional area and length c/\/lFr containing M
fluorochrome molecules per unit volume. The number
of molecules in the cylinder is:
Mc/VFT.
The number of excitations per second F' is proportional to A' and to the number of molecules
accessed, so that:
NMc _ Me
Ver ~Tv
(A3)
which gives the required relationship between the field
and the generated fluorescence.
Appendix 2
We believe that our bulk volume marker method may
have a vertical resolution better than 2nm, for the
reasons given in Numerical Results, third section. It
might be thought that resolution would be limited by
the dye concentration, since this determines the mean
separation between fluorochrome molecules. Since the
latter greatly exceeds 2nm it is important to show that
it is not a limiting factor.
The mean separation of fluorescein molecules /
in a l m g m l " 1 solution of fluoresceinated dextran
(/V/r = 4000) is =46 nm. Even at our top concentration
of 5mgml~' it falls to only 27 nm. Although these
distances are certainly not small compared with cellglass gaps or the evanescent decay depth x\ the
essential point is thato/ze fluorochrome molecule could
give a satisfactory 'report' of a small gap (given a
sufficiently sensitive fluorescence detector) as it diffuses in a random walk with components in the A*direction (perpendicular to the walls), radiating at a
rate or |.EV2|2. Even for the largest dextran we have used
(A/ r = 70000; D - 6 X 1 0 " 7 c m 2 s ~ ! ) the mean displacement S" in one second is 10 nm. This is far greater than
gap dimensions and implies many bounces from the top
and bottom walls during one photometric measuring
period. Thus:
F'(t)dt
(A4)
Further, even if the fluorescent molecules in the gap
were relatively fixed in position during the measuring
period, they would occupy random positions on the
A--axis, so that the use of a photometric measuring
area of much greater diameter than / would give a
GINGELL, D., TODD, I. & HEAVENS, O. S. (1982).
satisfactory measurement of the gap by generating
fluorescence proportional to:
Ev2\2dx.
The fact that we obtained coincidence between experimental and theoretical curves of fluorescence versus
angle of incidence provides a practical proof of the
correctness of the above arguments. Although the
system is unbounded (no gap) the same principles
apply, since/is not small compared with "x.
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Mapping cell—substratum topography
693