Spatially resolved fluorescence correlation spectroscopy using a spinning disk confocal microscope.

Abstract

We develop an extension of fluorescence correlation spectroscopy (FCS) using a spinning disk confocal microscope. This approach can spatially map diffusion coefficients or flow velocities at up to approximately 10(5) independent locations simultaneously. Commercially available cameras with frame rates of 1000 Hz allow FCS measurements of systems with diffusion coefficients D~10(-7) cm(2)/s or smaller. This speed is adequate to measure small microspheres (200-nm diameter) diffusing in water, or hindered diffusion of macromolecules in complex media (e.g., tumors, cell nuclei, or the extracellular matrix). There have been a number of recent extensions to FCS based on laser scanning microscopy. Spinning disk confocal microscopy, however, has the potential for significantly higher speed at high spatial resolution. We show how to account for a pixel size effect encountered with spinning disk confocal FCS that is not present in standard or scanning FCS, and we introduce a new method to correct for photobleaching. Finally, we apply spinning disk confocal FCS to microspheres diffusing in Type I collagen, which show complex spatially varying diffusion caused by hydrodynamic and steric interactions with the collagen matrix.

Full text

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Spatially Resolved Fluorescence Correlation Spectroscopy Using a
Spinning Disk Confocal Microscope
Daniel R. Sisan, Richard Arevalo, Catherine Graves, Ryan McAllister, and Jeffrey S. Urbach
Department of Physics, Georgetown University, Washington, District of Columbia
ABSTRACT
microscope. This approach can spatially map diffusion coefficients or flow velocities at up to ;105independent locations
simultaneously. Commercially available cameras with frame rates of 1000 Hz allow FCS measurements of systems with diffusion
coefficients D;10?7cm2/s or smaller. This speed is adequate to measure small microspheres (200-nm diameter) diffusing in
water,orhindered diffusionof macromoleculesincomplexmedia(e.g.,tumors, cellnuclei,or theextracellular matrix). Therehave
been a number of recent extensions to FCS based on laser scanning microscopy. Spinning disk confocal microscopy, however,
hasthepotentialforsignificantlyhigherspeedathighspatialresolution.Weshowhowtoaccountforapixelsizeeffectencountered
with spinning disk confocal FCS that is not present in standard or scanning FCS, and we introduce a new method to correct for
photobleaching. Finally, we apply spinning disk confocal FCS to microspheres diffusing in Type I collagen, which show complex
spatially varying diffusion caused by hydrodynamic and steric interactions with the collagen matrix.
We develop an extension of fluorescence correlation spectroscopy (FCS) using a spinning disk confocal
INTRODUCTION
Fluorescence microscopy is a powerful technique that has
given us a detailed view of the generation, maintenance, and
function of cellular organization. As our understanding of
subcellular and intercellular processes increases, there is a
need for more quantitative approaches to measuring intra-
and intercellular dynamics and transport, with high resolu-
tion in both space and time. A number of techniques based
on fluorescence correlation spectroscopy (FCS) and scan-
ning microscopy have been successful in addressing this
need. Here we report an extension of FCS to spinning disk
confocal microscopy, which can further push the limits of
temporal and spatial resolution to 1000 frames per second at
up to 105locations simultaneously.
FCS was developed in the 1970s and is widely used to
study intracellular dynamics (1). Using confocal or two-
photon techniques, light intensity fluctuations caused by
fluorescent molecules or microparticles moving through a
femtoliter measurement volume are recorded and analyzed.
The temporal autocorrelation of these fluctuations can be
used to determine the diffusion coefficients, concentrations,
mobile fractions, and flow velocities of the fluorescent
species.
FCS has been extended in many ways, including several
very recent applications of laser scanning microscopy
(LSM). In LSM, an image is constructed by collecting light
from confocal or two-photon optics point-by-point (creating
‘‘pixels’’) serially. We refer to the FCS techniques based on
LSM as ‘‘scanning FCS.’’ Among the variations of scanning
FCS are: image correlation spectroscopy (ICS) (2,3), image
cross-correlation spectroscopy (ICCS) (3), scanning fluores-
cence correlation spectroscopy (SFCS) (4), raster image
correlation spectroscopy (RICS) (5), spatiotemporal image
correlation spectroscopy (STICS) (6), and position-sensitive
scanning fluorescence correlation spectroscopy (PSFCS) (7).
Spinning disk confocal microscopy (SDCM) is a widely-
used extension of confocal LSM that allows for higher
imaging speeds (8,9). (The speed limitation in the most
common LSM implementation results from the time needed
to repeatedly accelerate and decelerate the scanning mirror,
which is eliminated in SDCM.) A rotating disk of ;10,000
pinholes, together with a corotating disk of microlenses,
produces an array of diffraction-limited spots that rapidly
scans the image field (Fig. 1). The fluorescent light is re-
focused on the pinholes and then projected on a CCD
camera.
ICS, ICCS, and STICS are approaches that can transfer
directly to SDCM, since they all start with an image time
series. ICS involves the calculation of either the spatial
autocorrelation (e.g., to establish the distribution of mole-
cules in a membrane) or the spatial average of the temporal
autocorrelation (to improve statistics when the dynamics are
slow and the region homogeneous). STICS extends ICS by
analyzing the generalized spatiotemporal autocorrelation for
an image or subimage. Both ICS and STICS have been used
to spatially map dynamics in cells (but only on a few sub-
images) (10). ICCS extends ICS to situations with different
fluorescent species. RICS, SFCS, and PSFCS, which use the
time delay between pixels in scanning microscopy to extract
more information, are not applicable to spinning disk mi-
croscopy.
In this article we describe the temporal version of ICS as
applied to SDCM, but without spatially averaging the results.
We address the effect of the pixel size on the FCS mea-
surement and calculate a numerical correction factor, which
Submitted February 28, 2006, and accepted for publication August 9, 2006.
Address reprint requests to J. Urbach, Tel.: 202-687-6594; E-mail:
urbach@physics.georgetown.edu.
? 2006 by the Biophysical Society
0006-3495/06/12/4241/12$2.00
doi: 10.1529/biophysj.106.084251
Biophysical JournalVolume 91December 20064241–42524241

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we validate with a stochastic simulation. We also introduce
an empirical method of minimizing the effect of photo-
bleaching, which can be severe for slowly diffusing fluores-
cent species. Finally, we use spinning disk FCS to map
hindered diffusion in hydrogels, a problem relevant to drug
and gene delivery, such as the dispersal of high molecular
weight agents in the tumor interstitium (11) or through the
extracellular matrix (12). Hindered diffusion in hydrogels
(e.g., collagen) is also a consideration in the design of tissue
substitutes and controlled release devices (13). We obtain a
spatial map at pixel resolution (128 3 128) of diffusion co-
efficients from diffusing fluorescent microspheres (210-nm
diameter) in Type I collagen and find diffusion coefficient
variation that is complex and directly related to the polymer
network structure. In addition, we observe deviations from
normal diffusion that are well described by spatially varying
anomalous diffusion.
Other approaches have been applied to hindered diffusion
in inhomogeneous media. NMR has been used to map ef-
fective diffusion coefficients and diffusion anisotropies (14).
Fluorescence recovery after photobleaching has been used to
measure diffusion coefficients of tracer molecules in various
media, for example, Type I collagen (15), agarose gels (16),
tumors (11), and the extracellular space in the central ner-
vous system (17). Fluorescence recovery after photobleach-
ing was also used to measure spatial inhomogeneities in
diffusion (18). An FCS-based nonscanning approach, sam-
pling-volume controlled FCS (19), was applied to diffusing
fluorophores in an aqueous hyaluronic acid solution. None of
these techniques provide the spatial resolution of spinning
disk FCS. Theoretical approaches to hindered diffusion also
typically spatially average, as for example the effective
medium model (20,21). More involved calculations (22) and
simulations (23), which are capable of incorporating micro-
structure, so far also have presented results as spatial
averages. However, in complex biopolymer networks,
diffusion can vary locally and dynamically, for example in
‘‘caging’’ effects (24). The fast, spatially resolved technique
described here will provide a powerful new tool for studying
hindered diffusion and other transport phenomena in com-
plex media.
THEORY
FCS
Fluorescence correlation spectroscopyisbasedon thetempo-
ral autocorrelation function,
GðtÞ ¼ AÆdFðtÞdFðt1tÞæ;
(1)
where the angled brackets indicate an average over the time
t,dFðtÞ ¼ FðtÞ ? ÆFðtÞæ, t is the delay time, and A is a norma-
lizationconstant.Therearetwocommonnormalizations:A¼
1/ÆdF2æ, so that G(t) is in the range [1, ?1]; and A ¼ 1/ÆFæ2,
in which case G(t ¼ 0) is the inverse of the average con-
centration.
The fluorescence fluctuation dF for a given detector
position depends on the details of the optical system, which
are incorporated in the point spread function PSF(r ? rd),
Z Z
where r is position in the object plane, rdis a point on
the detector mapped to the object plane, and dCðr;tÞ ¼
Cðr;tÞ ? ÆCðr;tÞæ is the zero-mean fluorophore concentra-
tion. (The analysis could be performed equivalently in any
plane conjugate with the sample, with appropriate factors to
account for magnification.) PSF(r ? rd) is the image of
a point particle located at r and depends on the spatial
variation of both the illumination and the light collection
efficiency. The standard FCS analysis assumes that PSF(r ?
rd) can be approximated as a separable product of Gaussians
dFðtÞ ¼
PSFðr ? rdÞdCðr;tÞdrdrd;
(2)
FIGURE 1
arrays of moving pinholes and microlenses scan a field of
view in one camera exposure, giving lateral and axial
resolutions typical of standard confocal microscopy (the
densities of the arrays are much higher than shown). The
image of a point source can then be approximated by a
product of Gaussians (Eq. 3) and standard FCS analysis
can be applied to each pixel time series.
Spinning disk confocal microscopy. Aligned
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Biophysical Journal 91(11) 4241–4252

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(25), which is generally true for confocal optics with a small
pinhole:
?
?2ðz ? zdÞ2
PSFðr ? rdÞ ¼ I0exp
?2ðx ? xdÞ2? 2ðy ? ydÞ2
w2
?
0
?
3exp
w2
z
?
:
(3)
The factor of 2 in the numerator is included for con-
sistency with the FCS literature, where w0refers to the e?1
radius of the illumination and the collection efficiency func-
tions, the product of which give PSF(r ? rd). The image of a
finite size microsphere can also be approximated by Eq. 3 (if
the sphere diameter is not much larger than w0), but w0will
be larger than the diffraction-limited w0.
In Eq. 2, the r integral is over all space, and for spin-
ning disk FCS, the rdintegral is over one pixel. Substituting
Eq. 3 and integrating over rdgives the apparent intensity
profile E(r). The x component, E(x), is (similarly for the y
dimension),
Z
pixel
PSFðx ? xdÞdxd¼ EðxÞ ¼ E0 Erf
ffiffiffi
??
2
p
w0
?
L
21x
?????
?
1Erf
ffiffiffi
2
p
w0
L
2? x
?
;
(4)
where L is the length of the pixel. When L is not too large
compared to w0, Eq. 4 can be approximated by a Gaussian,
with an e?2radius given by weff¼ b w0. The factor b de-
ends only on the ratio w0/L (see Materials and Methods and
Fig.2).Inthelimitofinfinitesimalpixels(L/0),b/1and
the result of the rdintegral is E(r) ¼ PSF(r) (i.e., Eq. 3 with
rd¼ 0).
After integrating PSF(r ? rd) to yield E(r), Eq. 1 can be
written as
Z Z
GðtÞ ¼ AEðrÞEðr9ÞÆdCðr;tÞdCðr9;t1tÞædrdr9 ;
(5)
where r and r9 are position at t and t1t, respectively, and
the brackets indicate a time average. With the Gaussian ap-
proximation to E(r), and the concentration autocorrelation
function for normal diffusion,
ÆdCðr;tÞdCðr9;t1tÞæ ¼
1
ð4pDtÞ3=2exp
?ðr ? r9Þ2
4Dt
??
;
Eq. 5 becomes (1)
GðtÞ ¼ Gð0Þ 11
4Dt
w2
eff
?????1
11
4Dt
g2w2
eff
?????1=2
;
(6)
where g ¼ wz/weffis the ratio of axial to (effective) lateral
e?2radii. (Correlations from dynamics of the fluorophore
triplet state are usually considered in conventional FCS, but
the timescale is much faster than our fastest frame rate and so
they are not considered here.) The autocorrelation with the
exact form of E(r) (Eq. 4) is too complicated (involving still
more error functions) to use with fitting procedures.
FIGURE 2
to account for the effect of finite pixels. (a)
The correction factor b, which depends
only on the ratio of the e?2radius w0to the
pixel length L, relates w0 to weff, the
effective e?2length that accounts for finite
pixel size. The solid line is a fit to the data
given by the equation at the top of the
figure. (b) When w0/L is ,0.5, the Gaussian
approximation to Eq. 4 starts to break
down, as seen by the best-fit Gaussian (line)
to Eq. 4 (?) for w0/L ¼ 0.4. (c) Results of a
stochastic simulation of spinning disk FCS
measurements, showing the relative error
with and without the finite pixel effect
correction. The top curve is the bottom
curve multiplied by b2.
Adjusting the FCS e?2radius
Spinning Disk FCS4243
Biophysical Journal 91(11) 4241–4252

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Hindered and anomalous diffusion
For normal diffusion in three dimensions, the mean-square
displacement of an ensemble of diffusers is proportional to
time,
Ær2ðtÞæ ¼ 6Dt;
(7)
where D is the diffusion coefficient. For a simple homoge-
neous solvent, D is given by the Stokes-Einstein equation,
D ¼
kBT
6pmrh;
(8)
where rhis the solute hydrodynamic radius, m is the solvent
viscosity, and kBT is the Boltzmann factor. Diffusion in a
dilute polymer network will have a D lower than the value
computed from Eq. 8 (using the solvent viscosity) because of
hydrodynamic and steric interactions of the solute with the
network. These interactions are said to hinder diffusion.
Hydrodynamic interactions with the polymer network
increase the drag on the solute and slow diffusion, and can be
calculated for a given configuration of fibers (23). These
interactions are long range, but are only significant within a
shielding length. Steric interactions occur only with direct
contact between the solute and the network, and their effect
depends on the geometry or structural properties of the
network (22).
These interactions can also change the character of diffu-
sion, such that Eq. 7 does not hold for any range of time-
scales. In hindered diffusion, the mean-square displacement
is often proportional to time to an exponent a,
Ær2ðtÞæ ¼ Gta;
(9)
where G is the anomalous transport coefficient (having
a-dependent dimensions). When a 6¼ 1, the diffusion is
called ‘‘anomalous’’, and when a , 1 (typical for hindered
diffusion), it is called ‘‘subdiffusive’’. The autocorrelation
function for anomalous diffusion is (26)
?
where Da (¼ G1=aw2?2=a
coefficient. Though fully characterizing anomalous diffusion
requires two coefficients, a map of Daalone can still charac-
terize the time to diffuse the fixed distance w: t ; w2/4Da.
Note that this generalization has normal diffusion as a special
case (a ¼ 1, Da¼ D), but still does not describe all possible
forms of diffusion in complex media.
GðtÞ ¼ Gð0Þ 11
4Dat
w2
eff
?a
=4) is an anomalous diffusion
???1
11
4Dat
g2w2
eff
??a
???1=2
; (10)
eff
MATERIALS AND METHODS
Spinning-disk confocal microscopy
Images were acquired with an inverted Nikon Eclipse TE2000-U micro-
scope and Nikon 603 oil-immersion (1.4 NA) objective (Nikon, Tokyo,
Japan), with 1.53 intermediate magnification (giving 903 total magnifica-
tion). The spinning disk confocal unit is a Yokogawa CSU21 scan head
(Yokogawa, Tokyo, Japan), part of a spinning-disk confocal package from
PerkinElmer (Wellesley, MA). The package also includes a Hamamatsu
cooled-CCD camera (ORCA-ER, Shimokanzo, Japan), a NEOS acoustic
optical tunable filter (AOTF) (Melbourne, FL), a proprietary sync box, and
software. Fluorescence excitation is provided by a Coherent Innova 300 gas
laser (Santa Clara, CA), which replaced the laser provided by PerkinElmer.
As described later in the section on imaging artifacts, the exposure of the
CCD must be precisely synchronized to the rotation of the spinning disk.
The PerkinElmer system achieves this with the AOTF, cutting the laser
power to the sample during the camera’s readout. The spinning disk spins at
1666.67 rpm, as controlled by a TTL pulse (at twice that frequency)
generated by the PerkinElmer sync box and connected to the scan head’s
BNC sync port. The spinning disk scans the field of view in 1/12 of a
rotation, so this rotation rate allows exposure times in steps of 3 ms. Only the
microspheres in pure glycerin were imaged using the full PerkinElmer
system.
The collagen and water-glycerin samples were imaged with a 512 3 512
Andor electron multiplied CCD (EMCCD) (DV-887, Andor Technology,
Belfast, Northern Ireland), acquired using Andor’s iXon software. With
these measurements, the spinning disk was controlled to run at its maximum
speed of 5000 rpm by a separate PC running Labview and a card with a
10 MHz timer (National Instruments, Austin, TX). The AOTF was con-
trolled by the camera’s TTL ‘‘exposure’’ signal, which was on only while
the camera was exposing (and not reading out). Cutting the laser power
when reading out is especially important for frame transfer cameras, like the
Andor EMCCD, as explained in the imaging artifacts section.
The pure water samples were imaged with a Cooke pco.1200 hs (high
speed) CMOS camera and Cooke’s Camware 2.12 (Cooke, Romulus, MI).
Here, the scan head was run at 5000 rpm and the AOTF allowed the laser
through continuously, since the camera does not have appropriate outputs to
synchronize illumination. Instead, the camera has a precise electronic
shutter, so the exposure time was adjusted to match a multiple of 1/12 of the
disk rotation rate.
Microspheres and PEGylation
Untreated and carboxylated microspheres adhere to collagen, so we em-
ployed spheres coated with polyethylene glycol (PEG), a molecule known
for its nonreactivity with proteins. Carboxylated fluorescent microspheres
(210-nm diameter, green-yellow FluoSpheres, Invitrogen, Carlsbad, CA)
were coated with amine-terminated PEG (5 kDa, Nektar, Huntsville, AL)
using a protein coupling kit (PolyLink Protein Coupling Kit for COOH
Microparticles) and protocol (Technical Note No. 644) from Polysciences
(Warrington, PA). In short, carboxylated beads were pelleted via centrif-
ugation and resuspended in a coupling buffer (50 mM MES, pH 5.2, 0.05%
Proclin-3). This step was repeated, and then EDAC (Carbodiimide) added to
form a 200 mg/mL solution. Next PEG was added and incubated for 30 min
to 1 h while gently mixing in coupling buffer. The beads were again
centrifuged and resuspended in a washing/storage buffer (10 mM Tris, pH
8.0, 0.05% Bovine Serum Albumin, 0.05% Proclin-300). For beads with
diameters below 1 mm, the protocol recommended using a Microkos micro-
filter (Spectrum,RanchoDominguez, CA) instead of centrifuging. However,
we found that many beads were lost in the filter, so we centrifuged instead.
The concentration of the sphere mixture, estimated from images of an un-
diluted drop, was ;5 3 1010spheres/ml, or 10% of the stock sphere
solution. Clumps were dispersed using continuous ultrasound for ;1 min
(Model 550 Sonic Dismembrator with 1/8" tip, Fisher Scientific, Pittsburgh,
PA), though a few (,1% of total) larger clumps remained.
Sample preparation
All samples consist of 50 ml of solution mounted between a microslide and a
No. 1.5 thickness glass coverslip, spaced with one layer of double-stick tape,
making a ;100-mm thick layer with low background fluorescence. The
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Biophysical Journal 91(11) 4241–4252

Page 5

sphere solutions were sonicated before mounting. Immediately after mount-
ing, the edges of the coverslip were sealed to the slide with nail polish to mini-
mize evaporation.
Glycerin samples were made by adding a drop of aqueous microsphere
solution (57-nm diameter, green fluorescent, G50, Duke Scientific, Palo
Alto, CA) to ;50 mL of glycerin on coverslips, which were left open to
evaporate the water. The samples were placed periodically on a slightly
warmed hot plate to speed evaporation. When the weight of the sample
returned to within 1% of its glycerin-only weight (typically overnight),
the evaporation was considered complete. The concentration of spheres was
;2 3 1011/ml.
Collagen samples were made from Type I Collagen stock derived from
rat tail (BD Sciences, Bedford, MA), and diluted with water to give a final
concentration of 2 mg/ml. Also added were 27 ml of a 7.5% sodium
bicarbonate solution per ml of original collagen stock, and 1/10 final volume
of 103 OptiMEM (Invitrogen). Last, the PEGylated sphere solution was
added to give a 6% final concentration by volume. (The sphere solution was
sonicated before adding to the mixture, since sonication damages collagen.)
The collagen solution was allowed to gel at room temperature after
mounting.
Water-glycerin samples were made with the same 210-nm diameter (un-
PEGylated) spheres, added to a 40% water by weight water-glycerin mix-
ture, which gives a diffusion coefficient similar to the collagen samples. The
sphere solution was added to be 6% of the final volume. Pure water samples
were made similarly.
Measurements
Samples were imaged ;5 mm above the coverslip. The tensioner on the
microscope focus was adjusted to minimize focus drift. Bright-field images
of the collagen looked identical before and after runs, suggesting there was
minimal sample drift during the measurement. Samples were measured at
ambient temperature, which was monitored near the sample with a thermo-
couple. Except for movies taken with the PerkinElmer software (pure
glycerin), all movies were taken on the same computer used for the autocor-
relation analysis.
The water-glycerin and collagen samples were both imaged at 903 with
the Andor EMCCD. 128 3 128 pixel (22.76 3 22.76 mm2) regions of
interest were selected and exposed for 12 ms at a frame interval of 12.2 ms,
for 8192 frames, and saved in a file (27), forming a ‘‘time chunk.’’ For each
sample, 16 time chunks were recorded.
The same procedure was used for the pure water samples, except the
images were recorded with a Cooke camera and a 3 ms frame interval. Also,
only three time chunks were recorded, and the images were stored in .tif
format. The pure glycerin samples were imaged at a frame interval of 50 ms
using the PerkinElmer system with Hamamatsu camera.
Autocorrelation analysis
All computer analysis was performed in MatLab R14 (The MathWorks,
Natick, MA) on a personal computer (3.0 GHz, 2GB RAM), using custom-
written C code compiledto run in the MatLab environment (using mex). The
mex C codes are ;20 times faster than comparable MatLab scripts, and can
use MatLab’s built-in functions.
The analysis began with loading a time chunk, consisting of 8192 128 3
128 images, into memory. The time-average for each pixel was subtracted,
and when the photobleaching correction was used, each time point was
divided by the spatial mean of the image for that frame. The autocorrelation
function was then calculated for each pixel and normalized by the variance
of the (corrected) time series, giving a curve in the range [?1, 1]. The
autocorrelation curves were then binned logarithmically. This procedure was
repeatedforeachtimechunk.Thebinnedautocorrelationcurvesforeachpixel
were averaged, and then fit using the procedure described in the next section.
The ratio of the mean-squared intensity to the intensity variance,
ÆF2æ=ÆdF2æ, was averaged for each pixel over time chunks; this ratio divided
by the extrapolated zero-time lag autocorrelation, G(0), gives the average
concentrationperobservationvolumeof ÆNæ ¼ ÆF2æ=ðÆdF2æGð0ÞÞ.Since our
experiments are performed at concentrations much less than one fluorescent
sphere per observation volume, it is essential to remove the background
intensity from the mean before constructing the ratio. We estimated the
background for each pixel and time chunk as the peak of the histogram of
intensity values for each time series.
Autocorrelation curve fitting procedure
The autocorrelations were performed using MatLab’s fft function (which
uses fftw) and the autocorrelation fits using MatLab’s nlinfit function (which
uses the Levenberg-Marquardt nonlinear least-squares fitting algorithm), to
the model,
?
GðtÞ ¼ Gð0Þ 11
t
tD
?a
???1
11
t
g2tD
??a
???1=2
1GðNÞ;
(11)
where G(0), G(N), tDand (sometimes) a were free parameters. When
diffusion was treated as normal, a was fixed at one. The diffusion coefficient
was found from the diffusion time, tD¼ w2
(effective) lateral e?2radii of the Gaussian profiles, g ¼ wz=weff, was found
from empirical estimates of wzand weff¼ bw0. First, two-dimensional
Gaussian fits (using fminsearch in MatLab) were performed on lateral slices
of a three-dimensional image of a 210-nm-diameter sphere mounted in a
viscous mounting medium (PS-Speck Microsphere Point Source Kit, P7220,
Invitrogen). The peak value (from the fits) for each z slice was then fit to a
one-dimensional Gaussian, giving wz. The e?2radius weighted by the peak
intensity was next averaged over slices, giving w0. Finally, multiplying w0
by the adjustment factor b (see Theory and Fig. 2 a) gave weff.
eff=4D. The ratio of the axial and
Photobleaching correction and pixel
size adjustment
The analysis involves two steps that are unique to spinning disk FCS. First,
to minimize the effect of photobleaching, which creates spurious time
correlations, particularly for slow diffusion, each pixel intensity time point
Fi(t) isdividedby thespatialaverageof the imageintensity,Favg(t). Thetime
dependence of Favg(t) and Fi(t) result from the product of the average
intensity per diffuser h(t)—which is generally a monotonically decreasing
concave-up curve resulting from photobleaching (what we are trying to
cancel)—and the fluctuating number of diffusers in the image Navg(t) and
pixel Ni(t), respectively:
Favg;iðtÞ ¼ hðtÞNavg;iðtÞ:
To the extent that Ni(t) and Navg(t) are uncorrelated, the ratio Fi(t)/ Favg(t)
is proportional to Ni(t). However, some number of diffusers leave the image
withinadiffusiontimeofleavingapixel(especiallyforpixelsneartheimage
edge), meaning some diffusive fluctuations of Ni(t) will be spuriously
cancelled in Fi(t)/ Favg(t). For diffusers that leave the image in the lateral
direction, this effect will be concentrated at the image edges, and uniform for
the z direction. This effect fortunately becomes insignificant when the
number of pixels Npis large. Since Navg(t) and Ni(t) both obey Poisson
statistics and because not all diffusers leave the image within one diffusion
timeafterleavingagivenpixel,therelativeimportanceofthiseffectisalways
,N?1=2
p
. The image size cannot be too large, however, because spatial
inhomogeneities in the illumination can cause h(t) to vary in space, causing
other spurious correlations. We find that 128 3 128 images work well.
The second step in the analysis is to replace the lateral e?2radius w0, with
an effective length scale weff¼ bw0that takes into account the finite size of
the pixels. Equation 4 (see Theory) is evaluated at 100 points over [?w0–L,
w01 L] and fit to a Gaussian using nlinfit in MatLab, from which b is
Spinning Disk FCS4245
Biophysical Journal 91(11) 4241–4252