Continuous Fluorescence Microphotolysis and Correlation Spectroscopy
Using 4Pi Microscopy
Anton Arkhipov,* Jana Hu ¨ve,yMartin Kahms,yReiner Peters,yand Klaus Schulten*
*Department of Physics and Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana,
Illinois; andyCenter for Nanotechnology and Institute for Medical Physics and Biophysics, University of Mu ¨enster, Mu ¨enster, Germany
surement of molecular mobility and association reactions in single living cells. CFM and FCS complement each other ideally and
microscope to the micrometer scale. However, cellular functions generally occur on the nanometer scale. Here, we develop the
theoretical and computational framework for CFM and FCS experiments using 4Pi microscopy, which features an axial resolution
of ;100 nm. The framework, taking the actual 4Pi point spread function of the instrument into account, was validated by mea-
surements on model systems, employing 4Pi conditions or normal confocal conditions together with either single- or two-photon
signal/noise ratio was small due to the small number of fluorophores involved.
Continuous fluorescence microphotolysis (CFM) and fluorescence correlation spectroscopy (FCS) permit mea-
Thermal molecular motion has been referred to as the restless
heartbeat of matter and life (1). This applies, in particular, to
living cells in which many processes are driven directly and
others are strongly influenced by diffusion. The experimental
study of diffusion in cellular systems was fundamentally ad-
vanced in 1974 when two complementary techniques were
introduced, fluorescence microphotolysis ((FM) also referred
to as fluorescence recovery after photobleaching) (2) and
fluorescence correlation spectroscopy (FCS) (3). In FM (2,4–
10), a small area of a fluorescent specimen is illuminated by a
fluorescence of the illuminated spot is then bleached ‘‘instan-
taneously’’ by increasing the power of the laser beam by
several orders of magnitude for a short time. Subsequently,
the redistribution of fluorescence by diffusion is followed at
the initial nonbleaching intensity and used to derive diffusion
coefficients. FCS also employs a focused laser beam to
illuminate a small spot of a fluorescent specimen at a small
nonbleaching intensity. However, the fluorophore concen-
tration is kept so small that fluctuations in the number of
illuminatedfluorophores becomeapparent and can be used to
derive diffusion coefficients by a correlation analysis. In the
meantime, FM and FCS have been further developed into
many directions (for review, see Day and Schaufele (11)). It
has become clear that both techniques can provide informa-
tion on molecular mobility and the association between
different molecular species. However, the techniques differ,
and, in fact, complement each other ideally, for instance with
respect to range of accessible diffusion coefficients and
fluorophore concentration. Thus, FM and FCS have become
indispensable tools of cell biology and are employed at an
In 1981, continuous fluorescence microphotolysis (CFM)
was introduced (12). It employs a focused laser beam to con-
tinuously illuminate a spot of a fluorescent sample at a laser
power inducing slow photobleaching. By these means the
illuminated spot is,atthesame time,depletedoffluorophores
diffusion. Because the laser power is much higher in CFM
experimental data. An intriguing property of CFM is that it
can be easily installed on a confocal microscope and com-
bined with FCS without any further instrumental modifica-
tions (13). Due to the complexity of the diffusion-reaction
processes studied using FM or FCS, it is often impossible to
extract the diffusion coefficient using an analytical descrip-
tion; in such cases, numerical calculations are employed.
Indeed, computational approaches proved to be reliable and
are widely used nowadays for analyzing the diffusion prop-
13,27), and correlation spectroscopy (28–30).
So far, in both CFM and FCS experiments the size of the
illuminated area was limited by diffraction to ;250 nm in
the focal plane and 600 nm in direction of the optical axis.
However, recently new light microscopy concepts have been
developed that improve the resolution up to ?30 nm. These
techniques include 4Pi microscopy (31–36), I5M microscopy
(37,38,35), and stimulated emission depletion (STED) mi-
croscopy (39–42,36). Emergence of these techniques has
already provided a new level of detail observed in biological
imaging (41,42,36,43), and promises to bring about a tre-
mendous improvement for the FM technology, but to take
advantage of the resolution improvement, an appropriate
Submitted February 28, 2007, and accepted for publication July 26, 2007.
Address reprint requests to Klaus Schulten, E-mail: email@example.com;
or Reiner Peters, E-mail: firstname.lastname@example.org.
Editor: Petra Schwille.
? 2007 by the Biophysical Society
4006Biophysical JournalVolume 93December 20074006–4017
theoretical and computational framework has to be created.
In this article, we develop the theory and computational
methods for the utilization of the 4Pi microscope in CFM and
FCS measurements. We also demonstrate the feasibility of
CFM and FCS experiments with a 4Pi microscope, and
analyze the ways of extracting the characteristics of diffusion
from the experimental data using numerical calculations.
The principle of 4Pi microscopy (31,33) is illustrated in
which are focused at the same point onto a sample. Con-
structiveinterferenceofthetwobeams enhances thefocusing
of the light, and the illuminated region is narrower along the
optical axis in the case of the 4Pi microscope than in the case
side lobes (see Fig. 2). Two-photon illumination is usually
employed to further narrow the excitation volume (31,33) by
mainly reducing the height of the side lobes. In this case, the
fluorescence signal (green beam in Fig. 1) is observed at the
wavelengths that are shorter than the illumination wave-
length. In 4Pi microscopy, various types of illumination and
detection are utilized (31): type A corresponds to the illu-
mination via two objectives with constructive interference
and detection through one of the objectives in a confocal
mode; in type B, illumination is performed by one beam, and
detection via two objectives; for type C, both illumination
and detection are performed using two objectives, with con-
copy, since it is the only one available commercially (Leica
Microsystems, Wetzlar, Germany).
The main challenge in analyzing CFM and FCS by means
of 4Pi microscopy is an adequate representation of the micro-
scope point spread function (PSF), which describes how a
point gets spread by the imaging process, due to the limited
resolution of the microscope (see Fig. 2 and Supplementary
Material).The 4PiPSFhasacomplex,nonanalytical shapein
three-dimensional (3D) space (see, e.g., Hell and Stelzer (31)),
which complicates the numerical solution required for sim-
ulation ofthesimultaneous diffusion andbleachingoffluoro-
phores in CFM. There exist programs that allow one to solve
the partial differential equations encountered in the descrip-
tion of CFM experiments (see, e.g., Schaff et al. (16)). How-
ever, to our knowledge, none of them is capable of handling
an arbitrary form of the PSF. Therefore, we developed our
own program to solve the arising differential equations
the experimental fluorescence signal. We find that in CFM
fluorescence is a poor signal/noise ratio, because of the small
number of fluorophores in the irradiated area due to the nar-
rowness of the PSF, a low detection efficiency of the optical
setup, and a low fluorescence yield in two-photon excitation.
Thiscan beimprovedbyreducing themeasurementrate inan
demonstrate that the characteristics of the diffusion can be
inferred nevertheless from a single CFM or FCS measure-
ment with a low signal/noise ratio. Overall, we show the
feasibility of two independent ways of obtaining the charac-
of the fluorescence signal and fluorescence correlation spec-
In the following,we will describe the experimentalsetup where fluorophores
diffuse in a 3D volume V, and the photons emitted by fluorophores are
recorded. The whole volume V is assumed to contain N fluorophore
molecules. For each molecule i (i ¼ 1;2;...;N), its position at time t is
denoted by r ~iðtÞ: Below, fluorophores will be referred to also as ‘‘particles’’.
Modeling of the 4Pi point spread function
The illumination PSF of the microscope describes the distribution of light
intensity in space and determines the strength of bleaching and fluorescence
at each given point. In the 4Pi microscope, the approximately spherical light
wavefronts from two opposing lenses are focused onto the sample (31), as
sketched in Fig. 2 a. The constructive interference of these wavefronts
sculpts the PSF (31,34,35,44,36) that generally extends over a narrower
region than that illuminated in a confocal microscope (45) (see Fig. 2, b and
c). The PSF specific for an instrument is expressed through the functions
hillðr ~Þ and hdetðr ~Þ; where hillðr ~Þ describes the illumination of the sample and
hdetðr ~Þ the recording of the fluorescent emission. The complete PSF hðr ~Þ is
theproduct ofthesetwo functions, i.e., hðr ~Þ ¼ hillðr ~Þhdetðr ~Þ:Bothfactorsare
wavelength dependent, but in experiments a range of wavelengths is
detected; for the purpose of analysis, we assume a single wavelength, which
corresponds to the weighted (in the detection channel) emission maximum
of the fluorophore.
Following the seminal work of Hell and Stelzer (31), one can calculate
the 4Pi PSF hillðr ~Þ; it is the result of the interference of two coherent light
beams, one coming fromthe top and another fromthe bottom (Fig. 2 a), with
their corresponding electric fields E~1ðr ~Þ and E~2ðr ~Þ: The function is given by
into two beams by a beam splitter (BS). Coming from opposite directions,
the two beams are focused at the sample, where constructive interference
of the two fields (E~1 and E~2) creates a narrow illumination spot. The
fluorescence signal is deflected by the dichroic mirror (DM), and focused on
a detector. Since two-photon excitation is employed (red), the fluorescence
(green) is at a shorter wavelength than the illuminating light. See the
literature (31,33,44) for details.
Schematic of a 4Pi microscope. Light from a laser is split
CFM Using 4Pi Microscopy 4007
Biophysical Journal 93(11) 4006–4017
hillðr ~Þ ¼ jE~1ðr ~Þ1E~2ðr ~Þj2nexc;
where nexc is the number of photons required for the excitation of a
fluorophore. We assume in the following two-photon excitation, and, thus,
choose nexc¼ 2. The field E~1ðr ~Þ corresponds to the spherical wavefront
focused around r ~¼ 0 (31,46,47), expressed as follows
E~1ðr ~Þ ¼ ðExðr ~Þ;Eyðr ~Þ;Ezðr ~ÞÞ
¼ ?iðI0ðr ~Þ1I2ðr ~Þcos2f;I2ðr ~Þsin2f;
? 2iI1ðr ~ÞcosfÞ:
The field E~2ðr ~Þ represents the coherent beam going in the opposite
direction, expressed through
Here, we used r ~¼ ðx;y;zÞ; f is the angle between the plane of oscillation
of the electric field in the beam and the plane of observation; the functions I0,
I1, and I2are given by
with u and v being optical coordinates (u ¼ (2pn sin2a)z/l, v ¼
=l), l the wavelength, n the refraction index, a the
aperture angle, and J0, J1, J2Bessel functions of the first kind. The resulting
4Pi illumination PSF (31) is given by
hillðr ~Þ ¼ ½ðReI0ðr ~ÞÞ21ðReI2ðr ~ÞÞ212ReI0ðr ~ÞReI2ðr ~Þcos2f
The detection PSF of the type A 4Pi microscope is given by hdetðr ~Þ ¼
jE~1ðr ~Þj2(because a confocal detection system is used), which leads to the
hdetðr ~Þ ¼ jI0ðr ~Þj212jI1ðr ~Þj21jI2ðr ~Þj2:
It should be noted, however, that because of the Stokes shift the wave-
length l is generally different for illumination and for detection.
Profiles of the total 4Pi PSF, hðr ~Þ ¼ hillðr ~Þhdetðr ~Þ; along the x and z axes
are shown in Fig. 2, b and c, and isointensity surfaces are shown in Fig. 2, d
and e (see also the movie in Supplementary Material). As can be seen from
Eqs. 4 and 5, as well as in Fig. 2, the 4Pi PSF is a function with a complex
shape in 3D that cannot be adequately reproduced by a step function or a
Gaussian, as commonlyused for confocalor other simple PSFsin theoretical
models for continuous photobleaching. Accurate reproduction of the PSF
shape in calculations is critical to take full advantage of the high resolution
provided by the 4Pi microscope or other emerging microscopy techniques in
Model for bleaching and diffusion of fluorophores
We assume that individual fluorophores diffuse and are subject to pho-
tobleaching,in whichcase aparticle disappearsfromthe volumeV. Foreach
particle, its trajectory r ~iðtÞ is a continuous function of t. Using a unified
objectives coming along the z axis interfere constructively at the sample. Calculated profiles of the PSF along the x axis (b) and z axis (c) are shown for the case
of the 4Pi (circles) and confocal (squares) microscopes. Experimental PSF profiles are shown as solid lines. The 4Pi PSF is substantially narrower in the
z-direction compared to the PSF of a confocal microscope (c), but has a complicated form (see Eq. 5). A 3D image of the 4Pi PSF is shown in panels d and e.
Isocontours of the PSF are drawn at three values: 1/20 (red), 1/2,000 (blue), and 1/20,000 (white) of the central peak height. The PSFs are symmetric in regard
to rotation around the z axis; ‘‘ripples’’ present in panel e are due to the rectangular grid used to visualize the PSF. Objectives with half-aperture angle of 68.5?
and n ¼ 1.46 were assumed. For the confocal PSF, the illumination wavelength is 488 nm, detection wavelength is 627 nm; for the 4Pi PSF two-photon
excitation is used, with light of 905 nm wavelength; the detection wavelength is 627 nm. The PSF width in the x-direction is the same for 4Pi and confocal
microscopes; the different widths observed in panel b are solely due to the different wavelengths. The 3D PSF images were created with VMD (64).
Point spread function. (a) Construction of the PSF of a 4Pi microscope is shown schematically. Light wavefronts from the top and bottom
4008Arkhipov et al.
Biophysical Journal 93(11) 4006–4017
description for all N particles, their positions at given time t are distributed in
space according to the probability distribution function pðr ~;tÞ: In the fol-
lowing, we will assume that the interactions between fluorophores can be
neglected and no external force is applied to the fluorophores, except the
interactions between the fluorophores and the medium in which they diffuse.
In this case, the distribution function, pðr ~;tÞ; describing the probability to
find the fluorophore at position r ~at time t, obeys the diffusion equation with
an additional term accounting for the bleaching
¼ D=2pðr ~;tÞ ? khillðr ~Þpðr ~;tÞ:
Here D and k are the diffusion coefficient and bleaching constant,
respectively; hillðr ~Þ is the illumination PSF introduced in Eq. 5. Interactions
between the fluorophores and the medium are accounted for by the diffusion
coefficient D. In general, D can be a function of r ~: We will assume that D is
constant in the whole region V, because usually one is interested not in
spatial distribution of diffusion characteristics in a given specimen, but
rather in an average characteristic of diffusion in the studied type of bio-
The initial distribution of particles pðr ~;t ¼ 0Þ is known (below we set
pðr ~;t ¼ 0Þ ¼ const:Þ; and we assume that reflective boundary conditions are
enforced at the boundary @V of the volume V. Also, pðr ~;tÞ is a one-particle
probability distribution function; i.e., it holds
For t . 0 the value of this integral decreases due to photobleaching. It is
important to note that pðr ~;tÞ is the same for all fluorophores.
The average fluorescence signal recorded from the sample is usually
pðr ~;t ¼ 0Þdr ~¼ 1:
Vdr ~hðr ~Þpðr ~;tÞ
Vdr ~hðr ~Þpðr ~;0Þ;
hðr ~Þ ¼ hillðr ~Þhdetðr ~Þ
is the total PSF, and hillðr ~Þ; hdetðr ~Þ are given by Eqs. 5 and 6. Obviously, it
holds ÆOðt ¼ 0Þænorm¼ 1: The function ÆOðtÞænormis obtained from the
experiments, and also can be computed once Eq. 7 is solved numerically.
Varying D and k in Eq. 7, one can fit the calculated ÆOðtÞænormto the ob-
served signal and, thus, find the diffusion coefficient.
The numerical solution of the diffusion-bleaching equation (Eq. 7) was
coded in C11. The solution is based on the method of finite differences
(48,49) on a uniform 3Dgrid (or effectively in a two-dimensional (2D) space
when symmetry allows one to reduce the 3D problem to a 2D one, as is true
in this case). The volume V is chosen as either a rectangular parallelepiped
or as a cylinder. The second-order expansion of the Laplacian is used and
the time derivative is represented by the first-order finite difference with
time step Dt. The diffusion-reaction Eq. 7 is propagated forward in time,
assuming reflective boundaryconditionsat the surface@V, incorporatedinto
the finite difference scheme with a regular lattice of step Dx. Typical values
for Dt and Dx are discussed below.
Computation of the PSFs hillðr ~Þ and hdetðr ~Þ can be demanding due to their
nonanalytical form and, therefore, we precompute these functions for a
given set of illumination and detection angles once, saving the PSFs and
reusing them when necessary. A Mathematica (Wolfram Research, Cham-
paign, IL) script was developed for this purpose. Before passing the
illumination PSF hillðr ~Þ to the solver of the diffusion-bleaching Eq. 7, it is
normalized so that the integral of hillðr ~Þ over the volume V is equal to one.
Similarly, the total PSF hðr ~Þ is normalized when used to calculate the
fluorescence signal (Eq. 9). The constant k in Eq. 7 incorporates then the
contributions from multiple factors, such as the light intensity, bleaching
efficiency for a single fluorophore, etc., and we report values of k in arbitrary
units (arb. u.). However, with the convention that the normalized PSF is
measured in mm?3, the ‘‘arbitrary units’’ for k are, in fact, mm3/s.
The discretized form of Eq. 7 reads
pðt1DtÞ ¼ ð11DtAÞpðtÞ;
where Dt is the time step, p is the vector representing the binned distribu-
tion pðr ~;tÞ; 1 the identity matrix, and matrix A represents the operator
D=2? khillðr ~Þ; including the boundary conditions. The integral IðtÞ ¼
IðtÞ ¼ h ? pðtÞ:
Equations 11 and 12 are applied iteratively to calculate the observable
The essential part of the 4Pi microscope PSF is usually spread over a
domain ,0.5–1 mm wide (see Fig. 2). Accordingly, to resolve this domain
we use a spatial discretization of Dx ¼ 20–30 nm. The region V described
in the calculations is usually of 20–30 mm size in each direction. For each
calculation, we check if the increase in considered volume changes the
function ÆOðt ¼ 0Þænorm; the result is considered to be the same if the average
deviation in this function is below 1%. For the experiments performed
(uniform diffusion of GFP in glycerin, D ; 1 mm2/s), a region of size
is usually large enough, where tM is the measurement time.
Experiments generally monitor diffusion on a timescale of tM# 100 s, i.e.,
in this case the computational diffusion domain should have a 20-mm width.
The maximum time step one can use in the numerical solution is inversely
proportional to the diffusion constant D; for the considered experiments
(D ; 1 mm2/s), the time step we used, accordingly, was Dt ¼ 10?4s.
A typical calculation covering 10 s of measurement time required 8 min
of computing time on a single core of an Intel Core 2 Quad CPU running at
2.66 GHz, using the Intel C/C11 compilers version 9.0. The computing
time can be reduced employing parallel processors that require, however,
professional support for their installation and operation. Fortunately, another
less demanding and much cheaper option is available through the most
recent generation of graphical processing units (GPUs) to be found soon in
commodity computers. These GPUs offer about a hundred cores on which
the needed computations can be carried out as fast as on a computer cluster
with as many processors offering nearly teraflops of computer power; a key
new feature of the new GPUs is the availability of general purpose com-
pilers. Indeed, such GPUs were already successfully employed for scientific
applications (50–52) with speed-ups of a factor 10–100 on a GPU in com-
parison with a CPU for various scientific computing tasks (52).
We adapted our code to work on GPUs as well as on a CPU. This was
possible in a rather straightforward manner since, according to Eqs. 11 and
12, the numerical solution of the diffusion-bleaching equation involves an
iterative matrix product, an operation commonly arising in the processing of
graphics, and, thus, particularly suitable for parallelization on GPUs. This
version of the GPU implementation of our code, on the NVIDIA GeForce
8800GTX GPU (Santa Clara, CA), performed 10 times faster than the CPU-
based version, carrying out the calculation mentioned above within 45 s
rather than 8 min on a CPU. The developed code is available from the
authors upon request.
Vdr ~hðr ~Þpðr ~;tÞ is in discretized form given by the scalar product
The GFP (S65T) was expressed in Escherichia coli and purified as described
previously (53). The CFM measurements were performed in a Leica TCS
4Pi microscope (Leica Microsystems) of type A equipped with a pair of
3100/1.35 N.A. glycerol objectives at 23 6 0.5?C. Samples were prepared
by diluting GFP at concentrations from 5 to 20 mM in 87% glycerol buffered
with phosphate buffered saline (PBS) and placed between two quartz
CFM Using 4Pi Microscopy4009
Biophysical Journal 93(11) 4006–4017
coverslips that were sealed with silicone. The embedding glycerol/PBS
mixture was also used as the immersion medium to ensure continuity in the
refractive index between the two objectives. The lower coverslip was
equipped with a mirror at the periphery, which allowed the adjustment of the
objectives’ correction collars for exact refractive index alignment.
Fluorescent beads of subresolution size with red-shifted emission wave-
length compared to GFP (TransFluorSpheres, 0.1 mm; Molecular Probes,
Eugene, OR) immobilizedat the upper coverslipwere used to align the focus
of the two objectives and the interference phase of the microscope. The
adjustment procedure was carried out for each individual sample. The beads
andtheGFPfluorescencewere excitedin two-photon modebya Ti:Sapphire
laser (MaiTai, Spectra Physics, Mountain View, CA) or in one-photon mode
signal detection was performed after passing a filter cube (SP700, BS560,
BP500–550, or BP607–683) by photon counting avalanche photodiodes
(Perkin Elmer, Foster City, CA). The detection pinhole was set to 0.74 Airy
To record the fluorescence with a nonscanning parked beam the
microscope was further equipped with an instrumental upgrade developed
for classical FCS applications (VistaFCS, ISS, Champaign, IL). In CFM
experiments the GFP was excited at 905 or 860 nm and the fluorescence was
recorded for 40–60 s with a sampling rate of 1–5 kHz. In FCS experiments
the wavelength was set to 860 nm and the signal was autocorrelated online
for 200 s at a sampling rate of 5 kHz.
As shown in Fig. 2, the illuminated volume of the type A 4Pi
microscope is much smaller than that of a conventional
microscope. Using such a small volume leads to a possibly
small number of fluorophores under the PSF, with the
fluorophoresbeing constantly bleachedinCFM experiments.
Indeed, for a GFP concentration of 10 mM that we normally
used in the experiments with uniform GFP solution, the
illuminated volume contains only ;200 fluorophores. In this
situation the continuum description of the fluorophore dif-
fusion, bleaching, and fluorescence may not be adequate, and
one may want to consider a stochastic description tracing
individual particles. Also, the signal/noise ratio is rather poor
for data obtained under such conditions. Therefore, we use
below descriptions of individual particles in CFM experi-
ments to derive the relation to a continuum representation.
Our results suggest that although noise due to individual
particles is indeed an important issue, the data from the 4Pi
CFM measurements can be safely accounted for through
a continuum description, and valid diffusion constants
Fluorescence signal calculated and recorded
The total PSF hðr ~Þ ¼ hillðr ~Þhdetðr ~Þ in Eqs. 9 and 10 is com-
posed of the illumination PSF hillðr ~Þ and the detection PSF
hdetðr ~Þ. The number of photons dNphotemitted by the photo-
excited particles during an infinitesimal time interval dt is
where A is an unknown constant. By Nphot(t0) we denote the
number of photons emitted by the whole system from time
dr ~hðr ~Þ +
dðr ~? r ~iðtÞÞdt;
t ¼ 0 to t ¼ t0. The experimental observable ÆOðtÞænorm(c.f.
Eq. 9) is proportional to the number of photons emitted by
the system during the device operational time window Tw.
Assuming that Twis small enough so that the rate of photon
emission does not change during Tw, one obtains the ex-
pression for the number of photons emitted from time t to
time t 1 Tw, i.e., for TwdNphot(t)/dt. Therefore, the signal
recorded in the experiment is directly proportional to the
function O(t), defined by
The ensemble average of a signal, ÆOðtÞæ; is an average
over separate experiments, with all conditions being the
same, denoted by k ¼ 1;2;...;M; with M/N: This signal
is given by
iðtÞ denotes the position of fluorophore i at time t
during the trial k. Assuming that all fluorophores i are the
same and that they do not interact, i.e., are described by the
same distribution function pðr ~;tÞ; one can write
and, using Eq. 15, one obtains
Normalizing ÆOðtÞæ to unity at t ¼ 0 results in an expres-
sion for ÆOðtÞænormgiven by Eq. 9; this normalized function
has been used by us to compare experimental and calculated
Free diffusion of GFP in an isotropic 3D solution has
been assessed using CFM (10 mM of GFP in 87% glycerol
solution in water). The experimental curves from single
measurement runs are well reproduced by the computed
ÆOðtÞænorm(Fig. 3 a) and an average over many independent
measurements is matched well, too (Fig. 3 b). The fittings
both of single measurements and of an average over multiple
measurements provide values of the diffusion coefficient D
for GFP in a range of 1.0–1.5 mm2/s (previous measurements
using different techniques suggested a value in the range
0.5–0.9 mm2/s for 90% glycerol and 2.0–3.0 mm2/s for 80%
glycerol (54,55)). The data from single measurements (Fig. 3
a) are noisy due to the small number of fluorophores within
the span of the PSF, as estimated above. Combining data
from multiple measurements (Fig. 3 b) helps one to reduce
the noise and makes it more adequate to use expression (9)
for the fluorescence signal, derived under the assumption of
an infinite number of trials.
dr ~hðr ~Þ +
dðr ~? r ~iðtÞÞ:
dr ~hðr ~Þ +
dðr ~? r ~ðkÞ
dr ~hðr ~Þ +
dðr ~? r ~ðkÞ
pðr ~;tÞ ¼
dðr ~? r ~ðkÞ
ÆOðtÞæ ¼ NA
dr ~hðr ~Þpðr ~;tÞ:
4010Arkhipov et al.
Biophysical Journal 93(11) 4006–4017
Fig. 3, c and d, illustrates the sensitivity of the fluores-
cence signal in CFM measurements to variation in the
diffusion constant D or bleaching constant k. Although
variation between, for example, D ¼ 1.0 mm2/s and D ¼
1.5 mm2/s leads to a significant difference in ÆOðtÞænorm; the
noise in the experimental data makes it difficult to pinpoint
a precise value for D. The fitting to experimental data
minimizes the least square deviation between the calculated
and measured curves. This procedure provides a single value
of D, however, one should keep in mind that ;60.2 mm2/s
variation in D would still reproduce the same measurement
quite well, due to the noise level. Averaging over multiple
measurements reduces the noise, fitting providing again a
single value for D, but making it difficult in this case to
estimate the error in D.
A comparison of the CFM curves obtained using 4Pi and
confocal PSFs is presented in Fig. 4. To our knowledge, 4Pi
microscopy has not been used for FM before, whereas
confocal microscopy is commonly employed in various
bioimaging applications, including FM, and has been
previously demonstrated to be a reliable tool for CFM in
living cells, when one-photon excitation is used (see, e.g.,
Wachsmuth et al. (13)). As mentioned above, the 4Pi PSF is
narrower in the z-direction than the confocal PSF (see Fig. 2),
which might be beneficial for many applications. As Fig. 4
demonstrates, the CFM curves for both PSF types show
similar trends, suggesting that the same analysis tools can be
used. However, a significant difference is that the confocal
PSF can be reliably approximated by a Gaussian, whereas
the 4Pi PSF has to be calculated numerically.
The CFM measurements with one-photon and two-photon
excitation for confocal and 4Pi illumination are shown in
Fig. 4 a. For the curves shown in this panel, the laser power
was scaled when changing from confocal to 4Pi PSF, so that
the count rate was approximately the same. In our analysis,
the illumination PSFs hillðr ~Þ are normalized (see ‘‘Numerical
Procedures’’); therefore, employing the same bleaching
constant k for different PSFs (see Eq. 7) corresponds to the
case when the laser power is scaled for each PSF to result in
the same count rate. Thus, the values of k used to fit the
measurements in Fig. 4 a are the same for the two different
PSFs. For the one-photon confocal PSF (13), as well as for
the one-photon 4Pi and two-photon confocal and 4Pi PSFs,
our calculations match experimental data well. The average
signal recorded in these measurements is approximately the
same for the 4Pi and confocal PSFs, despite their difference
in shape, suggesting that the amount of power pumped into
the sample is a primary factor determining the behavior of
the CFM curves.
It should be noted that the calculated curves in Fig. 4 a
were fitted to the measured ones only for the case of the 4Pi
PSF. Once a good fit was obtained, the calculation was
repeated with the confocal PSF, but with all other parameters
kept unchanged; still, a good match with observation was
obtained. An example of what happens when the laser power
is not changed, once the PSF is switched, is shown in Fig. 4 b.
Using the 4Pi PSF results in stronger bleaching, because
the light is more concentrated as a smaller volume is illu-
minated. Numerically, this corresponds to using different
values for k for the two PSF types, since the PSFs are
normalized. The main conclusion from comparing the CFM
curves obtained with the 4Pi or confocal PSFs is that it does
not matter which PSF type is used in a CFM experiment
designed to determine D; in either case the numerical
signal as a function of time. (a) A single 4Pi CFM
measurement (black) is compared with the calcu-
lated (fitted) signal (red). Best fitting is achieved
10 independent measurements are averaged (black)
and fitted to a calculated curve (red, D ¼ 1.0 mm2/s,
k ¼ 12.0 arb. u.). Curves in panels a and b are from
different series of measurements, performed at dif-
ferent laser intensities. The calculated signal with
panel c, from top to bottom, D ¼ 3.0, 1.5, 1.0, 0.5,
and 0.1 mm2/s. Curves for D ¼ 0.5 mm2/s, k ¼ 3.75
arb. u. (red) and D ¼ 2.0 mm2/s, k ¼ 15.0 arb. u.
(green) converge to the same level as the one with
in panel d, from top to bottom, k ¼ 3.5, 7.5, 10.0,
15.0, and 50.0 arb. u. The experimental curves were
obtained with 10 mM of GFP in 87% glycerol
solution in water; n ¼ 1.46, a ¼ 68.5?; l ¼ 905 nm
for illumination and l ¼ 500–550 nm for detection.
CFM Using 4Pi Microscopy4011
Biophysical Journal 93(11) 4006–4017
algorithm allows for a reliable estimation of D. The 4Pi PSF
should be chosen over the confocal one when the probed
volume has to be smaller than the spread of the confocal PSF.
A characteristic feature of both the calculated and
experimental curves in Figs. 3 and 4 is that the curves level
out for t/N. The signal’s value for t/N depends on D
(Fig. 3 c), but also can be the same for different D-values,
depending on k. Can this long-time behavior of the signal be
utilized to simplify the extraction of D from experimental
data? The leveling of the signal to a constant value is due to
the fact that the system assumes a quasistationary state, when
the rate of bleaching equals the rate of efflux of fresh fluo-
rophores from unbleached regions. This corresponds to
@pðr ~;tÞ=@t ? 0 in Eq. 7, and to the convergence of the fluo-
rophore distribution pðr ~;tÞ to the quasistationary solution
pqðr ~Þ; satisfying
0 ¼ =2pqðr ~Þ ?k
Dhillðr ~Þpqðr ~Þ:
Thus, the quasistationary solution pqðr ~Þ is determined by a
single parameter, k/D; since the signal is defined by pðr ~;tÞ
through Eq. 17, the quasistationary value of the signal at long
times isalsodetermined by k/D. As shownin Fig. 3 c, scaling
D and k simultaneously by the same value (which keeps k/D
constant) does not change the long-time level of the signal.
Moreover, the whole signal curve shrinks or expands as if
the only change was a scaling of time. The reason for this
behavior is that the diffusion-bleaching equation (Eq. 7) is a
first-order differential equation in time, and D and k enter the
equation linearly, so that if D and k are scaled by some
constant a, i.e., Dnew¼ aD, knew¼ ak, one can replace the
resulting diffusion-bleaching equation by one with t replaced
by tnew¼ at.
Fig. 5 compares the PSF, the fluorophore concentration,
and the fluorescence signal along the z axis during the CFM
experiment in Fig. 3 a. Fig. 5 b shows how the profile of the
fluorophore concentration Cðr ~;tÞ along the z axis changes
with time (Cðr ~;tÞ is defined through Cðr ~;tÞ ¼ Npðr ~;tÞ). One
recognizes that Cðr ~;tÞ is decreasing, following to some
extent the shape of the PSF, but at t ¼ 0.2 s it reaches a
quasistationarystate, after which moment the shape of Cðr ~;tÞ
remains largely unchanged, and the overall concentration
depletes slowly and uniformly. According to Eq. 17, one can
define the local emission probability from the volume dr ~at
around position r ~ as hðr ~Þpðr ~;tÞ: This quantity is plotted in
Fig. 5 c; it also reaches a quasistationary state at the same
pace as Cðr ~;tÞ does. As a result, time evolution of the
fluorescence signal follows the time evolution of Cðr ~;tÞ
Calculating the long-time signal level and matching it to
the experimental value can determine the value of k/D, but
the value of D remains unknown. Indeed, it is impossible to
obtain D from pqðr ~Þ: D would be known if one knew the
values of k corresponding to the laser power used in an
experiment, but to determine k one needs to perform separate
experiments. In principle, k can be determined from the
signal at t ? 0. Since pðr ~;t ¼ 0Þ ¼ const; Eq. 7 reduces to
? ?khillðr ~Þpðr ~;tÞ:
This suggests an initial exponential decay. One can
estimate the value of k by comparing the decay of the signal
suggested by Eq. 19 with that measured in experiments. To
do so, we used a least-squares fit of the numerically cal-
culated signal (with pðr ~;tÞ described by Eq. 19) to the
measured signal over the first 0.01 s of the recording.
photon excitation using 4Pi and confocal microscopy. Measurements with
a 4Pi PSF are shown in red, and with the confocal one in green; calculated
curves are in black, marked with circles for the 4Pi and with squares for the
confocal PSF. In panel a, results for two-photon illumination are the curves
at the top, and for one-photon those at the bottom. The illumination laser
power was scaled to give approximately the same count rate for measure-
ments with 4Pi and confocal PSFs. (b) Only two-photon excitation is used,
with the same laser power for 4Pi and confocal measurements, resulting in
significant difference between the recorded signal decays. The experimental
curves are averages over 10 measurements each. All measurements were
done with the same parameters as in Fig. 3, but with l ¼ 860 nm for
illumination in the two-photon case, l ¼ 488 nm for illumination in the one-
photon case, and l ¼ 600–650 nm for detection. Calculations were done
with D ¼ 0.5 mm2/s, and with k ¼ 1.7 arb. u. for the two-photon case in
panel a, k ¼ 9.0 arb. u. for the one-photon case, k ¼ 2.5 arb. u. for the 4Pi
PSF and k ¼ 1.2 arb. u. for the confocal PSF in panel b.
Fluorescence signal recorded with one-photon and two-
4012Arkhipov et al.
Biophysical Journal 93(11) 4006–4017
Unfortunately, due to the high noise level in the experimental
signal, the values of k obtained from fitting the signal around
t ¼ 0 are highly unreliable.
Even though the quasistationary levels of the signal can
be extracted with good fidelity, providing a good estimate for
k/D, using Eq. 18 to match k/D involves considerable com-
putational effort that might be best invested into computing
the numerical solution of the time-dependent Eq. 7 up to the
point when pðr ~;tÞ assumes the quasistationary state charac-
terized by pqðr ~Þ: Such computation requires about the same
effort as solving Eq. 18, but furnishes the initial decay of
pðr ~;tÞ and, hence, of ÆOðtÞæ that can be matched then to the
observation to yield both k and D. Therefore, the most
practical route for determining D is to compute the signal as a
function of time for an arbitrary starting value of D0, to vary
k, calculating the signal curve for each new value of k, and to
match experimental and calculated ÆOðtÞæ in the quasista-
tionary regime. We judged the quality of the match by the
mean square deviation between the calculation and mea-
surement, using the t . 0.5 s part of the signal for this
purpose. Then one scales the time to match the calculated
signal to the experimental signal, using now the least-squares
matching for all values of time t. The scaling factor a is the
same for D as well as k, and one arrives at the best D- and
k-values, D ¼ aD0, k ¼ ak0. These values can be checked
by a direct calculation of ÆOðtÞæ and comparison with the
The signal for CFM consists of an initial (t ? 0) regime
dominated by k, an intermediate regime determined by both
parameters, D and k, and a long-time regime dependent on
k/D. The most interesting regime for determining D is the
intermediateone, whichbears thesignature ofbothbleaching
and diffusion. The CFM technique resolves the intermediate
regime and, hence, permits measurement of D.
in the 4Pi microscope by working in the nonbleaching limit.
Without bleaching (k ¼ 0 in Eq. 7), the expression for the
correlation function (Eq. 26) is simplified, as the operatorˆLr
describes now free diffusion. The conditional probability
(Green’s function) for free diffusion is a Gaussian, namely,
˜ pðr ~;t1jr ~9;t0Þ ¼ ½4pDðt1? t0Þ??3=2exp ?ðr ~? r ~9Þ2
Using this expression one can rewrite Eq. 26, normalizing
the correlation function by ÆOðtÞæ2(58),
GðTÞ ¼ÆOðtÞOðt1TÞæ ? ÆOðtÞæ2
Vdr ~dr ~9hðr ~Þhðr ~9Þexp ?ðr ~?r ~9Þ2
Vdr ~hðr ~ÞÞ2
where ÆCæ is the average concentration of the fluorophores,
and we chose t0¼ t, t1¼ t 1 T.
We measured the correlation function G(T) in the non-
bleaching limit, and, because of the simplification brought
about by setting k to zero, we were able to also calculate G(T)
cence along the z axis. (a) The PSF profile is shown. (b) The local fluoro-
phore concentration (normalized to unity at t ¼ 0) is shown for successive
time moments, and the local emission probability hðr ~Þpðr ~;tÞ (normalized) is
shownin panelc. Thetimes for whichcurvesare drawninpanelsb andc are,
from top to bottom, 0.0001, 0.0002, 0.0005, 0.001, 0.002, 0.005, 0.01, 0.02,
0.05, 0.1, 0.2, 0.3, 0.4, 0.5, and 1.0 s. All parameters are the same as in the
calculation shown in Fig. 3 a.
Time evolution of fluorophore distribution and local fluores-
CFM Using 4Pi Microscopy 4013
Biophysical Journal 93(11) 4006–4017
numerically. The experimental and calculated correlation
functions are compared in Fig. 6. Two features of the cor-
time. One can express GðT ¼ 0Þ ¼ 1=ðVeffÆCæÞ; where Veff
is the effective focal volume (58) defined through Veff¼
of particles in the effective focal volume is ;100 (which
would become 200 for the 10-mM concentration used in the
measurements oftheCFM signalabove).Fig.6demonstrates
a good agreement between the calculated curves and the
experimental one near T ¼ 0. The decay of the correlation
function is also reproduced with acceptable diffusion con-
one that matched the CFM signal (1.0–1.5 mm2/s; see Fig. 3),
but includes the latter range.
The FCS measurements can be compared also for the 4Pi
and confocal PSFs, as shown in Fig. 7. Numerically calcu-
lated correlation functions reproduce the experimental ones
well. As was the case for the CFM signal (Fig. 4), parameters
for the numerical solution were varied only to match the
curve corresponding to the 4Pi case; once a good match was
obtained, switching the numerical solver to employ the con-
focal PSF instead of the 4Pi one resulted in a good match for
the confocal case, too. The laser power was scaled in these
experiments to have the same count rate for the 4Pi and
confocal PSFs (see also Fig. 4).
The effective focal volume differs substantially for 4Pi
and confocal PSFs, being only half the size for the 4Pi PSF
Vdr ~hðr ~Þ?2=R
Vdr ~½hðr ~Þ?2: Thus, G(T ¼ 0) is an inverse
number of particles in the effective focal volume. According
(see also Fig. 2), as demonstrated in Fig. 7 by the values of
G(T ¼ 0). Indeed, according to these values, the number of
particles in the effective focal volume for the confocal PSF
is ?550, and for the 4Pi PSF is ?300, with the same
fluorophore concentration of 15 mM (in case of Fig. 6, the
number of particles in the effective focal volume for the 4Pi
PSF was ;100 at a 5-mM concentration). When the two
curves in Fig. 7 are scaled to start at the same value at T ¼ 0,
it appears that the 4Pi correlation function decays faster. This
difference is prominent for intermediate times 0.001–0.1 s.
The faster decay in the 4Pi case is due to the smaller effective
focal volume. The smaller the volume, the shorter is the time
spent by a fluorophore in that volume, and, thus, the cor-
relations in the signal are limited to shorter average times that
the fluorophores spend in the focal spot.
Although the 4Pi correlation function decays faster on
a timescale of 0.001–0.1 s, for longer, and especially for
shorter times, the confocal and 4Pi correlation functions are
quite close (when scaled to coincide at T ¼ 0). Due to the fact
that the diffusion studied here is rather slow, and, accord-
ingly, the acquisition rate is slow too, fast photophysical
phenomena such as transition into a nonradiating triplet state
(which can happen even though the laser power is very low)
are not resolved, and the correlation functions do not exhibit
a difference in their short-time behavior. In principle, such a
difference could be used to distinguish between photo-
physics and diffusion because the ratio of the correlation
functions for the patterned 4Pi PSF and the confocal PSF
should be independent of the spectroscopic properties of the
particles (34). In the case studied here, the short-time
behavior of the two correlation functions is similar; however,
the numerical calculations are able to distinguish between the
two cases, in regard to both decay rate and absolute values of
the correlation function.
perimental data (circles with error bars) are the average of correlation func-
tions from five separate measurements. Error bars represent the averaging
root mean-square deviation. Setup parameters for these measurements were
the same as for those presented in Fig. 3, besides the GFP concentration,
which was 5 mM for these measurements, and the excitation wavelength,
which was switched to 860 nm. Calculated correlation functions are shown
as continuous lines (D ¼ 0.5 mm2/s, 1.0 mm2/s, and 2.0 mm2/s, from top to
Correlation function for the case k ¼ 0 (no bleaching). Ex-
(two-photon in both cases), measured on the same sample. Numerical fits for
both cases are shown as solid black lines, corresponding to D ¼ 0.5 mm2/s
(k ¼ 0). Experimental data are the average of correlation functions from 10
separate measurements. The GFP concentration is 15 mM; all other param-
eters are the same as in Fig. 6.
Correlation function for the 4Pi (d) and confocal (n) PSF
4014Arkhipov et al.
Biophysical Journal 93(11) 4006–4017
CONCLUSIONS AND PERSPECTIVES
The biological cell features a complex and sophisticated
nanostructure that is dynamic and subject to frequent and
rapid reorganization. An overwhelming body of data sup-
ports the contention that it is this dynamics on the nanometer
scale that plays a crucial role in cell function. For these
reasons one needs techniques that capture dynamic processes
in living cells at the highest possible spatial resolution.
of dynamic processes in single living cells. However, spatial
resolution remains a central issue. One exemplary case is the
lateral mobility of proteins in the plasma membrane. Already
some time ago (60) it has been observed that the lateral
mobility of membrane proteins as measured by FM depends
on the size of the illuminated area, pointing to an intricate
It has been suggested (61) that the plasma membrane features
nanoscopic lipid aggregates (‘‘rafts’’) that serve as a basis for
organizing membrane proteins in clusters and, thus, have far
reaching functional implications, particularly in signal trans-
duction. However, so far the spatial resolution of FM and
FCS has been insufficient to unambiguously verify the exis-
tence of lipid rafts and their existence remains debated (62).
Similar arguments for the need of improved resolution of FM
and FCS methods hold for most other cellular processes.
In this study the spatial resolution of CFM and FCS were
increased by combining the techniques with 4Pi microscopy.
The 4Pi microscope features a point spread function (Fig. 2)
with a central peak of ;$100 nm width in the axial direction
and $220 nm width in the focal plane. In addition, the point
spread function has smaller secondary maxima spaced on the
optical axis at a distance of about half a wavelength from the
main maximum. The essential point here is (c.f. Fig. 5 c) that
.90% of the signal measured in both CFM and FCS
experiments derives from the main peak. Thus, the obser-
vation volume is virtually coincident with that of the main
maximum of the point spread function.
This does not imply that 4Pi CFM and FCS measurements
have an axial resolution of $100 nm. In fact, the environ-
ment of the observation volume affects diffusion in the
observation volume and is an integral part of the system. In
CFM measurements the depletion of fluorophores reaches
further and further into the environment with time (Fig. 5 b).
Nevertheless, the spatial resolution in CFM and FCS mea-
surements scales with the size of the observation volume and,
therefore, can be said to be improved by 4Pi microscopy by a
factor of 5–7 as compared to normal confocal microscopy.
A major problem in the adaptation of CFM and FCS to
4Pi microscopy was to account for the complicated point
spread function of the 4Pi microscope. For that purpose a
theoretical and computational framework was developed for
data analysis. The numerical approach developed and de-
scribed here is characterized by flexibility; i.e., the compu-
tation scheme can be adopted to point spread functions of
arbitrary shape. This is particularly relevant with respect to
the fast development of high-resolution light microscopy
culminating recently in a resolution of ?30 nm (63). The
numerical approach suggested here permits one also to take
other conditions into account that are relevant in cell bio-
logical studies. For example, in addition to the pure and
unrestricted diffusion of one molecular species one can take
the association of the diffusing species with immobile binding
sites into account. Furthermore, one can consider situations in
which the diffusion space is limited. The approach can also be
extended to two-color microscopy, a powerful means to
quantify reversible bimolecular association reactions.
Altogether this study provides the tools for installing FM
and FCS on light microscopes that are able to extend or even
completely overcome classical resolution limits. Accord-
ingly, the theoretical and computational framework will
contribute to the further unraveling of the cellular nano-
machinery that lies at the heart of life and thus holds the key
for future progress in biomedicine.
APPENDIX A: CORRELATION FUNCTION IN A
ÆOðt1ÞOðt0Þæ ? ÆOðt1ÞæÆOðt0Þæ; in the general case of t16¼ t0 and non-
negligible bleaching. The average signal ÆOðt1Þæ or ÆOðt0Þæ is given by
Eq. 17. The correlation term, ÆOðt1ÞOðt0Þæ; is expressed as follows
dr ~dr ~9hðr ~Þhðr ~9Þ +
seek toderivethe expression for thecorrelationfunction
This can be rewritten as
ÆOðt1ÞOðt0Þæ ¼ A2
dr ~hðr ~Þ +
dðr ~? r ~iðt1ÞÞ
dr ~9hðr ~9Þ +
dðr ~9 ? r ~lðt0ÞÞ
dðr ~? r ~iðt1ÞÞdðr ~9 ? r ~lðt0ÞÞ
dr ~dr ~9hðr ~Þhðr ~9Þ +
dðr ~? r ~ðkÞ
iðt1ÞÞdðr ~9 ? r ~ðkÞ
dr ~dr ~9hðr ~Þhðr ~9Þ
dðr ~? r ~ðkÞ
iðt1ÞÞdðr ~9 ? r ~ðkÞ
CFM Using 4Pi Microscopy 4015
Biophysical Journal 93(11) 4006–4017
The product of delta-functions arising here corresponds to a probability that
at time t1particle i is found at r ~; and at time t0particle l is found at r ~9: In the
case of i 6¼ l, these two events are independent (particles do not interact), and
the sum over k with M/N becomes the product of two probability
distributions, pðr ~;t1Þpðr ~9;t0Þ; the same probability distribution p is used
because the particles behave identically. In the case of i ¼ j, the probability
of the two events is conditional, so that in the limit M/N the sum becomes
˜ pðr ~;t1jr ~9;t0Þpðr ~9;t0Þ; where ˜ pðr ~;t1jr ~9;t0Þ is the probability for a single par-
ticle to move from r ~9 at time t0to r ~at time t1(Green’s function). Inserting the
relationpðr ~;t1Þ ¼R
Vdr ~9 ˜ pðr ~;t1jr ~9;t0Þpðr ~9;t0ÞintoEq.7, andusingtheinitial
condition ˜ pðr ~;t1¼ t0jr ~9;t0Þ ¼ dðr ~? r ~9Þ; one finds that ˜ p can be expressed
˜ pðr ~;t1jr ~9;t0Þ ¼ e
r? khillðr ~Þ is the diffusion-bleaching operator in Eq. 7, in
this case acting on the vector r ~. Using these results in Eq. 23, one obtains
ÆOðt1ÞOðt0Þæ ¼ A2Z
pðr ~;t1Þpðr ~9;t0Þ ? +
ˆLrðt1?t0Þdðr ~? r ~9Þ;
dr ~dr ~9hðr ~Þhðr ~9Þ
pðr ~;t1Þpðr ~9;t0Þ
ˆLrðt1?t0Þdðr ~? r ~9Þpðr ~9;t0Þ
The elements under the sums over i and l depend neither on i nor on l, so the
sums reduce to the multiplication of corresponding terms by N or N2.
Combining this with the expression for ÆOðtÞæ (Eq. 17), one obtains the final
result for the correlation function
ÆOðt1ÞOðt0Þæ ? ÆOðt1ÞæÆOðt0Þæ
The correlation function in Eq. 26 could be used for the analysis of diffusion,
but we found that in its general form, t16¼ t0, the utilization of the correlation
function is not practical, due to the difficulties with its computational
implementation and high noise level in the measurements.
dr ~hðr ~Þe
ˆLrðt1?t0Þhðr ~Þpðr ~;t0Þ
dr ~hðr ~Þpðr ~;t1Þ
dr ~9hðr ~9Þpðr ~9;t0Þ
To view all of the supplemental files associated with this
article, visit www.biophysj.org.
The authors are grateful to Nathan A. Baker, Petros Koumoutsakos, Ivo F.
Sbalzarini, and Ingo Lepper for useful suggestions and discussions. We also
thank John Stone and Kirby Vandivort for help with numerical algorithms,
and Peter Freddolino for suggestions on visualizing the PSF.
This work was supported by a grant from the National Institutes of Health
(PHS-5-P41-RR05969). K.S. was supported by the Humboldt Foundation.
The work was further supported by the National Institutes of Health (grant
GMO 71329), the Deutsche Forschungsgemeinschaft (grants PE138/19 and
PE138/21), and the Volkswagenstiftung (grant I/79 248).
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