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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

ABSTRACT

surement of molecular mobility and association reactions in single living cells. CFM and FCS complement each other ideally and

canberealizedusingidenticalequipment.Sofar,thespatialresolutionofCFMandFCSwasrestrictedbytheresolutionofthelight

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

excitation.Inallcasesexperimentaldatacouldbewellfittedbycomputedcurvesforexpecteddiffusioncoefficients,evenwhenthe

signal/noise ratio was small due to the small number of fluorophores involved.

Continuous fluorescence microphotolysis (CFM) and fluorescence correlation spectroscopy (FCS) permit mea-

INTRODUCTION

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

focusedlaserbeamataverylow,nonbleachingintensity.The

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

ever-increasing pace.

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

byphotobleachingandreplenishedwithfreshfluorophoresby

diffusion. Because the laser power is much higher in CFM

thaninthemeasuringphasesofFM,theCFMsignalqualityis

better.Thisistradedinbythenecessitytoextractnotonlythe

diffusioncoefficientbutalsothephotobleachingratefromthe

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-

ertiesobservedinconventionalFM(14–24),CFM(12,25,26,

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

doi: 10.1529/biophysj.107.107805

Submitted February 28, 2007, and accepted for publication July 26, 2007.

Address reprint requests to Klaus Schulten, E-mail: kschulte@ks.uiuc.edu;

or Reiner Peters, E-mail: petersr@uni-muenster.de.

Editor: Petra Schwille.

? 2007 by the Biophysical Society

0006-3495/07/12/4006/12$2.00

4006Biophysical JournalVolume 93December 20074006–4017

Page 2

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

Fig.1.Coherentlight(red)fromalaserissplitintotwobeams,

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

ofthecommonconfocalmicroscope,butaccompaniedbytwo

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-

structiveinterferenceinbothcases.WeusetypeA4Pimicros-

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

numerically.

Asshownbelow,numericalcalculationsreliablyreproduce

the experimental fluorescence signal. We find that in CFM

andFCSwiththe4Pimicroscope,theprominentfeatureofthe

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

experiment,which,however,reduces thetimeresolution.We

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-

teristicsofdiffusionfromthe4PiFMmeasurements:analysis

of the fluorescence signal and fluorescence correlation spec-

troscopy.

METHODS

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

the expression

FIGURE 1

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

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hillðr ~Þ ¼ jE~1ðr ~Þ1E~2ðr ~Þj2nexc;

(1)

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

(2)

E~2ðx;y;zÞ¼ðExðx;y;?zÞ;Eyðx;y;?zÞ;?Ezðx;y;?zÞÞ: (3)

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

Za

I1ðr ~Þ¼

0

sina

Za

with u and v being optical coordinates (u ¼ (2pn sin2a)z/l, v ¼

ð2pnsinaÞ

I0ðr ~Þ¼

0

cos1=2usinuð11cosuÞJ0

Za

cos1=2usinuJ2

vsinu

sina

?

iucosu

sin2a

??

exp

iucosu

sin2a

?

du;

?

du;

cos1=2usin2uJ1

vsinu

?

?

?

?

exp

iucosu

sin2a

?

?

I2ðr ~Þ¼

0

vsinu

sina

exp

?

du;

(4)

ffiffiffiffiffiffiffiffiffiffiffiffiffi

x21y2

p

=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

14ðImI1ðr ~ÞÞ2cos2f?nexc:

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

expression

(5)

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

photobleaching schemes.

(6)

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

FIGURE 2

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

Page 4

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

@pðr ~;tÞ

@t

¼ D=2pðr ~;tÞ ? khillðr ~Þpðr ~;tÞ:

(7)

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-

logical medium.

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

Z

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

written (12,25)

R

where

V

pðr ~;t ¼ 0Þdr ~¼ 1:

(8)

ÆOðtÞænorm¼

Vdr ~hðr ~Þpðr ~;tÞ

R

Vdr ~hðr ~Þpðr ~;0Þ;

(9)

hðr ~Þ ¼ hillðr ~Þhdetðr ~Þ

(10)

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.

Numerical procedures

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;

(11)

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

R

IðtÞ ¼ h ? pðtÞ:

Equations 11 and 12 are applied iteratively to calculate the observable

ÆOðtÞænorm:

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

2

DtM

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

(12)

ffiffiffiffiffiffiffiffi ffi

p

Experimental procedures

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

Page 5

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

bythe488-nmlineofanargonlaserwithbeamexpander3ofthemicroscope;

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

units.

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.

RESULTS

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

extracted.

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

Z

where A is an unknown constant. By Nphot(t0) we denote the

number of photons emitted by the whole system from time

dNphot¼ A

V

dr ~hðr ~Þ +

N

i¼1

dðr ~? r ~iðtÞÞdt;

(13)

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

Z

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

Z

N

A

M

V

k¼1

where rðkÞ

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

1

M+

k¼1

and, using Eq. 15, one obtains

Z

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

fluorescence signals.

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.

OðtÞ ¼dNphot

dt

¼ A

V

dr ~hðr ~Þ +

N

i¼1

dðr ~? r ~iðtÞÞ:

(14)

ÆOðtÞæ ¼

A

M+

?

M

k¼1

V

dr ~hðr ~Þ +

N

i¼1

M

dðr ~? r ~ðkÞ

iðtÞÞ

??

?

M/N

¼ +

i¼1

Z

dr ~hðr ~Þ +

dðr ~? r ~ðkÞ

iðtÞÞ

M/N

;

(15)

pðr ~;tÞ ¼

M

dðr ~? r ~ðkÞ

iÞ

??

M/N

;

(16)

ÆOðtÞæ ¼ NA

V

dr ~hðr ~Þpðr ~;tÞ:

(17)

4010Arkhipov et al.

Biophysical Journal 93(11) 4006–4017

Page 6

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

FIGURE 3

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

withD¼1.5mm2/sandk¼7.5arb.u.(b)Resultsof

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

constantk¼7.5arb.u.andvariedDarepresentedin

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

D¼1.0mm2/s,k¼7.5arb.u.Datafromcalculations

withconstantD¼1.5mm2/sandvariedkareshown

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.

Calculatedandobservedfluorescence

CFM Using 4Pi Microscopy4011

Biophysical Journal 93(11) 4006–4017

Page 7

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:

(18)

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

closely.

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

@pðr ~;tÞ

@t

? ?khillðr ~Þpðr ~;tÞ:

(19)

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.

FIGURE 4

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

Page 8

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

measurement.

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.

Correlation function

Fluorescentcorrelationspectroscopy(56–59)canbeemployed

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

4Dðt1? t0Þ

??

: (20)

Using this expression one can rewrite Eq. 26, normalizing

the correlation function by ÆOðtÞæ2(58),

GðTÞ ¼ÆOðtÞOðt1TÞæ ? ÆOðtÞæ2

ÆOðtÞæ2

R

¼

V

R

Vdr ~dr ~9hðr ~Þhðr ~9Þexp ?ðr ~?r ~9Þ2

ÆCæð4pDTÞ3=2ðR

4DT

hi

Vdr ~hðr ~ÞÞ2

;

(21)

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)

FIGURE 5

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

Page 9

numerically. The experimental and calculated correlation

functions are compared in Fig. 6. Two features of the cor-

relationfunctionrequireattentionwhenonefitsthecalculated

curvetotheexperimentalone,thevalueatT¼0andthedecay

time. One can express GðT ¼ 0Þ ¼ 1=ðVeffÆCæÞ; where Veff

is the effective focal volume (58) defined through Veff¼

½R

toFig.6(forboththeexperimentandcalculation),thenumber

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-

stantsintherange0.5–2.0mm2/s.Thisrangeiswiderthanthe

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.

FIGURE 6

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

bottom).

Correlation function for the case k ¼ 0 (no bleaching). Ex-

FIGURE 7

(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

Page 10

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.

BothFMandFCShavemuchcontributedtotheunraveling

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

organizationoftheplasmamembraneonthenanometerlevel.

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

GENERAL CASE

We

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

? Z

dr ~dr ~9hðr ~Þhðr ~9Þ +

i;l¼1

seek toderivethe expression for thecorrelationfunction

This can be rewritten as

ÆOðt1ÞOðt0Þæ ¼ A2

? Z

?Z

V

Z

dr ~hðr ~Þ +

N

i¼1

dðr ~? r ~iðt1ÞÞ

?

V

dr ~9hðr ~9Þ +

N

l¼1

dðr ~9 ? r ~lðt0ÞÞ

?

? ??

¼ A2

VV

N

dðr ~? r ~iðt1ÞÞdðr ~9 ? r ~lðt0ÞÞ

:

(22)

ÆOðt1ÞOðt0Þæ ¼A2

M

Z

V

Z

Z

V

dr ~dr ~9hðr ~Þhðr ~9Þ +

Z

M

k¼1

+

i;l¼1

?

N

dðr ~? r ~ðkÞ

iðt1ÞÞdðr ~9 ? r ~ðkÞ

lðt0ÞÞ

????

M/N

¼ A2+

N

i;l¼1

VV

dr ~dr ~9hðr ~Þhðr ~9Þ

1

M+

M

k¼1

dðr ~? r ~ðkÞ

iðt1ÞÞdðr ~9 ? r ~ðkÞ

lðt0ÞÞ

?

M/N

:

(23)

CFM Using 4Pi Microscopy 4015

Biophysical Journal 93(11) 4006–4017

Page 11

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

N

pðr ~;t1Þpðr ~9;t0Þ ? +

ˆLrðt1?t0Þdðr ~? r ~9Þ;

(24)

whereˆLr¼ D=2

V

Z

V

dr ~dr ~9hðr ~Þhðr ~9Þ

3 +

i;l¼1

N

N

i¼1

pðr ~;t1Þpðr ~9;t0Þ

?

"

1 +

i¼1

e

ˆLrðt1?t0Þdðr ~? r ~9Þpðr ~9;t0Þ

:

(25)

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

¼ A2N

V

Z

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.

Z

dr ~hðr ~Þe

ˆLrðt1?t0Þhðr ~Þpðr ~;t0Þ

? Z

?

dr ~hðr ~Þpðr ~;t1Þ

?

V

?

V

dr ~9hðr ~9Þpðr ~9;t0Þ

? ??

:

(26)

SUPPLEMENTARY MATERIAL

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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