INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

Fluorescence correlation spectroscopy (FCS) (Magde et al., 1972, 1974; Elson and Magde, 1974) is an elegant and sensitive technique for measuring dynamic processes that manifest themselves in a fluorescent signal on the submicrosecond-to-second time scales. With origins in quasielastic light scattering (Cummins and Swinney, 1970) and a basis in the statistical thermodynamics of fluctuations in solution, FCS has been used to measure diffusion coefficients, chemical kinetics, excited-state molecular dynamics, picomolar concentrations (Eigen and Rigler, 1994), and the dynamics of the interaction of fluorescent molecules in vitro (Koppel et al., 1976; Magde et al., 1972). Recently there has been rapidly increasing use of FCS for biological applications (Hess et al., 2002; Rigler and Elson, 2001; Schwille et al., 1996; Maiti et al., 1997) including living biological systems (Schwille et al., 1999a; Brock et al., 1998, 1999; Hink et al., 2000; Wachsmuth et al., 2000; Cluzel et al., 2000; Politz et al., 1998).

What are the best optical operating conditions for FCS? That question comprises the focus of this study. Many FCS systems allow changes in the type and numerical aperture (NA) of the objective lens, the size of the detector aperture, and the degree of underfilling of the back-aperture of the objective. Previous theoretical work has investigated the signal-to-noise ratio (S/N) for FCS systems (Koppel, 1974; Qian and Elson, 1991), and confocal microscopes (Gu and Gan, 1996; Sheppard et al., 1991; Sandison and Webb, 1994; Gu and Sheppard, 1993), but most previous work has made significant approximations in the treatment of the focal volume (Rigler et al., 1993), particularly at high-NA (Sheppard and Matthews, 1987) and in the treatment of underfilling, except in the case of two-photon excitation (W. Zipfel, personal communication). It is commonly assumed that the observation volume in FCS is a Gaussian in three dimensions. However, it will be shown that this is not sufficiently accurate under many measurement conditions and may lead to inaccurate results. To attempt to improve upon current FCS methodology we use a more sophisticated description of the focal volume (Wolf, 1959; Richards and Wolf, 1959), which treats the polarization of the excitation and is well-suited to high-NA optics. Furthermore, we present theoretical predictions and measurements together. We also attempt to determine the values of the detector aperture and underfilling fraction that yield the highest S/N and minimize artifacts in the FCS autocorrelation function that result from the non-Gaussian nature of the observation volume.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

Although FCS is an excellent technique for extracting a variety of quantitative information from biological systems, the influence of the illumination and collection optics on the measured autocorrelation must be considered to ensure accurate results. This section delineates the relationship between the spatial profile of the focal volume and parameters measured by FCS.

Introduction to FCS

Fluorescence correlation spectroscopy (Magde et al., 1972), which is based on fluctuation analysis of the fluorescence from an observation volume on the order of a femtoliter containing a small number (<1000) of molecules, provides quantitative physical and chemical kinetic information on time scales from 10⁷ to >10² s. FCS has been demonstrated to be quite useful for photophysical characterization of sparse fluorescent molecules (Mertz et al., 1995) and measurement of dynamics of those molecules that give rise to fluctuations in their fluorescence (Schwille et al., 1996; Heikel et al., 2000). Technological innovations, adapted particularly in the Rigler and Eigen groups (Eigen and Rigler, 1994), have resulted in a recent revival of the technique, the fundamentals of which were first developed in the early 1970s in the Webb group. The basis of the technique is the observation of the fluorescence F, produced by dilute fluorescent species (~nM concentrations) that diffuse and react chemically according to

(1)

where the concentration fluctuation C_j of the jth species from the mean (temporal average) C_j is given by C_j(r, t) = C_j(r, t)  C_j(r, t), as a function of time (t), the diffusion coefficient D_j, and the tensor T_jk, which expresses the chemical reaction kinetics and stoichiometry between species.

The concentration fluctuations result in fluorescence fluctuations F(t), which are related to the experimental illumination and collection profiles (now considering a single species) by

(2)

where I_O is the illumination intensity profile, S  I(r_O)/I(r_O = 0) is the normalized illumination profile,  is the collection efficiency profile, r_O is position in object space, and q, , and C are the effective quantum yield, extinction coefficient, and concentration, respectively, of the observed fluorescent species. Clearly based on Eq. 2, the spatial dependence of the product of S and , the normalized observation volume profile

(3)

is crucial in determining the spatial distribution of fluorescence fluctuations that are to be observed. In practice, kinetic information is extracted from the fluorescence fluctuations by autocorrelation (Webb, 1976):

(4)

where F(t) = F(t)  F(t), and  is time delay. G() contains diffusion, chemical kinetic, molecular brightness, concentration, and photophysical information by quantifying the temporal decay (as a function of ) of the fluorescence fluctuations.

A solution to the autocorrelation function may be formulated in the case of ideal solutions and concentration- and position-independent diffusion coefficients using Green's function for classical diffusion kinetics:

(5)

where  is time, N is the number of particles, D is the diffusion coefficient, and r is position. The diffusion autocorrelation function can be calculated using a double integral:

(6)

Autocorrelation function assuming a Gaussian observation volume

The standard autocorrelation function for a single fluorescent species freely diffusing in an ellipsoidal 3D-Gaussian observation volume O(r_O) at low intensity (well below saturation) has the solution (Schwille et al., 1996):

(7)

where _D is the characteristic (diffusion) time molecules spend on the average in the observation volume,  is the axial ratio (ratio of axial to radial dimensions of the observation volume), and N is the average number of molecules. Autocorrelation and volume are clearly related in the limit   0, where

(8)

and C(r) = C₀ is a spatially constant concentration of molecules in the observation volume of size V. Note that both x and y (z is axial) contribute a factor of (1 + D/²)^0.5 to G_D(), where  is the transverse 1/e² radius of O(r_O), but that Eq. 7 is the special case where _x = _y, and hence only two diffusion time constants exist: _D ²/4D, ' = ()²/4D.

For anomalous diffusion, where r² = t and 4Dt  t, the standard fitting function is (Schwille et al., 1999b)

(9)

which also assumes a Gaussian observation volume, where the temporal exponent  has the possible range 0 <  < 1, and the transport coefficient  is generally analogous to 4Dt¹.

The autocorrelation of multiple diffusing species is just a linear combination of the autocorrelations for each species separately, weighted by their fluorescence intensity (Hess et al., 2002):

(10)

where _i and G_i() are the average fluorescence and autocorrelation for the ith species, respectively, and m is the total number of species. Processes occurring faster than the diffusion time _D can be resolved as additional factors that generate shoulders added to G() at  < _D. Because time scales are involved that do not depend on the concentration of fluorescent molecules, FCS can be used as a "ratiometric" means of measurement, i.e., it can also extract concentration-independent information from a system.

Consider now transitions of the form B*  B, between bright (B*) and dark (B) states of a molecule or mobile object, which occur on time scales faster than diffusion and are hence observable by FCS, as in triplet-state intersystem crossing, molecular conformational changes, or certain chemical reactions. If such a reaction has an equilibrium constant K, forward and backward reaction rate constants k₊ and k, respectively, and a free energy G, as related by

(11)

the Boltzmann constant k_B, temperature T, and respective concentrations [B*] and [B], then a characteristic time constant (_B)¹ = k₊ + k, and molecular dark fraction F_B = [B]/([B*] + [B]) may be defined. These quantities _B and F_B describe the reaction and are introduced as an exponential factor F_B exp(/_B) in the autocorrelation G(), resulting in

(12)

Exploiting the fact that G_D(0)¹ = N and C₀ = N/V, the physical volume of the observation volume (V) can be determined from an FCS measurement at known concentration C₀, or for a known observation volume an absolute concentration can be obtained.

For multiple independent chemical kinetic processes that occur on well-separated time scales, the fitting function is a product of the diffusion and m chemical kinetic factors (Hess et al., 2002):

(13)

with amplitude F_i and time scale _i.

The above derivation assumes a Gaussian profile for O(r). However, if the profile is non-Gaussian, G_C(), G_D(), and G() will all be distorted, and the kinetic parameters obtained from measurements will be subjected to significant systematic errors. An understanding of the effects of the optics on the functional form of G() are thus crucial. Therefore, we consider the optical problem of calculation of O(r) that depends on the illumination and collection point spread functions of the FCS system, and which allows us to calculate G() from O(r).

Calculation of the point spread function

The point spread function for a lens system is defined as the 3D image of a point source. It is also the intensity distribution in the vicinity of the focal plane resulting from a point source of monochromatic light in the image plane of the lens system. The paraxial approximation (Born and Wolf, 1991) is used to simulate confocal optical systems (Sheppard et al., 1991; Sandison and Webb, 1994), which is adequate for sin  < 0.8 (Sandison and Webb, 1994), or roughly NA = 1.06 in water, where  = sin¹ (NA/n) is the half-angle of opening of the objective lens in radians, and n is the refractive index. Here we consider a non-paraxial, high-NA description (Richards and Wolf, 1959), which is a better approximation for  > 0.8 and ideal lenses (no aberrations).

The dimensionless optical coordinates used are  and u, corresponding to radial and axial coordinates, respectively:

(14A)

(14B)

where k is wavenumber in the medium,  is wavelength, and r² = x² + y². The diffraction theory by Richards and Wolf uses the complex integral representation:

(15)

for the energy density (intensity):

(16)

where A₀ is a constant, J_i are Bessel functions of ith order, _i are the integrals (Eqs. 15),  is the integration angle, and  is the azimuthal angle around the longitudinal axis ( = 0 corresponds to the x-axis).

Illumination optical geometry: the underfilling fraction

Underfilling the back-aperture of the objective can be used to dramatically elongate and enlarge the illumination profile at the focus of the objective and to reduce the effective NA of illumination, which produces a nearly Gaussian illumination profile. Specifically, for a Gaussian beam with 1/e² intensity radius r₀ and objective back-aperture radius r_BA, the underfilling fraction  = r_BA/r₀ is just the ratio of the back-aperture radius to the beam radius. For  < 1 the objective is "overfilled," and for  > 1 the objective is "underfilled." The apodization function in Eqs. 15

(17)

(W. Zipfel, unpublished data) is the factor that accounts for the effect of  on the illumination profile.

Collection optical geometry and observation volume profile

After illumination with confocal optics, the second factor determining the optical geometry in FCS is the collection profile, which accounts for the confocal detector aperture and diffraction of collected fluorescence propagating from the object space into the image space, and is combined with the illumination profile to calculate the observation volume profile. The relative spatial collection efficiency can be expressed as the contribution to the measured fluorescence signal as a function of position in object space. Therefore we consider a point r_O in object space and the contribution to the image it produces (a point spread function centered at the corresponding image point r_i, weighted by the illumination at r_O). The collected portion of the fluorescence emitted from this point gives the collection efficiency for this point relative to other points in the object space. We assume that the detector collects all of the fluorescence that passes through the confocal detector aperture, which is by definition in the image plane.

In the case of two-photon excitation, where there is typically no confocal detector aperture, the observation volume is determined exclusively by the squared illumination intensity:

(18)

where S(r_O) is the normalized illumination intensity point spread function, which depends on the illumination wavelength, intensity, and objective properties.

In confocal FCS, the collection through the detector aperture must also be considered. The observation volume is defined not only by the illumination profile factor S(r_O), but also by a collection profile factor (r_O) resulting from the objective and tube lens properties, fluorescence emission spectrum, and the confocal detector aperture in the image plane, assuming incoherent emission (Born and Wolf, 1991; Sandison et al., 1995). The observation volume is thus defined as:

(19)

where r_i is the position of the element of area in the image plane, (r_O) is the centrosymmetric collection point spread function, M is the total magnification of the system, and r_O is the object-space position. Note that an infinitesimal detector is represented by a delta-function at r_i = 0 and in this case the integral reduces to

(20)

This form reduces to the point spread function of a two-photon (2P) microscope with no detector aperture, where (r_O )  S(r_O), yielding O(r_O)  [S(r_O)]². The detector aperture size in optical units is calculated using

(21)

where R_d is the detector aperture radius in real space, and r_d is its radius in dimensionless optical units.

Calculation of collected fluorescence for incoherent emission

The time-averaged collected fluorescence F is given by an integral over the object space

(22)

where O(r) is the observation volume spatial profile, and  is a constant proportional to overall detection efficiency, dye fluorescence excitation cross section, and quantum yield.

Observation volume

We define the volume of an illumination or observation profile (note that volume is to be distinguished from profile) by an integral over the appropriate space:

(23)

where W(r) is the spatial profile normalized to unity at its maximum, i.e., W(r) = O(r)/O(r = 0). This definition is similar to established definitions (Thompson, 1991; Mertz et al., 1995; Xu and Webb, 1997). Equation 23 is used to calculate the effective volume of the calculated observation volumes.

Signal-to-noise ratio in FCS

Forming a correlation requires the observation of correlated photons. The S/N ratio in FCS depends simply on being able to observe pairs of photons emitted by a single molecule within a correlation time . There must be an abundance of photons within the time window  for the correlation to have a reasonable S/N ratio, and hence the count rate per molecule ( = F/N) has been shown to be directly related to signal-to-noise in FCS (Koppel, 1974). In the regime () 1, the S/N ratio is proportional to  (Koppel, 1974). The average fluorescence F is measured routinely during FCS, and the zero-time value of the autocorrelation function, G(0), gives the inverse of the mean number of fluorescent molecules in the focal volume (N), if properly corrected for background fluorescence. Corrections to N and  for measured (noncorrelating) background (Koppel, 1974) are made using

(24)

Therefore,  is an accessible experimental parameter that is commonly used to optimize the S/N ratio. Fortunately,  is also accessible to our calculations once we have obtained fluorescence and number of molecules (i.e., the physical volume of the observation volume). Because the maximization of S/N with minimal experimental artifacts is of great interest, the dependence of  and artifacts on  and r_d are calculated and examined under the same conditions.

Dependence of count rate per molecule on detector aperture

Small detector aperture limit

Large detector aperture limit

(25)

No peak in (r_d) with a 3D-ellipsoidal Gaussian focal volume

If a 3D-ellipsoidal Gaussian profile is assumed for illumination and collection functions, as well as a Gaussian profile for convolution with the pinhole, then the observation volume is Gaussian. With a Gaussian observation volume and spatially constant concentration C₀ the ratio W²(r)dr/ W(r)dr is constant, and hence the count rate per molecule (r_d) is proportional simply to O(r_O = 0), which is proportional to the collected fluorescence from a point source at the origin (object plane focus) and will only increase with increased detector aperture. Therefore, using a Gaussian approximation for the observation volume will never give a peak in count rate per molecule as a function of detector aperture.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

Numerical implementation

The integrations in Eqs. 15 are done numerically using Bessel functions J₀(x), J₁(x), and J₂(x) evaluated from x = 0 to 200 in steps of 0.02 and interpolated linearly. Illumination profiles agree well at low NA with paraxial results in the literature (data not shown) and with paraxial calculations (Sandison and Webb, 1994) at intermediate NA. Furthermore, results as a function of  at high NA agree with independent calculations by the method of Richards and Wolf and measurements (W. Zipfel, personal communication, 2001). Numerically, the observation volume profile (Eq. 19) is calculated as a sum:

(26)

where r = (x, y, 0) and n is the number of steps in the grid approximating the detector face, which is contained in the x-y plane. The theta function excludes values of r outside the detector radius. To accelerate calculations, the values for the normalized illumination intensity S(u, ) and collection efficiency (u, ) are calculated once and stored in 1000 × 1000 element floating-point arrays. Values are then interpolated from the array values for use in the observation volume calculations. Typical parameters used are  = 1.12 (=1.2 NA in water),  from 0 to 200, u from 0 to 200, 1000 steps in both u and ,  = 0,  = 0.002, and   1 for overfilled or  > 1 for underfilled. Typically a 20 × 20 detector grid is used.

Calculated correlation curves

The double integral in Eq. 6 is solved numerically as a convolution, followed by a product, followed by a single integral in three dimensions, using a discrete three-dimensional array of (128)³ = 2,097,152 elements. The theoretical G() in Eq. 6 then becomes (for discrete elements)

(27)

where T_ijk is a three-dimensional array representing the result of the first spatial integral, calculated from a product of the discretized spatial observation profile O_ijk, and the result of the convolution, A_ijk. The subscripts i, j, k denote spatial position along the three Cartesian axes. The convolution is between the concentration function C_ijk and a second (identical) spatial observation profile, O'_ijk. C_ijk is the solution to the diffusion equation from Eq. 5 evaluated at the discrete grid positions defined by the three orthogonal Cartesian coordinates i, j, and k. Note that the Einstein summation convention is not used below.

(28)

(29)

where

(30)

The theoretical G_T() is calculated at 50 logarithmically spaced  values spanning ~4 orders of magnitude in time. G_T() can also be used to obtain a diffusion coefficient by fitting G_T() to the measured G(), using the diffusion coefficient as a fitting parameter. The theoretical G_T() is normalized such that G(  0) = 1. The two-photon autocorrelation was calculated using the square of the illumination profile for O_ijk and O'_ijk, and using the same C_ijk as in the one-photon case. Because there is usually no detector aperture in two-photon FCS, the illumination profile is the only factor determining the profile of the observation volume.

Experimental methods

Autocorrelation curves were measured with two different systems.

One-photon FCS with aperture

Two-photon FCS

1

1

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

Calculated illumination and observation volume profiles are non-Gaussian and will lead to artifacts

We delineate the effects of the observation volume profile O(r_O) on the FCS autocorrelation function and the resulting consequences on the measured dynamics and analytical FCS parameters. Here we will present calculated illumination and observation profiles for a high-NA objective, as a function of the back-aperture underfilling () and the confocal detector aperture (for observation profiles only), first as two-dimensional pseudocolor intensity bitmaps, then as plots with attempted fits to analytic functions. It will be shown that the observation profile is predicted to be non-Gaussian under many experimental conditions. The detector-aperture dependence of the count rate per molecule, which is directly related to the signal-to-noise ratio, provides direct evidence for the non-Gaussian nature of the observation volume.

Next, the effect of a non-Gaussian observation volume on the FCS autocorrelation function is presented, demonstrating that inaccuracy or artifacts frequently arise as a result of using the standard fitting function (i.e., Eq. 7). The form and magnitude of these artifacts are then explored; their effects on the measured diffusion time, diffusion coefficient, axial ratio, and exponential fraction(s) and time constant(s) are shown. Methods for avoiding artifacts are outlined. Finally, the collected fluorescence and volume of observation (number of molecules at a known concentration) are analyzed as a function of  and r_d and provide evidence for an actual observation volume that is larger than predicted.

Calculated illumination and observation volume profiles

View larger version (42K): [in this window] [in a new window]   FIGURE 1   Observation profile images in log-scale pseudocolor, for a 1.2 NA objective, plotted for various values of the detector aperture rd and underfilling fraction . Images show the axial direction (u) horizontally and the radial direction (v) vertically away from the focus. The logarithmic color scale emphasizes the low-amplitude fringes away from the focus. Closing the detector aperture (from left to right) constrains the observation volume, reducing fluorescence collection from the tails of the illumination profile that are particularly periodic and non-Gaussian for an overfilled back aperture (top row). Underfilling the back aperture (from top to bottom) elongates and smoothes the illumination profile and hence the observation volume, but the effect is not drastic until  > 1.25. Using small rd and an underfilled back aperture ( > 1) results in a more nearly Gaussian observation volume. An exactly Gaussian profile is shown (bottom row left) with a 20 ou scale bar (center) and intensity color scale (right) for reference.

Bu²

View larger version (19K): [in this window] [in a new window]   FIGURE 2   Using conventional analytical functions to fit calculated axial illumination profiles is ineffective. The calculated profile for an overfilled 1.2 NA water objective (solid line) is fit with a Lorentzian (dashed line) and Gaussian (dotted line) on a linear (A) and logarithmic (B) intensity scale.

View larger version (18K): [in this window] [in a new window]   FIGURE 3   Deviation of calculated radial illumination profile from a Gaussian profile. The calculated illumination profile for an overfilled 1.2 NA water objective is shown along the radial axis (A and D), with a linear intensity scale (top) and a nonlinear scale (bottom), [log(intensity)]0.5, which should yield a straight line for all positions if the profile is Gaussian. Note that the profile begins to deviate from Gaussian (B and C, red dashed line) at less than the 1/e2 radial width (dotted black line).

View larger version (20K): [in this window] [in a new window]   FIGURE 4   Attempt to fit calculated observation profiles for an overfilled 1.2 NA water objective with a 4.7 ou detector aperture (corresponding to the top center panel in Fig. 1) using analytical functions. Left: The radial profile O(r) = O(u, ) = O(0, ) is shown on a linear scale (A) with Gaussian fit (B), and on a nonlinear scale (C) with the same fit (D). Right: The axial profile O(r) = O(u, ) = O(u, 0), (top right, E) and its Gaussian fit (F), is also shown with nonlinear vertical axis (G) and fit (H). The nonlinear axis transforms any Gaussian function into a straight line.

Measurements confirm observation volume is non-Gaussian

Peak in count rate per molecule versus detector aperture

View larger version (23K): [in this window] [in a new window]   FIGURE 5   Experimental evidence for a non-Gaussian observation volume. Measurements and diffraction-theory-calculated count rate per molecule () versus detector aperture (rd) for a 1.2 NA 40× water objective as a function of underfilling fraction (). A Gaussian observation profile predicts no maximum in (rd) at finite rd (thin green curve).

View larger version (27K): [in this window] [in a new window]   FIGURE 6   Underfilling reduces count rate per molecule. The effect of underfilling fraction () and detector radius rd on count rate per molecule () for a 1.2 NA 40× water objective. The curves are normalized such that the total rate of excitations in the x-y (focal) plane is constant.

View larger version (29K): [in this window] [in a new window]   FIGURE 7   Comparison of measured and calculated autocorrelation versus detector aperture. (Top) Measured autocorrelation (points) and diffraction theory fits (lines) for 1P-FCS (left column) with detector aperture diameter (A) 2.4 ou, (B) 3.9 ou, (C) 5.8 ou, and for 2P-FCS (right column) with no detector aperture (D). Residuals are shown below for fits with calculated autocorrelation (middle row) and with Eq. 7 (Gaussian volume; bottom row). Residual curves (E-G) correspond to measured data in A-C, respectively. Residuals for 2P-FCS are shown on the middle and bottom right plots. The number of molecules and diffusion coefficient were used as free parameters for the theoretical curve-fitting.

Simulation of the autocorrelation function using diffraction-based observation profiles

Comparison of calculations with measured autocorrelation and diffusion coefficients