## Abstract

In this work we systematically explored performance of an EM-CCD as a detector for spatially resolved total internal reflection image correlation spectroscopy (TIR-ICS) with respect to adjustable parameters. We show that variations in the observation volume (pixel binning) can be well described by a simple structural term ω. To test the sensitivity of camera-based TIR-ICS we measured diffusion coefficients and particle numbers (PN) of fluorescent probes of different sizes (Fluorospheres, GFP and labeled antibodies) at varying viscosities, concentrations, and sampling rates. TIR-ICS allowed distinguishing between different probe concentrations with differences in PN of 5% and differences of 6% in D by acquiring only 15 independent measurement runs.

© 2010 OSA

## 1. Introduction

Diffusion constants (D) are suitable indicators of different properties of fluorescently labeled molecules as they can be easily assessed by a variety of methods [1,2]. They can be measured for molecules such as proteins in solutions or on membranes. From diffusion measurements it is also possible to derive other properties such as particle size, viscosities of solutions, or strength of intermolecular interactions [3]. Thus a number of methods have been developed measuring diffusion constants including particle tracking (PT), fluorescence recovery after photobleaching (FRAP), molecular dynamics (MD) simulations and fluorescence correlation spectroscopy (FCS). Among them particularly fluorescence correlation spectroscopy (FCS) [4,5], has become a widely used method. It is used for instance in medical diagnostics [6,7], for quantifying cell surface interactions [8], or to measure flow rates [9]. In FCS the autocorrelation function (ACF) e.g. provides information about particle transitions [10] and therefore the residence time which is a gauge for its diffusion constant. As in a measuring period many of these transits occur and the signal may be hardly over the noise, multi-curve fitting or manual evaluation of the mean signal burst duration may be inefficient. The shapes of ACFs are highly dependent on the geometry of the observation volume. Thus mathematical models for different microscopy types have been derived [11,12] working with sample volume parameters often condensed in a structural term ω. In confocal FCS e.g. ω describes the ratio of the lateral and axial extensions of the detection volumes and depends on the excitation profile, i.e. the numerical aperture of the objective and the confocal pinhole size. Diffusion times are drawn from the best ACF fitting function at defined ω. Aside from conventional confocal FCS and TIRFCS variations of the technique e.g. fluorescence cross-correlation spectroscopy (FCCS) have been described [13]. In fluorescence cross-correlation spectroscopy (FCCS) two detectors are used to cross-correlate spatially or spectrally [11] separated signals. In TIRFCS a laser beam is totally reflected at the coverslip surface and generates an exponentially decaying, evanescent wave over the coverslip. TIRFCS is used to measure events at or near an interface e.g. a biological membrane [14] as the contrast of signals nearby the coverslip to those a few nanometers away is enhanced allowing for sensitive studies of surface-near diffusion processes. Most TIRF microscopy studies use fast EM-CCD cameras to monitor surface-near particle dynamics [15,16]. Recently it has been shown that with the very same equipment it is possible to perform spatially resolved FCS and FCCS [17] and that camera-based TIR-ICS is an alternative to single particle tracking in determining diffusion constants [18]. In the present study we examined and evaluated the discriminatory performance of camera-based TIR-ICS.

## 2. Materials and Methods

#### 2.1 Materials

The Nikon-TE2000 microscope is the basis of our TIR-ICS system. The beam coming out of an argon ion laser [National Laser Company; 800 Laser system] is enlarged and then refocused by a condenser [Nikon, TIRF] to the back focal plane (BFP) of an oil immersion objective [Nikon, Plan Apo TIRF 100x, NA 1.49]. The maximum achievable angle of 79 ° between light beam and optical axis is calculated by

where θ is the maximally achievable angle, the refractive index*n*of oil = 1.516 and the numerical aperture NA is 1.49. Thus, the evanescent field depth

*d*, calculated by

*n*= 1.33) at the excitation wavelength λ = 488 nm and with the

_{solution}*n*of glass = 1.5255. The illumination intensity profile showed a Gaussian shape in the x-y plane parallel to the cover slip and an exponential decay in z-direction along the optical axis.

The emitted sample signal passing the dichroic mirror and an emission filter [AHF, F36-149] was detected by an EMCCD-camera (Andor, DV860AC) with a computer memory-limited readout area of 16 x 64 pixels and a corresponding detector size of 2.6 x 10.4 µm. With this camera 16 lines can be read out at 1.2 kHz, i.e. an exposure time of 0.78 ms. Signals from the camera were transformed into ASCII-files by the acquisition software (Andor Solis) and analyzed offline by custom-built routines in Matlab [Mathworks, Matlab 7.5.0].

#### 2.2 Observation volume

To specify the shape and the size of the observation volume, we calculated the normalized molecule detection efficiency function (MDE) [11]. The MDE defines the observed area in the sample region by the camera. Contrary to using a photodiode or photomultiplier tube where the MDE function is laterally limited by the diameter of a pinhole, the observation volume in TIR-ICS is determined by an array of pixels in x-y direction, in the simplest case of quadratic shape by binning n x n pixels. Thus, it should be possible to calculate the lateral part of the MDE by the convolution of the point spread function (*PSF*) of the system and the physical detector size at defined binning.

The vector *q* denotes coordinates in the focal plane and q’ are the corresponding shift values of detector and PSF in the convolution term. The variable Det_{size} represents the diameter of interconnected pixel area in the sample. We determined the PSF of our imaging array by measuring settled particles on the cover slip. The PSF was then approximated by a Gaussian function for convolution.

The lateral part of the MDE, shown in Eq. (3), is approximated by a Gaussian function. Due to the evanescent wave excitation an exponentially decaying part has to be multiplied to the lateral part. With the approximation of a z-independent photon collection efficiency the entire MDE leads to

_{xy}. To summarize, the lateral part of the MDE is a Gaussian approximation of a quadratic area of interconnected pixels convolved with the PSF. The axial part describes evanescent field decay with the penetration depth

*d*. The Gaussian width ω

_{xy}and the depth

*d*affect the shape of the ACF and therefore have great importance to Eq. (6).

All further fitted data are based on this approximation, especially the definition of the structure parameter of the fitting model.

With the MDE and

where V is the half-space of the sample region and r denotes a vector in the focal plane, the effective observation volume V_{eff}can be calculated. It is defined by [11,20]:For a typical penetration depth of 100 nm and 1x1 binning (ω

_{xy}= 316.9 nm) the numerical calculation for an effective volume of 0.063 fl give W

_{1}= 0.015 fl. This equals a ratio of 4:1.

#### 2.3 TIR-ICS data analysis

The mathematical model used to analyze the ACF comprises a model of diffusion in a 3D volume [11]. The emission profile of fluorophores is assumed to be uniform in the x-y direction and exponentially decaying in z direction. Therefore the observation volume in z direction is described by the field depth d and laterally limited by the camera pixel size and the selected pixel, condensed in a factor ω_{xy}. With shown modifications the elemental autocorrelation function

*F*denotes the acquired signal, $<F>$ represents the mean of the signal

*F*and δ

*F*with$\delta F=F-<F>$its fluctuation, becomes [11,20]

Here G (τ) is the autocorrelation function depending on the lag time τ, the correction factor γ, the number of particles in the observation volume N and the diffusion time τ_{z}, which is strongly related to the diffusion coefficient D. The geometry of the observation volume ω is denoted by the parameters ω_{xy} and d. The factor γ corrects the amplitude of *G* for the observation volume and describes the deviation of the effective volume V_{eff} from the volume of the MDE (W_{1}). It is defined as γ = W_{2} /W_{1} and therefore has an influence to the particle number [19]. From Eqs. (4) and (5) it follows that γ equals 1/4.

The acquired images are correlated offline with adjusted binning. The diffusion coefficient D can be estimated by fitting the above model to the normalized autocorrelation functions. For particle number measurements the recorded signal *F*(t) has to be separated from the camera-specific background or offset [22]. The number of particles and therefore the amplitude of *G* is then calculated by:

## 3. Results

#### 3.1 Definition of the observation volume

In optical spectroscopy spatial resolution is limited by the diffraction of light. Since photons were detected by an EM-CDD camera, signals detected at one pixel will therefore contain contributions of neighboring objects. Thus the lateral extend of the observation volume of one pixel is larger than the pixel size itself. Mathematically this effect can be described by a convolution of pixel size and the point spread function (PSF). To specify this lateral extend, we first estimated the size of the PSF. Although the lateral PSF can also be described by a Lorentzian function [23] we for simplicity fit a Gaussian function (since a Gaussian is assumed in the MDE, Eq. (4) to settled 43-nm fluorescent beads yielding a width of 299.2 nm (Fig. 1a
). Lateral observation volumes were first estimated from the convolution of the estimated PSF with the physical detector size at different pixel connections (binnings) by subsequent fitting with Gaussian functions as described [11]. The lateral extend of the observation volume (ω_{xy}) is then defined as the double root mean square deviation of this fit, corresponding to the first part of the MDE (Eq. (4), and varied from 316.9 nm for no (1x1) binning to 1838.9 nm for 16x16 binning (Fig. 1b). For large lateral observation areas the importance of the shape of the PSF decreases in the convolution term which results in a box-shaped extension. Therefore the Gaussian approximation deteriorates for higher binning. To analyze this error in the MDE estimate on diffusion coefficient (D) measurements we varied the detection area from 160 nm to 2560 nm, corresponding to 1x to 16x binning, and estimated D for 43 nm beads as well as for GFP-molecules. As expected the estimated D varies by a factor of 1.4 for the largest and smallest lateral observation area for GFP shown in Fig. 1c.

In confocal FCS the structural parameter ω describes the ratio between lateral and axial widths of the Gaussian approximations of the PSF, and it is typically determined empirically by calibration with a fluorophore with known diffusion constant (e.g. rhodamine 6G): the recorded autocorrelation is fit by a simple diffusion model with fixed D, such that an apparent ω can be easily extracted. Thus, in analogy we first determined D for the smallest lateral observation area (1x binning), where the error due to the Gaussian approximation should be smallest and negligible. Assuming this D as the best determined one we calculated the corresponding ω_{xy} for the different binning sizes. This revealed a linear relationship between ω_{xy} and bin size, yielding an empirical setup-specific correction factor, which yields an apparent ω_{xy} for all detection areas, shown in Fig. 1d. The linear behaviour can be expressed by the following equation and depends only on the PSF and the lateral detection area:

Contrary to the difficulties in describing the lateral observation volume by a simple function the axial extension determined by the evanescent wave can be well described by an exponential. To demonstrate this we estimated D for 43 nm fluorospheres in 60% glycerol-water solutions using different angles in the range of 73° to 68.7° corresponding to changes in the evanescent field penetration depth from 107 nm to 253 nm. Figure 2 shows the invariance of fit D values to these changes.

#### 3.2 Particle number estimation

After having determined the parameters for the sample volume, we were interested in the applicability of an EM-CCD as detector for estimating particle numbers. Following Eq. (7) the estimated particle number is strongly related to the background level, the camera gain, camera shot noise, and the molecule brightness. The parameters constant during an experiment like camera gain, shot noise, and the molecule brightness can be lumped in a factor α. Thus, α can be determined by an initial calibration measurement and has no effect on the analysis of relative particle numbers. However, the level of the readout noise and therefore the background counts can vary from acquisition to acquisition (Fig. 3a ), which necessitates background measurements for each acquisition to enhance the precision.

This can be accomplished by shading a small part of the EM-CCD chip to readout the dark counts of the camera simultaneously with the measurement (Fig. 3b).

To test whether EM-CCD TIR-ICS is capable for differentiating between different particle numbers in the observation volume and to proof the reliability of this approach we performed a dilution series starting from a concentration of fluorospheres of 39.6 nM in 50% glycerol at 84 nm depth. This results in a range of particle numbers N = 1.25 to 0.02 in a corresponding sample volume of 0.0527 fl (1x binning, 84 nm evanescent field depth). We found that measured and calculated numbers of particles were matching when using the above described method (Fig. 3c). In contrast to a measurement with background subtraction from a preceding image acquisition of dark counts the precision enhancement with simultaneous background readout as described here, determined by a comparison of the relative standard deviation, was about 56% for a sampling volume of 0.0527 fl.

Furthermore the number of particles can be altered by adjusting the size of the observation volume. Due to the huge number of available pixels on a CCD-chip the lateral extend of the observation volume can be easily changed offline. This allows highly comparable measurements with different observation volumes as the very same image stack can be used. To test whether particle numbers are correctly estimated irrespective of binning, i.e. detection area, we changed the total observation volume from 0.05 fl to 1.78 fl corresponding to an alteration of the lateral part ω_{xy} from 316.9 nm to 1839 nm. As expected these changes correlate linearly with estimated particle numbers, as shown by the red line fit in Fig. 3d.

#### 3.3 Diffusion coefficient estimation

We next wanted to know whether TIR-ICS is able to differentiate D values of particles of different sizes and in media of different viscosities in agreement with the Stokes-Einstein relationship. To address this question, we measured diffusion coefficients of GFP-molecules and 43 nm fluorospheres in glycerol-water solutions of different concentrations (40% - 80%). As these solutions have different refractory indices, the resulting sample volume varied between the concentrations. A readout area of 64x16 pixels was used and D fitted to the Stokes-Einstein equation plus an additive term to care for systematic errors, displayed in Fig. 4 . For GFP-molecules we estimated an effective hydrodynamic radius of 4.2 nm and for the fluorospheres of 58.1 nm in the glycerol solution, equivalent to a radius difference of a factor of 13.8.

To test the performance of TIR-ICS for resolving acute particle size alterations we measured antibody binding using a species specific alexa-labeled anti IgG-antibody. IgG from different species (mouse and rabbit) were presented as antigen to the fluorescent secondary antibody whose diffusion coefficients was measured in an area of 64x16 pixels and an observation volume of 68.7 al in 50% glycerol. When combined with anti-mouse IgG the D of the secondary antibody increased indicating binding. The binding was specific as it was not observed for rabbit-IgG (Figs. 5a , 5b). As a result the estimated diffusion coefficients of the bound and unbound state show a variation by a factor of 2. This would fit to that the majority of the fluorescent anti-mouse-IgG antibody binds to only one mouse-IgG.

After having determined parameters for the choice of the sample volume and the absolute numbers for diffusion coefficients, we were interested in the discriminatory power of the method for determination of particle numbers and diffusion constant differences. In general the precision of differentiating between measured values depends on their variation coefficient and the number of measurements.

#### 3.4 Resolving power of TIR-ICS

To test the discriminatory power for particle number estimation we used a dilution series, as shown in Fig. 3c. Given a continuous course of standard deviation and values between the reading points we were able to interpolate reading points and their standard deviation in order to test smaller intervals for significance.

It was then possible to show the dependence of the discriminable number of particles and the total number of particles in a defined volume (Fig. 6
). For 43 nm fluorophore diffusion with an concentration interval from 39,6 nM to 0.3 nM within an effective volume of 1.78 fl (16x binning, d = 84 nm) we found that for a 95% confidence interval in the t-test the minimal differentiable difference in estimated particle numbers is better than 16% for the complete range of 21 to 0.7 total number of particles. With a total number of only 15 independent measurements and a binned array of 8 pixels, an acquisition length of 10^{4} images in the range of 2 to 21 particles in the observation volume the precision is 8% or better. As shown in Fig. 6 with a total number of 15 particles in the observation volume the precision is one particle.

To measure the discriminatory power of TIR-ICS for diffusion coefficients we used the series of experiments shown in Fig. 4. Therefore we interpolated acquired data points and their standard deviations and used again the t-test with 95% confidence interval as a boundary to test for significant difference. Thereby the precision of diffusion coefficient measurements depends on the ratio of sampling time (ST) and observation volume transition time (τ_{xy} and τ_{z}). For increasing ST/τ_{xy} and ST/τ_{z} the number of fit able ACF data points and therefore the precision of diffusion coefficient estimations decreases (Fig. 7a
). Here the minimal relative resolvable difference in diffusion constant (RRdDmin) for GFP was 6% for a total amount of 10.24*10^{6} data points, which were achieved by 15 independent measurements within an array of 16x64 pixels and 10^{4} images. As an example to resolve GFP-molecules in 40% glycerol-water solution within an observation volume of ω_{xy} = 316 nm and d = 100 nm a sampling time of 1.6 ms or better is required. Surprisingly with under-sampling in x-y direction by a factor of 2 and in z direction by a factor of 40 RRdDmin is still at a level of better than 20%.

The autocorrelation analysis relies on averaging over the behavior of many molecular encounters within the observation volume. Thus the total number of observed fluctuations depends on the total acquisition time, the diffusion coefficient, the geometry of the observation volume and the concentration of the fluorospheres. We investigated the effect of the total number of observation volume transitions (signal fluctuations) to the precision of estimating diffusion coefficients (Fig. 7b). In this experiment we used a GFP concentration of 18.5 nM. The mean diffusion time was approximated to be a weighted sum of τ_{xy} and τ_{z}:

^{6}nm

^{2}/sec the mean diffusion time is τ = 1.6 ms. The number of transitions is then calculated from τ, the total acquisition time, and the average number of particles in the volume (equals approximately 1 particle for 1 pixel for the shown concentration). Figure 7b shows a hyperbolic dependence (red fit) of precision on the total number of volume transitions resulting in a maximal achievable precision of 2.9% for GFP and 2.01% for 43 nm beads (data not shown).

## 4. Discussion

In the present study we have shown camera-based TIR-ICS to be a valuable method for measuring diffusion constants. In addition. we introduced a simple approach to correct for the deteriorating Gaussian approximation of the lateral observation volume for increasing binning. Estimated diffusion coefficients do not vary with observation volume changes. Furthermore a correct particle number measurement is hardly possible without calculating the corresponding background from the acquisition itself due to changes in the camera read out level. In this work we describe a way to take care of this effect and estimated more precise particle numbers. However, FCS or TIR-FCS is a valuable method to analyze binding kinetics and rates [7,8]. With this camera-based approach we had the ability to differentiate between small particle size alterations, indicating antibody binding, as well as small viscosity changes reflecting the Stokes-Einstein regime. We affirm a 1:1 binding of an anti-mouse-IgG antibody to mouse-IgG and measured a hydrodynamic radius of 4.1 nm for GFP, which is in a good agreement to another FCS result of about 3 nm [21]. It was recently shown that TIR-ICS is an accurate method [17,18]. But when using a new method, besides its accuracy, its precision and resolving power, are important parameters. In this study we have probed the resolving power of TIR-ICS in discriminating between different particle numbers and diffusion constants. With a smallest resolvable relative PN difference of 5% TIR-ICS is a relatively precise method to analyze small particle number changes. However, since measurements of discriminatory power have not been performed to our knowledge in other FCS studies [11,17–20] we cannot easily compare the resolution achieved here with other FCS configurations. With a resolvable difference in diffusion constants of 6% TIR-ICS is a useful method to distinguish diffusion constants not only in membrane-bound processes as described [14,18] but also in solutions. This makes typical FCS measurements possible on standard TIRF microscopes if camera readout is fast enough. With TIR-ICS of the here described sensitivity protein-protein interactions such as antibody binding could be accessed rapidly. Furthermore TIR-ICS is a suitable tool to resolve diffusion dynamics on surfaces with a high precision [18].

What are intrinsic advantages of TIR-ICS in comparison to other TIRFM diffusion measurement techniques such as evanescent wave fluorescence recovery after photobleaching (EW-FRAP), particle tracking (PT) or TIRFCS? In TIR-FRAP the diffusion coefficient and the transport process of cytosol molecules are evaluated by the fluorescence recovery rate after cell surface fluorescence photobleaching [24]. In contrast to TIR-ICS a particular measurement requires far more time as the experiment is performed in two steps as bleaching and recovery. Moreover this technique only applies to living probes such as cells in culture whereas TIR-ICS is also suitable for measurements in solutions. Here changes in protein-sizes and molecule-molecule interactions such as antibody-binding can be assessed (Fig. 5) [8]. This and the fast time track of measurements makes TIR-ICS a high potential tool for medical diagnostics [6]. Besides EW-FRAP, PT applies for measurements of molecular dynamics. Here, in contrast to TIR-ICS the individual particle has to be identified and tracked. This, as dependent on peak-detection, however requires high signal-to-noise ratios and inter-particle distances [25]. Moreover calculation-times and readouts are far more time consuming. The main advantage of PT is the measurement of individual molecules in contrast to the other here discussed methods. Since in TIRFCS the diffusion coefficient is measured by the means of correlation of only a few photodiodes signals, TIR-ICS can be regarded as its follower. In contrast to TIRFCS, TIR-ICS allows simultaneous adjustments of different observation volumes due to the possibility of their offline adjustment. Moreover the spatial resolution makes cross-correlation easily accessible as it does not require for the exact positioning of photodiodes. TIR-FCS, however has, when compared to an actual camera-system the advantage of a higher time resolution and therefore provides access to smaller molecules and faster processes. Thus the experiment has a big impact to the choice of the method. However this study provides the necessary basis for researches when considering TIR-ICS for measurements of molecular diffusion dynamics, including molecule-molecule interactions as well as measuring particle concentrations.

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