Successful FCS Experiment in Nonstandard Conditions - Langmuir

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Successful FCS Experiment in Nonstandard Conditions Ewa Banachowicz,† Adam Patkowski,†,‡ Gerd Meier,§ Kamila Klamecka,∥ and Jacek Gapiński*,†,‡ †

Molecular Biophysics Department, Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland NanoBioMedical Center, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland § Institute of Complex Systems, ICS-3, Forschungszentrum Jülich, D-52425 Jülich, Germany ∥ Lehrstuhl für Angewandte Physik und Center for NanoScience, Ludwig-Maximilians-Universität, Amalienstr. 54, 80799 München, Germany ‡

S Supporting Information *

ABSTRACT: Fluorescence correlation spectroscopy (FCS) is frequently used to measure the self-diffusion coefficient of fluorescently labeled probes in solutions, complex media, and living cells. In a standard experiment water immersion objectives and window thickness in the range of 0.13−0.19 mm are used. We show that successful FCS measurements can be performed using samples of different refractive index placed in cells having windows of different thickness, even much thicker than nominally allowed. Different water, oil, and silicon oil immersion as well as long working distance dry objectives, equipped with the correction collar, were tested and compared. We demonstrate that the requirements for FCS experiments are less stringent than those for high resolution confocal imaging and reliable relative FCS measurements can be performed even beyond the compensation range of the objectives. All these features open new possibilities for construction of custom-made high temperature and high pressure cells for FCS.

1. INTRODUCTION

Moreover, since the size of the focal spot gets bigger, the diffusion times τ and the numbers N of particles per confocal volume, measured by fluorescence correlation spectroscopy (FCS), also increase, leading to a drop of imaging resolution in microscopy applications and to difficulties in FCS experiments due to reduced signal-to-noise ratio. In ref 5 the authors have presented calculations of the point spread functions (PSF) and FCS correlation functions distorted by spherical aberration due to refractive index mismatch between the immersion liquid and the mounting medium as well as the window thickness different from the nominal value for the ConfoCor2 (Zeiss) instrument with water immersion objective 40×/1.2W (#1 in Table 1). They showed that both the PSF and the FCS correlation functions (CFs) are strongly distorted even by a small deviation from the nominal values of refractive index and window thickness. Unfortunately, those calculations do not correspond to the real experiment situation, since the correction of these spherical aberrations by the correction collar of the objective was not taken into account. In this way the authors presented an overpessimistic evaluation of the FCS experiment performed for a nonexisting objective, which has the parameters of the Zeiss objective mentioned above but is not equipped with the correction collar. Here we show that the spherical aberration resulting from the refractive index mismatch and window thickness can be

Fluorescence correlation spectroscopy (FCS) is a powerful technique used to measure the self-diffusion coefficients of fluorescently labeled probes in solutions, complex media, and living cells.1−3 Since the signal-to-noise ratio in FCS improves with the decreasing size of the observation volume, confocal microscopes became natural components of FCS setups. Very often the same microscope is shared between FCS and laser scanning microscopy (LSM)−confocal imaging. Both high resolution imaging and FCS can be performed when the incident laser beam is focused to the smallest possible volume of a well-defined shape. This requires the use of water/ oil immersion objectives of high numerical aperture (NA) amounting to 1.2−1.4. These objectives have a very small working distance of the order of 0.2 mm. If the refractive index of the sample is the same as that of the immersion liquid, then the confocal volume has a Gaussian shape of dimensions σx, σy, and σz in the x, y, and z directions, respectively, where z is the direction of propagation of the incident beam. Usually σx is equal to σy. The shape of the confocal volume is defined by the structure parameter, SP = σz/σx which in the optimal case amounts to about 5. In many cases the stringent requirements regarding the refractive index and the nominal window thickness of the sample cannot be met. It has been shown4,5 that in the case when the refractive indexes of the sample and immersion liquid are different or/and the window thickness deviates from the nominal value, the confocal volume is strongly distorted and elongated.6 © 2014 American Chemical Society

Received: May 22, 2014 Revised: June 30, 2014 Published: July 3, 2014 8945

dx.doi.org/10.1021/la5015708 | Langmuir 2014, 30, 8945−8955

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Table 1. Technical Data of the Used Objectivesa

a

obj. no.

objective description

MF

NA

WD [mm]

CR [mm]

1 2 3 4 5 6 7 8

Zeiss, C Apochromat 40×/1.2W, water immersion Olympus, SLC plan FL 40×/0.55 Olympus, LUCPLFLN 40×/0.6 Zeiss, LD Achroplan 20×/0.40 Zeiss, Plan Apochromat 63×/1.40, oil immersion Olympus, UAPO N 100×/1.49 oil immersion, TIRF Olympus, UplanSApo 60×/ 1.30 Sil, silicon oil immersion Olympus UplanSApo 30×/1.05 Sil, silicon oil immersion

40× 40× 40× 20× 63× 100× 60× 30×

1.2 0.55 0.60 0.40 1.40 1.49 1.30 1.05

0.28 7.7 3.0−4.2 10.2 0.19 0.09 0.3 0.8

0.13−0.19 0−2.6 0−2.0 0−1.5 no 0.13−0.19 0.15−0.19 0.13−0.19

MF, magnification; WD, working distance; CR, correction range.

with and one without the correction collar, and two new silicon oil immersion objectives (ni = 1.4) with the correction collars (3.4.3 and 3.5.2). The application of silicon oil immersion objectives to study samples in water is of particular interest, since here the spherical aberration due to the excess thickness of the glass window (over the maximum nominal value) can be compensated by the corresponding layer of water, and in this way much thicker windows can be used with full compensation. At the end we discuss the limits of the thick window compensation by the correction collar of dry LD objectives which is necessary for custom-made experimental setups for temperature dependent and pressure dependent FCS measurements.

compensated in a range discussed in ref 5 and even broader by the correction collar and undistorted correlation functions can be measured. Additionally, we have determined the range of compensation for different objectives. We also show that beyond the compensation range the relative FCS measurements analyzed using the standard form of the CF result in correct values of the self-diffusion coefficients. In some applications when thicker windows of the sample cell are necessary, i.e. in temperature and pressure dependent FCS measurements, the use of dry objectives of longer working distance of the order of 3−7 mm7 is required. Due to their much smaller numerical aperture of about 0.4−0.6 they produce much larger and longer focal spot, which has often been a reason to disqualify them as objectives suitable for FCS due to strongly reduced molecular brightness of fluorescent molecules. Moreover, in addition to the problems met in the case of water immersion objectives, the shape and size of their focus depend also on its position in the sample, even for pure water solutions. In some objectives, the spherical aberration effects introduced by window thickness d, sample refractive index ns, and the focus position in the sample can be partially compensated by means of so-called correction collar. The optimum position of the correction collar corresponds to the maximum number of counts per molecule. It is strongly recommended to use only such objectives for FCS measurements. The crucial questions are (i) can the distortion of the confocal volume in nonoptimum experimental conditions be compensated, and if not (ii) can FCS experiments be performed successfully when the confocal volume (PSF)8 is distorted and has a non-Gaussian shape? This paper is organized in the following way: In section 3.1 we discuss the mathematical features of the standard CF derived for the Gaussian confocal volume. In sections 3.2 through 3.6 we investigate the effects of the (i) refractive index mismatch between the immersion liquid and the mounting medium, (ii) depth of the focus in the sample, and (iii) window thickness on the experimental FCS CFs in cases when the distortion of the confocal volume is introduced in a controlled way and can/cannot be compensated with the correction collar of the objective or balance between some optical parameters (e.g., depth of focus and glass thickness). The experimental results include a water immersion objective (3.2 and 3.5.1) and three dry LD objectives (3.3, 3.4.1, and 3.6) with different numerical apertures all equipped with the correction collar. At the end we will also show the performance of oil-immersion objectives (3.4.2 and 3.5.2): two of the standard type (refractive index of immersion medium ni = 1.51) offered by many companies as the best solution for high-resolution imaging, one

2. EXPERIMENTAL SECTION Rhodamine 6G was purchased from Sigma-Aldrich. Water solutions were made using deionized water. Alexa 488 was obtained as a sideproduct after protein labeling reaction using Alexa 488 succinimidyl ester (Invitrogen, USA). Hydrolyzed molecules of this compound turned out to be very stable and efficient fluorophores. Silica particles of 164 nm diameter were synthesized as Rhodamine B core surrounded by silica layers added according to a modified Stöber method.9 The core size of 30 nm allows to treat the silica particles still as point-like fluorophores, especially when using dry objectives with large focal spots. Glycerol was purchased from Riedel-de Han (99%, puriss). Viscosity values of glycerol/water mixtures were measured using Contraves LS40 rheometer and compared to literature data.10 Alexa488 stock solutions (in water) for measurements in water/ glycerol mixtures had a concentration of 1000 and 100 nM for immersion and dry objectives, respectively. The actual samples were made by mixing 2 μL of respective Alexa488 stock solution with 198 μL of the proper water/glycerol mixture, which resulted in obtaining optimum conditions of N ≈ 1. In order to see the effect of distorted shape of the confocal volume on the form of the correlation function experiments were performed using the Zeiss ConfoCor 2 (based on Axiovert 100 M microscope) and ConfoCor 3 (based on LSM 780 system) instruments for eight different objectives listed in Table 1, different window thickness and different refractive index of the medium. Most of the objectives were equipped with the correction collar (see Table 1) which allows for the compensation of the spherical aberration due to window thickness within the correction range given in Table 1. Each measurement was repeated several times to ensure reproducibility and to minimize the errors. It took up to 10 min for the immersion objectives and up to 30 min for LD objectives. The measured CFs have been analyzed using the standard software of the microscope systems. The error resulting from the statistical noise of the experimental data was in the range of 2−5% for τdiff and N, and amounted to 30% for SP. In all experiments the confocal pinhole size was set to 1 AU (Airy unit11). 8946


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3. RESULTS AND DISCUSSION 3.1. Standard Correlation Function. The image of a point light source in the focal plane is the Airy diffraction pattern. The radius of the first dark ring in the Airy diffraction image in the specimen plane is11 rAiry = 0.61

λ0 NA

increase and 2 comes from elongation). The correlation time of the FCS CF should increase by a factor of 22 = 4 compared to the time measured for the water immersion objective (eq 5a). Although the actual values of the discussed ratios will be different because they depend on the pinhole size and the degree of filling of the objective aperture, the estimation presented above may serve as the first approximation of the confocal volume dimensions, useful e.g. in sample preparation. As will be shown later, such approximation gives reasonable agreement with experimental results. The effect of the value of SP on the contribution of Gxy(t) and Gz(t), to the total CF and on its overall shape, eq 4a, can be seen in Figure 1. For clarity, the two CFs, first with SP = 1 (full

(1)

where λ0 is the wavelength and NA is the numerical aperture of the objective. The first minimum in the axial (z−) direction is located at a distance zmin from the center of the diffraction pattern: zmin =

2λ 0ns (NA)2

(2)

where ns is the refractive index of the object medium. The ratio of the axial to lateral resolution: n zmin /rAiry = 3.28 s (3) NA The FCS CF has been analytically derived,12 for the approximation when the confocal volume has a Gaussian shape with dimensions σx = σy = σxy and σz and SP = σz/σxy: G (t ) − 1 =

1 Gxy(t )Gz(t )GT (t ) N

(4a)

Gxy(t ) = (1 + t /τxy)−1

(4b)

Gz(t ) = (1 + t /τz)−1/2

(4c)

⎛ ⎞ T GT (t ) = ⎜1 + e−t/ τt⎟ ⎝ ⎠ 1−T

(4d)

τxy = σxy 2/D

(5a)

τz = τxySP 2

(5b)

Figure 1. Total FCS correlation functions and their components (eqs 4a−4d) using T = 0 and N = 1 for τxy = 20 μs, SP = 1 (full symbols and solid line) and τxy = 1000 μs, SP = 10 (open symbols and dashed line). Note that only the green lines represent the actually measurable (total) CF, G(t). The components of G(t), Gxy(t) (circles) and Gz(t) (squares), have a different shape and cannot be superimposed by a horizontal shift.

symbols and solid line) and second with SP = 10 (open symbols and dashed line) have been simulated for different τxy (20 and 1000 μs, respectively). For low value of SP = 1, Gxy(t) (squares) and Gz(t) (circles), describing the diffusion in the xy plane and in the z-direction, respectively, decay on a comparable time scale. The total CF (lines), i.e., the product of these two terms, decays on a bit faster time scale, different from that of both components. In the case of a relatively high value of SP = 10 the two terms of the CF decay on time scale different by more than two orders of magnitude due to quadratic dependence of τz on SP (eq 5b). During the time when Gxy(t) decays practically to zero, Gz(t) stays almost constant and equal to 1. As a result, the total CF is practically indistinguishable from Gxy(t), i.e., Gz(t) practically does not contribute to the total CF. Since, formally speaking, Gxy(t) describes the 2-dimensional diffusion in the xy plane, one can say that for large values of SP > 10, the total 3-dimensional CF is reduced to a 2-dimensional CF. For the same τxy increasing SP slows down the decay of the CFs calculated using eq 4a (Supporting Information, Figure S1). The higher the value of SP the smaller the difference between the subsequent CFs. Thus, for a strongly elongated confocal volume the time dependence of the CF is defined only by σx = σy and not by σz any more. Consequently, it should be possible to measure the FCS CF and analyze it using eq 4a (with SP = 20) in the case when the confocal volume is strongly distorted in the zdirection but its xy cross-section in the focal plane is welldefined and circular. This conclusion, of course, is limited to conditions, where the elongation of the confocal volume does

D is the particle translational self-diffusion coefficient, T is the triplet contribution amplitude, and τt is its characteristic decay time. The triplet contribution arises from occasional transitions of fluorophores to the triplet state which relaxes to the ground state in a nonradiative way. Fluorescence fluctuations produced in this way contribute to the FCS CF as an additional exponential decay with relaxation time of the order of single microseconds, typically much shorter than the diffusion times. Although the topic of this paper is not related to the triplet contribution, its presence had to be taken into account in the analysis of our experimental data. A fit of eq 4a to experimental data measured with an immersion objective typically gives 4 < SP < 6, depending on the wavelength, NA value and the size of the pinhole defining the confocal geometry. The refractive index mismatch or the use of dry objectives result in an increase of the confocal volume and elongation of its shape (SP > 5). For example, taking eqs 1−3 as an approximate measure of the confocal volume, a dry objective with NA = 0.6 should produce a focal spot twice broader than a water immersion objective with NA = 1.2 (eq 1). At the same time this focal spot is 22 = 4 times longer compared to the one from immersion objective (eq 2) and hence, its aspect ratio SP is twice larger (SP ≈ 10). Total confocal volume of the dry objective is 23 × 2 = 16 times larger than that of the immersion objective (23 comes from isotropic 8947


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not reduce the molecular brightness of fluorophores to the noise level. However, here we would like to make clear that we do not claim that distortions of the focus may result in deviations of measured diffusion time by not more than 5%. It is obvious that increase of the focal spot may cause a substantial increase of fluorophore diffusion time τxy. 3.2. Effect of Refractive Index−Water Immersion Objective. The influence of the refractive index of the mounting medium of the sample on the FCS CFs has been theoretically studied on the basis of exact wave-optical calculations.4,5 It has been shown in Figure 6 of ref 5 that the change of refractive index of the sample solution from 1.333 to 1.36 results in a decrease of the apparent diffusion coefficient by almost 50%, an increase of the apparent concentration of the fluorescing particles by a factor of 7 and a corresponding decrease of G(0). In these calculations the effect of the correction collar was ignored. In order to check these theoretical predictions and the effect of the correction collar experimentally, we studied diffusion of a dye molecule Alexa488 in water/glycerol mixtures of glycerol concentration increasing from 0 to 50% (by weight) and a corresponding refractive index changing from 1.33 (water) to 1.41 (50% glycerol). A water immersion Zeiss objective (#1) and three long distance objectives, a Zeiss (#4) and two Olympus objectives (#2, #3) listed in Table 1, all equipped with a correction collar, were used for these studies. Since only inverted microscopes were applied, the solutions were contained in a standard NUNC Lab-Tek chambered coverglass cell (Thermo Scientific, USA) of the window thickness of about 0.15 mm. In the current report by depth h we denote the elevation of the objective above the point where the focal spot was at the bottom of the cell and not the real position hr of the focal spot. When the refractive indices of the immersion medium ni and of the sample, ns, are identical, hr and h are equal. In general, h and hr are related by eq 6. ⎡⎛ n ⎞ 2 ⎤ ⎡⎛ n ⎞ 2 ⎤ hr = h ⎢⎜ s ⎟ − 1⎥ /⎢⎜ i ⎟ − 1⎥ ⎝ ⎠ ⎝ ⎠ ⎣ NA ⎦ ⎣ NA ⎦

Figure 2. Normalized number of particles N/N0 calculated from G(0), measured for Alexa488 in water/glycerol mixtures at h = 200 μm using Zeiss water immersion objective (#1) without (bullets) and with (circles) collar correction as well as corresponding values of the normalized diffusion coefficient D/D0 (red boxes and rectangles), respectively. At this depth h full compensation was not possible beyond n = 1.355. The crossed symbols refer to these conditions with only partial compensation.

theoretical calculations of Enderlein et al.,5 who obtained a similar change of both parameters for the refractive index change from 1.333 to 1.36. The same solutions were measured with collar correction; the position of the correcting collar was optimized for each solution separately (Supporting Information, Figure S3c). The collar correction is performed by finding the maximum of the counts per molecule (CPM) at given experimental conditions. In this case G(0) is practically constant, opposite to the CFs measured without collar-correction. The CFs were normalized with the value N obtained from the fit and plotted vs t/η (Supporting Information, Figure S3c-d). This normalization is sufficient for the superposition of all CFs. That means that by the proper adjustment of the correction collar one can fully compensate the distortion of the confocal volume due to the refractive index change in the range 1.33 ≤ ns ≤ 1.355 for measurements at the depth of 200 μm in the sample. This conclusion is corroborated by data presented in Figure 2, where the values of N/N0 and D/D0 obtained with the collar correction are constant and independent of refractive index in contrast to the theoretical calculations5 and measurements performed without collar correction. Molecular brightness expressed in counts per molecule (CPM) without collar correction rapidly decays with increasing spherical aberration, i.e. increasing glycerol concentration (Supporting Information, Figure S3c). With the collar correction CPM decays slowly instead of remaining constant, but this small decay is primarily due to decreasing fluorescence efficiency of Alexa488 with increasing concentration of glycerol. For glycerol concentrations above 15% resulting in values of the refractive index above 1.355 the correction range of the collar of the Zeiss water immersion objective was insufficient and a complete correction of the distorted confocal volume was not possible at this depth of 200 μm. This resulted in increasing values of N/N0 and decreasing D/D0 in a similar way as in the case without compensation. The compensation range of the collar depends on the depth of the focus position h in the sample above the window surface, as shown in Figure 3. As is clearly seen in Figure 3 the largest range of the values of the refractive index that can be compensated using the water immersion Zeiss objective can be obtained when the incident laser beam is focused close to the glass surface. For the lowest value of h = 20 μm used the refractive index of the sample in

(6)

Equation 6 can be easily derived from the Snells law of refraction and from the definition of NA. The experimental CF can be very well fitted using eq 4a and the deviation plot is random (Supporting Information, Figure S2). Correlation functions were measured at a depth of 200 μm above the glass surface for Alexa488 in water/glycerol solutions using the Zeiss water immersion objective (#1) (Supporting Information, Figure S3) without collar correction−the collar was optimized for the pure water solution and left unchanged for all other measurements. The values of the normalized number of fluorescent particles in the confocal volume N/N0 (which is inversely proportional to the amplitude of the CF, see eq 4a) and the normalized diffusion coefficients of the dye D/D0 obtained from the fit to the CFs (Supporting Information, Figure S3a) measured without collar correction are shown as solid squares and circles, respectively, in Figure 2. Parameters N0 and D0 are the number of particles and diffusion coefficient measured for undistorted confocal volume at initial experimental conditions. One can clearly see that the change of the refractive index of water/glycerol Alexa488 solutions from 1.33 to 1.41 results in an increase of N/N0 by a factor of 7 and a decrease of the D/D0 value by 40%. This is in semiquantitative agreement with the 8948


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70% and normalized diffusion coefficient D/D0 remained constant (Supporting Information, Figure S5). The increase of N/N0 by 70% is entirely due to the (intended) increasing misalignment of the correction collar. One can clearly see (Supporting Information, Figure S5) that with the correction collar adjusted for the water solution of ns = 1.335 one can measure the diffusion coefficient for all water/ glycerol mixtures with refractive index in a broad range of 1.335 < ns < 1.402 with an accuracy of ±2%. It is also possible to correct the distortion of the confocal volume and to obtain comparable values of N/N0 using the correction collar. Since for the dry LD objectives the value of the structure parameter is high, SP > 10, the correlation time (diffusion coefficient) is defined practically only by the xy cross-section of the confocal volume, as it has been shown above. We can conclude from Figure S5 of Supporting Information that for the LD objectives the change of refractive index in a broad range of 1.335 < ns < 1.402 does not result in a change of the xy crosssection of the confocal volume (diffusion coefficient) but the confocal volume is getting more elongated, as indicated by the substantial change of the number of molecules, if the correction collar is not adjusted for each refractive index. Thus, one can obtain the correct relative value of the diffusion coefficient from FCS experiments performed using dry LD objectives with the correction collar initially adjusted for the refractive index of 1.335 and not readjusted for the other solutions. The correct value of the number of particles requires a tiny readjustment of the correction collar. Similar measurements were performed using the other two LD objectives (#2, #4 in Table 1). The experimental and normalized with N and viscosity CFs measured using the Zeiss LD and Olympus 40 × 0.55 LD objectives collapse on a master curve when plotted versus (t/η), indicating that reliable relative measurements of the diffusion coefficient can be performed also using these two objectives for samples of refractive index in the range of 1.33−1.40. The effect of refractive index of the sample on the FCS correlation functions measured using the LD objective #3 was also studied by comparing the results obtained for ATTO633 solutions in water (n = 1.333) and dimethylformamid (DMF, n = 1.430). Using the correction collar one can compensate the spherical aberration caused by the high refractive index sample and the same values of the correlation time and number of particles were obtained. Additionally, shape of the correlation functions was the same as the model function (eqs 4a−4d) with no systematic deviations. The correlation times of Alexa488 in water were measured using different objectives of numerical aperture in a broad range: 0.4 < NA < 1.2, thus we can check if the experimental data concerning the values of τ and SP are in agreement with theoretical expectations given by eqs 1 and 3 combined with the general feature of Brownian motion: ⟨x2(t)⟩ ∝ t. As discussed above, the FCS correlation time τ measures the diffusion time through the xy cross-section of the focused incident beam (rAiry in eq 1) if the pinhole size is adjusted properly to 1 AU. Thus, τ should be proportional to (NA)−2. The structure parameter SP obtained from the fit should be proportional to (NA)−1 (eq 3). In order to check these dependencies, τ and SP should be plotted versus (NA)−2 and (NA)−1, respectively. The theoretically predicted dependencies of τ and SP on are quite well fulfilled for these objectives (Supporting Information, Figure S6). That means that reliable measurements of the relative diffusion coefficients can be made

Figure 3. Dependence of the refractive index compensation range of the water immersion Zeiss objective on the depth h of the focal point in the sample. The corresponding refractive index of the mixtures is also given.

the range from 1.335 to 1.40 can be compensated with the correction collar. For h = 200 μm only the range from 1.335 to 1.36 is available with full compensation. The positive effect of using a correction collar has been shown in a study of Chattopadhyay et al.,13 where some dynamic properties of a certain protein were studied by means of FCS as a function of denaturating agent (guanidine hydrochloride, GdnHCl) concentration. Since increasing GdnHCl concentration was changing the sample refractive index and hence also distorting the optical geometry, the use of correction collar was necessary (water immersion objective). The authors also found that for a given GdnHCl concentration, optimum correction collar setting changed with the height of the focal spot above the bottom. 3.3. Effect of Refractive Index−LD Objectives. We have also studied the effect of the refractive index of the sample on the parameters of the measured FCS correlation functions and the compensation range of the collars for the long working distance (LD) objectives (#2−#4) listed in Table 1. These LD objectives are characterized by a much lower NA from 0.4 to 0.6 and much larger confocal volume and SP parameter. In the case of these dry LD objectives the mismatch of the refractive index is very large and results in a very strong spherical aberration and distortion of the PSF.4 This can be compensated with the correction collar but only for a single plane. Thus, any change of the depth in the sample results in a distortion of the PSF and a decrease of intensity. Fortunately, these distortions are smaller for dry objectives with NA < 0.6 which are not very useful for high resolution scanning imaging purposes. However, as we will show here, they are very useful for FCS experiments which require a long working distance and standard methods can be used to analyze the FCS data. In order to demonstrate that, we have performed FCS experiments for the three LD objectives with compensation collars: (i) at different depth− distance of the focus from the glass surface, (ii) for different window thickness, and (iii) for a broad range of refractive index of the sample. The FCS correlation functions were measured at the depth of 200 μm for Alexa488 in water/glycerol solutions of glycerol concentration from 0 to 50% using the Olympus 40 × 0.6 LD objective (#3 in Table 1) (Supporting Information, Figure S4a). The same data normalized using the values of N from the fit were plotted versus time normalized with the solvent viscosity (Supporting Information, Figure S4b). The values of the normalized number of particles N/N0 increased by about 8949


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Figure 4. Obtained from the fit values of the diffusion times of the colloid (red circle) and Alexa488 (■) in water (a) and their ratio (▲) (b) for different apparent penetration depth h of the incident beam into the sample measured with the Olympus long distance objective, NA = 0.6 (#3 in Table 1) with the proper setting of the correction collar. The inset in (b) shows the constant molecular brightness (CPM) of both colloidal particles and dye molecules.

0.6) for the colloid (Silica/Rhodamine B, 164 nm diameter) and Alexa488 in water, Figure 4. One can see that the diffusion times of both samples (Figure 4a) do not depend on the penetration depth and their ratio (Figure 4b) is constant within ±2.5%, also the CPMs are constant, provided that the correction collar is properly adjusted for every penetration depth. 3.4.2. Oil Immersion Objectives. Most of the oil immersion objectives have no correction collar, but they can be used for any window thickness in the range of their working distance, provided that the refractive index of the glass is equal to that of the immersion oil ni ≈ 1.51. The disadvantage of such objectives is that the spherical aberration resulting from the refractive index difference between the immersion oil and the mounting medium (usually water) cannot be compensated. This distortion, acceptable near the glass wall, increases with the depth of the focal point in the sample. A new generation of oil immersion objectives with a very high NA >1.46 was designed for the TIRF applications and is equipped with the correction collar. In order to test how useful are the oil immersion objectives for an FCS experiment, the CFs were measured for Alexa488 solutions in water and a 50% glycerol/water mixture and a suspension of a fluorescently labeled silica colloid in water using Zeiss 63×/1.40 oil (#5) and Olympus 100×/1.49 oil (#6) objectives. The fit parameters obtained for the FCS CFs measured for Alexa488 in water and 50% water/glycerol mixture using the Zeiss oil immersion objective (#5 in Table 1) clearly show that only the CFs measured very close to the surface of the window (h < 3 μm) are not distorted substantially (Supporting Information, Figure S8). With increasing depth in the sample both σxy ∝ √τ ∝ √1/D and σz ∝ SP√1/D increase, indicating an increasing distortion/growth of the confocal volume. Despite this distortion, the CFs can still be fitted reasonably well using the standard form of the CF with an increasing value of the SP. The important question at this point is whether a meaningful FCS experiment can be performed at such conditions. In order to check that FCS CFs were measured for Alexa488 solutions and silica colloid suspensions in water at different depth h using the Olympus oil immersion objective equipped with a correction collar. For all water solutions the correction collar was set to one of the limiting positions and no correction was possible. As one can see in Figure 5, the confocal volume

by means of FCS both with immersion and dry LD objectives equipped with the correction collar. Long working distance objectives suffer from small values of NA (0.4−0.6). The molecular brightness of fluorophores measured with such objectives is by more than one order of magnitude smaller than brightness measured with a water immersion objective of NA = 1.2 using the same (low) laser power. As two major reasons of such an effect we point (i) smaller laser power density in the larger focal spot of LD objectives and (ii) worse efficiency of fluorescence collection due to smaller solid angle of collection. These two features explain the common opinion that LD objectives are not suitable for FCS experiments. Our results clearly show that such experiments are possible and can be very accurate on the cost of slightly longer measurement time (30 min) and careful collar adjustment. 3.4. Effect of the Depth of the Focus in the Sample. 3.4.1. Long Working Distance Objectives. For the systems where the refractive indices of the immersion medium and that of the sample are different, the distortion of the focal volume of the incident beam increases with increasing penetration depth of the beam into the sample.7 This distortion can, to some extent, be compensated using the correction collar of the objective, which has to be adjusted for each depth. In order to check if a complete compensation is possible we have measured this effect for the Olympus 40×/0.55 LD objective (#2 in Table 1). The experiment was performed in the following way: First the correction collar was optimized for a measurement at the depth of 200 μm into the sample and with this fixed position of the collar the FCS CFs were measured for Alexa488 in water at different depth from 10 to 600 μm. The correlation times τ and number of particles in confocal volume N extracted from the fits to the CFs were different for different depths (Supporting Information, Figure S7). Then similar measurements were performed at the same depths after adjusting the correction collar for each depth separately (Supporting Information, Figure S7). The individual adjustment of the correction collar for each penetration depth results in constant values of τ and N for all penetration depths h. Thus, reproducible constant values of not only the diffusion coefficient but also the number of particles can be obtained using LD objectives for a broad range of penetration depths h, if the correction collar is adjusted properly. Similar depth dependent measurements were performed using the other Olympus LD objective (#3 in Table 1, NA = 8950


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Figure 5. Relative changes of the diffusion coefficient D/Dh=0 (■, □, ▲) and number of particles N/Nh=0 (red solid diamond, red open diamond, red triangle) for different apparent penetration depth h of the incident beam into the sample for (a) Alexa488 in water (solid symbols) and in 50% water/glycerol mixture (open symbols), and (b) the silica colloid in water (▲, red triangle) measured with the Olympus (TIRF) oil immersion objective, NA =1.49 (#6 in Table 1) with the correction collar set to one of the limiting positions.

Figure 6. Residua of the fit to CFs measured for a) Alexa488 in water, b) silica particles in water, for different apparent penetration depth h of the incident beam into the sample measured with the Olympus (TIRF) oil immersion objective, NA = 1.49 (#6). For clarity, the data obtained at h = 30 and 60 μm have been shifted by 0.01 and 0.02, respectively. c) The ratio of the diffusion coefficients (■) and number of particles of the colloid to that of Alexa488 in water (red circle) for different apparent penetration depth h of the incident beam into the sample. To avoid crowding effects the colloids were diluted to volume fraction ϕ = 0.005, resulting in N0 ≈ 0.3.

parameters extracted from the fit using the standard form of the CF are changing in a very similar way, as functions of the depth h, both for Alexa488 and the colloid samples. The decreasing values of D/D0 reflect the increase of the beam waist diameter, while much faster increase of (N/N0) values results both from the thickening and relative elongation of the confocal volume confirmed by increasing fitted SP values (from SP ≈ 4 to SP ≈ 20, data not shown). It is very interesting to verify how the strongly distorted confocal volume influences the form of the CFs. In Figure 6 we show the residua of the fitting procedure for CFs measured at three depths h: 5, 30, and 60 μm for Alexa488 in water (a) and silica particles in water (b). The stochastic form of the residua clearly shows that the standard analytical form of CF given by eqs 4a−4d is still valid even for distortion stronger than those analyzed in ref.5 In common opinion TIRF objectives are not recommended for FCS measurements in water solutions. The results presented here show that close to the window they perform even better than water immersion objectives provided that the user has a good control over the focal spot position in the sample and the sample refractive index does not change. Moreover, even when the focal spot is set deeper in the sample (up to 60 μm), the CFs measured with such objectives will have a correct analytical form (eqs 4a−4d), however with reduced signal-to-noise ratio. In the studied h range the TIRF objective effectively performed like an objective with a decreasing NA

value. Of course, we do not recommend to buy TIRF objectives as dedicated to FCS studies in water solutions. We simply suggest that if it is the only available immersion objective, it can be safely used for FCS measurements with some precautions concerning the position of the focal spot in the sample. Finally, we would like to remind that the effective NA value of any objective cannot exceed the sample refractive index. That means that our TIRF objective performed as it would have NA ≈ 1.33 for water samples and NA ≈ 1.41 for 50% water/ glycerol mixtures. As a support for this suggestion, in Figure 6c we plot the ratio of the diffusion coefficients and number of particles for both samples (Alexa488 in water and silica colloid in water), which stays practically constant and independent of the depth of the focal point. This means that a meaningful relative FCS measurement can be performed for sample with the mounting medium of a refractive index much lower (e.g., ns = 1.33) than that of the oil (ni = 1.51) using the oil immersion objectives at a depth from 0 to about 60 μm and the measured CFs can be analyzed in the standard form, despite substantial distortion/ increase of the confocal volume and correlation time. 3.4.3. Silicon Oil Immersion Objectives. A good choice for samples of refractive index ns > 1.4 is the silicon oil immersion (ni = 1.4) objective of Olympus (objectives #7, #8 in Table 1) equipped with the correction collar. Using objective #7 we measured the FCS CFs of Alexa488 in water (ns = 1.33) and in 8951


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Figure 7. Correction range measured for silicon oil immersion objectives. a) 60 × NA 1.3 (#7) for the window thickness: d = dNUNC,d = dNUNC + 0.13 mm and d = dNUNC + 0.16 mm; b) 30 × NA 1.05 (#8) for the window thickness: dNUNC,d = 0.5 mm (Hellma). In b) the long lines (blue and black) correspond to actually measured compensation ranges, while the short ones (yellow and cyan) correspond to compensation ranges limited by the nominal edge settings of the correction collar of objective #8.

volume at a corresponding depth h. The advantage of this optical configuration is that the compensation can be obtained for a much thicker window than originally recommended. When using the correction collar we obtain a range of depths for each n. In order to get some quantitative characteristics of this optical arrangement we have studied water solutions (ns = 1.33) of Alexa488 using the silicon oil immersion (ni = 1.40) objective (#7 in Table 1) with correction collar, where condition (eq 7) is fulfilled (ng = 1.51). First, we measured the FCS CFs using the nominal window thickness of 0.15 mm at different depths h in the sample with the optimum adjustment of the correction collar (Supporting Information, Figure S10). The correct fit parameters of the CFs can be obtained only very close to the window surface (h < 20 μm) and the distortion is increasing with the increasing depth in the sample. However, if we increase the window thickness beyond the nominal value by g = 0.13 mm (total thickness 0.28 mm beyond the nominal range of 0.15− 0.19 mm), the correct values of the fit parameters can be recovered e.g. at a substantial depth of 160 μm although it can not be done close to the window. Such behavior, when the position of the focus close to the window is not optimum for its correct shape is rather unusual, so we performed a series of measurements to find the range of depth at which the correlation function recovers its optimum shape. The limits of this range were determined by the limits of adjustment of the correction collar. The results of such a procedure obtained for two kinds of windows are shown in Figure 7a. Similar measurements were performed using the silicon oil immersion objective of NA = 1.05 (#8). Since, as mentioned above, the correction collar of this objective has a much broader correction range than nominally specified by the manufacturer (0.13−0.19 mm), it was possible to compensate the spherical aberration of water up to a depth of h = 300 μm, using the standard window (dNUNC = 0.15 mm), Figure 7b. The short lines in Figure 7b refer to the nominal limits of collar compensation marked by the manufacturer on the scale (0.13 and 0.19 mm). The long lines reflect the possibility to turn the collar beyond those limits and compensate much stronger optical aberration. This range is not calibrated by the scale on the collar, nevertheless correction is effective. Taking the advantage of the mutual cancellation of the spherical aberration of excess window thickness g and the layer of water h, it was possible to obtain a full compensation of the distortion of the confocal volume even for d = 0.5 mm (0.31 mm above the

50% water/glycerol mixture (ns = 1.4) as a function of the depth of the confocal volume in the sample in the range of 0− 200 μm (Supporting Information, Figure S9). For water solutions the deviations of the measured diffusion coefficient D and number of particles in the confocal volume N from the correct values were increasing with the increasing depth up to 30% and a factor of 5, respectively, for the depth of 150 μm. These deviations could not be compensated with the correction collar. For water/glycerol mixture the values of D and N did not change with the depth and no correction was needed, as expected, since the refractive indexes of the mounting medium and immersion liquid were equal. The correction collar was used only to compensate for the thickness of the glass window. We have seen that in the case of water immersion objective the correction collar is used to compensate the spherical aberration due to the window thickness in the compensation range and/or the excess refractive index of the mounting medium over that of the immersion liquid. However, the correction collar cannot be used to compensate the distortion of the confocal volume when the refractive index of the mounting medium ns is lower than that of the immersion liquid ni (e.g., oil (#6) and silicon oil (#7) immersion objectives with water-based samples). An exception is one of the silicon oil immersion objectives (#8 in Table 1), where the correction collar allows for compensation of the window thickness far below and above the nominal range of 0.13−0.19 mm. The correction range below 0.13 mm opens a possibility to compensate the spherical aberration in the case when ns < ni if a window thickness in the nominal range is used. That allows us to measure water-based samples with full compensation even at a large depth. The features of the objective #8 will be discussed in detail later. Here we show that in some cases there is an interesting possibility to study samples of ns < ni when using the window (of refractive index ng) substantially thicker than the nominal value. This possibility appears when the following criteria are met for an immersion objectives equipped with a correction collar:

ng > ni > ns

(7)

In such a case the additional spherical aberration due to the layer of the mounting medium of thickness h and refractive index ns is compensated by the opposite spherical aberration due to the excess thickness of the window g of refractive index ng. For every g (gmax limited by the working distance WD) we obtain a full compensation of the distortion of the confocal 8952


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3.6. Thick Glass Compensation−LD Objectives. LD objectives equipped with a correction collar usually have a very broad range of glass thickness compensation. It was interesting to verify the actual performance of their compensation system using the decay time and shape of FCS CFs as a measure of their quality. We have prepared dye solutions (Alexa488 and Rhodamine 6G) at appropriate concentration and a set of glass plates of different thickness: NUNC cells 0.15 mm, cover glass 0.15 mm, cover glass 0.19 mm, microscope slide 1 mm, and a diamond plate 1 mm. Glass windows of different thickness were made by assembling different combinations of glass plates using mineral oil as immersion medium. Due to some imperfections of this procedure (e.g., not uniform layer of oil between the plates), the FCS CFs did not overlap perfectly after adjusting the correction collar, but the differences among the individual CFs lied only in their amplitude equal to the reciprocal value of N. In order to account for this effect, in Figure 9 we have plotted the CFs after

maximum nominal compensation range) at a depth h in the range of 200 to 400 μm. 3.5. Effect of Window Thickness−Immersion Objectives. 3.5.1. Water Immersion Objective. The effect of the window thickness on the values of the diffusion coefficient D and number of particles in the confocal volume N extracted from the FCS CFs measured with the Zeiss water immersion objective (40×/1.2W) was studied theoretically.5 It was shown that a deviation of the window thickness from the optimal value by 10 μm results in a decrease of the diffusion coefficient D by 30% and an increase of N by 110%. The effect of the correction collar was not taken into account in the calculations. However, this objective is equipped with a correction collar which should compensate the distortions resulting from different window thickness in the range 0.13 to 0.19 mm. Corresponding results of experimental test of this effect using the same water immersion Zeiss objective (#1) are shown in Figure 8.

Figure 8. R elative changes of the diffusion coefficients D/D0 (●, ○) and number of particles N/N0 (■, □) for window thicknesses deviating by Δd from the nominal value of 150 μm: solid symbols correspond to values obtained with fixed position of the correction collar adjusted for the nominal slide thickness and open symbols represent values obtained when the correction collar was individually adjusted for each window thickness. Figure 9. Normalized FCS CFs of Alexa488 (a, c) and Rhodamine 6G (b) solutions measured using LD objectives (#2: b, c; #3: a) with different window thickness and adjusted collar position. The legends explain the kind and thickness of the window. In multiple layer windows oil was used as immersion medium between the layers. For clarity lines in (b) and (c) have been moved up by 0.5 and 1, respectively.

In this figure we show that the deviation of the window thickness from the reference value of 150 μm results in a decrease of the diffusion coefficient by about 30% and an increase of the number of particles by about 250% (solid symbols) if the correction collar is adjusted for the nominal thickness and remains unchanged for all other window thicknesses. The changes of D and N (solid symbols) are in semiquantitative agreement with the results of theoretical calculations.5 These distortions can be fully compensated and the values of D and N remain constant for all window thicknesses (open symbols) if the correction collar of the objective is individually adjusted for each window thickness. 3.5.2. Oil Immersion Objectives. The refractive index of the immersion oil used for the oil immersion objectives is equal to that of the glass window (usually ni = 1.51) so they can be used for any window thickness in the range of their working distance. Most of these objectives have no correction collar, the distortions due to the difference of the refractive indexes between the sample and immersion oil cannot be compensated. In case of silicon oil immersion (ni ≈ 1.4) the interplay between negative and positive aberration of the sample with ns < ni and glass window opens space for broadening the range of glass thickness as has been shown above.

normalizing them with N. Figure 9 shows the results of three series of experiments performed with different dyes (Alexa488 and Rhodamine 6G) and different ranges of total window thickness. As one can see from the almost perfectly overlapping CFs, the correction system of the two studied objectives performs very well in the whole range of declared correction. Moreover, in Figure 9c we show that it is also possible to use a 1 mm thick diamond plate (ng = 2.415) and still obtain a CF with the same decay time and overall shape as the one measured using a regular glass plate.

4. CONCLUSIONS In this paper the optimal conditions for a successful FCS experiment and possible distortions of the measured CFs are discussed and compared with the theoretical calculations.5 In an imaging experiment using a confocal scanning microscope it is 8953


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objectives, i.e. the excess refractive index of the sample over that of the immersion liquid and the thickness of the window can be compensated up to the nominal maximum value. When ns < 1.4, it is possible to compensate the spherical aberration resulting from window thickness much larger than the nominal value since the aberration introduced by the layer h of the mounting medium of ns < 1.4 is compensated by an aberration of opposite sign resulting from the excess thickness g of the window of ng ≈ 1.5. The use of the correction collar allows to obtain compensation for a range of depths for each g, which is limited by the working distance of the objective. Thus, using the silicon oil immersion objectives for solutions with ns < 1.4 we can use cells with much thicker windows than the maximum value (usually 0.19 mm) and obtain full compensation of the spherical aberration for a range of depths in the sample. This is a very useful possibility for samples dissolved in organic liquids, for which the commercial NUNC (or other cover glass - based) cells cannot be used. Additionally, since the correction range of the silicon oil immersion objective of NA = 1.05 allows to compensate the distortion due to both much thinner and much thicker windows than the nominal range (of 0.13−0.19 mm), it is possible to perform undistorted measurements using cell of the nominal window thickness in water-based samples (ns < ni). The LD objectives equipped with the correction collar can be used to measure undistorted CFs for the nominal glass thickness up to 2.3 mm, depending on the objective, and a diamond window thickness of 1 mm. Thus, they can be used for custom-made FCS cell which allow temperature and pressure change in a broad range and require long working distance and possibly high NA. We have shown that even for strongly distorted confocal volume relative FCS measurements are possible, that is eq 4a can be safely used for estimation and comparison of the diffusion times of different fluorophores, as long as the optical conditions defined by the glass thickness, sample refractive index, correction collar settings, and the distance from the glass remain constant.

very important that the confocal volume and the PSF are optimal. Analyzing the mathematical form of the CFs we show that a successful FCS experiment can be performed and the measured CFs can be analyzed in the standard form even for a very elongated confocal volume. In this case practically only the xy cross-section of the confocal volume defines the shape of the CF. In our experiments one water-immersion, two oil-immersion, and two silicon oil-immersion objectives as well as three long working distance dry objectives were used. We have shown that the distortion of the CFs resulting from the mismatch of the refractive indexes n of the mounting medium and the immersion fluid for the water immersion objective predicted from calculations in ref 5 can be compensated in a relatively broad range of ns using the correction collar of the objective. The compensation range depends on the depth of the confocal volume in the sample: the smaller the depth the larger the range. In the case of full compensation the correct correlation time (diffusion coefficient) and number of particles can be obtained using the conventional analytical form of the fit function. In the case of the long distance dry objectives equipped with the correction collar the correct values of the correlation times of the CFs, which depend mainly on the xy cross-section of the confocal volume, can be obtained for a broad range of the refractive index of the mounting medium from 1.33 to 1.4 even without using the correction collar. In order to obtain the correct value of the number of particles in the confocal volume, N, the correction collar has to be used. The use of LD objectives reduces substantially the molecular brightness of fluorophores. For equal total laser power the loss compared to a water immersion objective is more than 10-fold (change of NA from 1.2 to 0.6). However, it can be partially compensated through the increase of the laser power to the level corresponding to comparable power density in the beam waist in both cases. As can be seen in Figure S7 of Supporting Information, the quality of CFs measured with LD objectives is satisfactory after a few minute measurement. Both the correlation time τ and structure parameter SP measured with different objectives scale with the numerical aperture NA in the expected way: τ ∝ NA−2 and SP ∝ NA−1. For long working distance objectives we show that full compensation of distortions, i.e., a constant correlation time τ and number of particles N can be obtained from CFs measured at a different depth in the sample if the correction collar is adjusted properly. Also the relative diffusion coefficient can be determined correctly at different depths by fitting the CFs in a standard form. The oil immersion objectives without the correction collar can be used for any glass thickness provided that the refractive indexes of the glass and immersion oil are equal. However, substantial distortions of the CFs occur when these objectives are used for FCS measurements on samples of refractive index ns smaller than ni. In this case the distortion increases with the increasing depth of the confocal volume in the sample. Nevertheless, also in this case the CFs can be analyzed in the standard form and correct relative measurements of the diffusion coefficient and number of particles can be performed at any depth. Silicon oil immersion objectives (ni ≈ 1.4) equipped with the correction collar when used for samples in a mounting medium of refractive index ns > 1.4 behave like water immersion

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ASSOCIATED CONTENT

S Supporting Information *

Additioinal figures (S1−S10) are presented with calculated and experimental FCS correlation functions used to obtain the results presented in the paper and additional dependencies of the parameters of the CFs on the sample refractive index, penetration depth, and window thickness for some objectives. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Dr. Johan Buitenhuis for providing us with the labeled silica colloid samples, Dr. Anna MarcinkowskaGapińska for viscosity measurements of water/glycerol mixtures. J.G. and A.P. acknowledge financial support of the Grant PBS1/A9/13/2012 of the National Centre for Research and Development and partial support by the Polish National 8954


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Science Centre (Grant No. 2011/01/B/ST3/02271 and 2013/ 09/B/ST3/01678).

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REFERENCES

(1) Elson, E. L.; Magde, A. Fluorescence correlation spectroscopy. I. Conceptual basis and theory. Biopolymers 1974, 13, 1−27. (2) Fluorescence Correlation Spectroscopy. Theory and Applications; Rigler, R., Elson, E. S., Eds.; Springer-Verlag: Berlin, 2001. (3) Ries, J.; Schwille, P. Fluorescence correlation spectroscopy. Bioessays 2012, 34, 361−368. (4) Egner, A.; Hell, S.W. In Handbook of Biological Confocal Microscopy; Pawley, J. B., Ed., Springer: New York, 2006; pp 1−16. (5) Enderlein, J.; Gregor, I.; Patra, D.; Dertinger, T.; Kaupp, U. B. Performance of Fluorescence Correlation Spectroscopy for Measuring Diffusion and Concentration. ChemPhysChem 2005, 6, 2324−2336. (6) Hess, S.T.; Webb, W.W. Focal Volume Optics and Experimental Artifacts in Confocal Fluorescence Correlation Spectroscopy. Biophys. J. 2002, 83, 2300−2317. (7) Keller, E. In Handbook of Biological Confocal Microscopy; Pawley, J. B., Ed.; Springer: New York, 2006; pp 145−160. (8) Jukaitis, R. In Handbook of Biological Confocal Microscopy; Pawley, J. B., Ed.; Springer: New York, 2006; pp 239−250. (9) Verhaegh, N. A. M.; van Blaaderen, A. Dispersions of Rhodamine-Labeled Silica Spheres: Synthesis, Characterization, and Fluorescence Confocal Scanning Laser Microscopy. Langmuir 1994, 10, 1427−1438. (10) Cheng, N.-S. Formula for the Viscosity of a GlycerolWater Mixture. Ind. Eng. Chem. Res. 2008, 47, 3285−3288. (11) Inou, S. In Handbook of Biological Confocal Microscopy; Pawley, J. B., Ed.; Springer: New York, 2006; pp 1−16. (12) Nägele, G.; Dhont, J. K. G.; Meier, G. Diffusion in Colloidal and Polymeric Systems, in Diffusion In Condensed Matter, 2nd ed.; Heitjans, P., Krger, J., Eds.; Springer-Verlag: Berlin, 2005; pp 619− 715. (13) Chattopadhyay, K.; Saffarian, S.; Elson, E. L.; Frieden, C. Measuring Unfolding of Proteins in the Presence of Denaturant Using Fluorescence Correlation Spectroscopy. Biophys. J. 2005, 88, 1413− 1422.

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