Anal. Chem. 2004, 76, 1963-1970
Systematic Error in Fluorescence Correlation Measurements Identified by a Simple Saturation Model of Fluorescence Goro Nishimura* and Masataka Kinjo
Research Institute for Electronic Science, Hokkaido University, N12W6, Sapporo 060-0812, Japan
The distortion of the fluorescence correlation function of a dye solution becomes larger with the increase in the excitation power, and eventually the parameters, such as the number of molecules and the diffusion time, obtained by the fluorescence correlation function systematically change. The most fundamental reason for this change is thought to be the distortion of the Gaussian excitationdetection field due to the saturation of the photocycle of the chromophore. The deviation from linearity of the fluorescence intensity causes the distortion of the fluorescence correlation function. Consequently, a smaller excitation power reduces the distortion and ensures an accurate measurement of the absolute value of these parameters. At the same time, the measurements at a fixed excitation power can be used to quantitatively determine the relative value of concentration and of the diffusion time. The deviation in the linearity of the fluorescence intensity and the deviation of the parameters show a high degree of correlation, and a 10% deviation of the intensity results in a prediction of a ∼10% deviation in the number of molecules and a ∼5% in the diffusion time.
The number of molecules and the diffusion time are obtained by FCS in solution, and eventually FCS can be directly applied to the analysis of the chemical reaction with a dilute sample at a very minuscule volume. In particular, the amplitude of FCF directly gives information on the concentration, and therefore, a simple amplitude analysis has been proposed.7-9 The amplitude, however, was strongly affected by many factors, such as the photochemical process and the experimental artifacts. In the experiments, we have found a systematic change in the amplitude of the correlation function with the excitation power, and this has not been adequately explained by results in other studies. Therefore, we aim to clarify and explain the FCFs with the excitation power dependence and discuss the systematic error in the number of molecules and diffusion time, which are obtained from the FCFs. In this paper, we discuss the intensity dependence of the correlation function by experiments and Monte Carlo (MC) simulation. An MC simulation including a three-state photocycle is very simple and straightforward but at the same time accurately traces the emission from the molecules. Then, we show that the saturation effect is a crucial factor in the measurement of the correlation function and discuss the systematic error in the measurement under this condition.
Fluorescence correlation spectroscopy (FCS) is well known as a sensitive, accurate, and convenient analytical tool at the singlemolecule level by means of confocal optics and has been applied to the analysis of various fluorescent labeled molecules in solutions and in much more complex systems, such as membranes, cytosols, and organelles in cells.1,2 FCS has become a major movement within the fields of biophysics, medical science, and analytical chemistry. The theory of the fluorescence correlation function (FCF) of dyes diffusing in an open volume defined by optics was previously established in the 1970s.3,4 Later, the theory was extended to dyes diffusing in the three-dimensional (3D) Gaussian field with photophysics.5,6 FCS data have been since analyzed under this framework.
EXPERIMENT The correlation functions were measured by FCS systems (ConfoCor2, Carl Zeiss, Jena, Germany) without any modification. The pinhole was adjustable in the range from 30 to 100 µm in diameter, and the 50-µm diameter was primarily used. A fluorescein dichroic mirror set was used for excitation and detection in rhodamine 123 measurements, so that all samples were excited by a 488-nm line of Ar+ laser and the fluorescence emission was detected with a 505-550-nm band-pass filter. The excitation power was varied with a variable ND filter and a laser current. The laser power was measured at the output of an objective lens (CApochromat 40×/1.2W, Carl Zeiss) with a photodiode power meter calibrated at 490 nm (TQ8215, Advantest, Tokyo, Japan). Dyes, rhodamine 123 (R123) and rhodamine 6G (R6G), were purchased commercially (Sigma, St. Louis, MO). No further
* To whom correspondence should be addressed. E-mail:
[email protected]. hokudai.ac.jp. (1) Rigler, R., Elson, E. S., Eds. Fluorescence Correlation Spectroscopy: Theory and Applications; Springer-Verlag: Berlin, 2001. (2) Hess, S. T.; Huang, S.; Heikal, A. A.; Webb, W. W. Biochemistry 2002, 41, 697-705. (3) Magde, D.; Elson, E.; Webb, W. W. Phys. Rev. Lett. 1972, 29, 705-708. (4) Elson, E. L.; Magde, D. Biopolymers 1974, 13, 1-27. 10.1021/ac034690b CCC: $27.50 Published on Web 03/05/2004
© 2004 American Chemical Society
(5) Rigler, R.; Mets, U ¨ lo.; Widengren, J.; Kask, P. Eur. Biophys. J. 1993, 22, 169-175. (6) Widengren, J.; Mets, U ¨ lo.; Rigler, R. J. Phys. Chem. 1995, 99, 13368-13379. (7) Nishimura, G.; Rigler, R.; Kinjo, M. Bioimaging 1997, 5, 129-133. (8) Kinjo, M.; Nishimura, G. Bioimaging 1997, 5, 134-138. (9) Kinjo, M.; Nishimura, G.; Koyama, T.; Mets, U ¨ lo.; Rigler, R. Anal. Biochem. 1998, 260, 166-172.
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purification was performed. The dyes were dissolved in filtered distilled water, and the concentrations were determined by a spectrometer (UV3000, Shimadzu, Kyoto, Japan) with the molar absorption coefficients 7.5 × 104 M-1 cm-1 for R123 and 1.0 × 105 M-1 cm-1 for R6G. The sample, 50-100 µL, was dropped on a coverslip chamber (Labtek 155411, Nalge Nunc, Naperville, IL) at ∼25 °C. The evaporation of the sample was negligible in our experiments. The chamber was treated with a protein blocker (BlockAce, Snow Brand Corp., Tokyo, Japan) to measure the absolute value of the number of molecules to reduce the unspecific adsorption of dyes onto the glass surface.10 In all measurements, the blocker greatly reduced the adsorption of rhodamine dye. However, in the experiment using the R6G solution, the adsorption to the chamber was still significant and the blocker prevented the adsorption only partially. Therefore, the main experiments were carried out with R123 solution. The correlation amplitude and diffusion time were obtained by the fitting with the FCF for the three-dimensional Gaussian model
〈I(t)I(t + τ)〉
g(τ) )
〈I(t)〉2
-1)
1 n
[
1
(1 + τ/τc)x1 + τ/τc/q2
+
]
fT exp(-τ/τT) (1)
where n and τc are the number of molecules in the open volume defined by optics and the diffusion time defined by ωxy2/4D, respectively. The structure parameter q was the ratio of the e-2 position at optical axis z and the e-2 radius in focal plane, ωz/ωxy. The amount of the open volume can be obtained by the e-2 lengths as π3/2ωxy2ωz. fT and τT are the fraction and time constants of the photocycle due to the triplet relaxation, respectively. The expressions of the parameters, fT and τT, are described in the literature.6 The diffusion time and the structure parameter varied only slightly (less than 10%) in the experiments day by day. The diffusion coefficient, 3.2 × 10-6cm2/s, of R123 solution was obtained by the following ratio of the diffusion times: DR123/DR6G ) τR6G/τR123. SIMULATION A simple classical simulation, including the photocycle, was carried out by the MC method to obtain the FCF. The excitation field at position r was defined by a Gaussian field, Iex(r) ) IexΦ(r), with Φ(r) ) exp{-2(x2 + y2)/ω02 - 2z2/ωz2}, where ω0 () ωi)x,y) and ωz are the radius of e-2 intensity on the xy plane and distance of it on the z axis, respectively. The simulation space was defined by a rectangular box of 20ω0 × 20ω0 × 20ωz in dimension centered in x, y, and z coordinates (the wall positions were Li)x,y,z ) (10ωi)x,y,z). The unit of length was defined by ω0 ) 1. The unit of time was defined by the diffusion time τ0 ) 1/4. The isotropic diffusion, D ) Di)x,y,z, was assumed. The simulation was initiated from a homogeneous distribution of the ground-state molecules in the simulation space. The simulation step consisted of two stages. The first stage was the calculation of the positions of all molecules with the (10) Pack, C.-G.; Nishimura, G.; Tamura, M.; Aoki, K.; Taguchi, H.; Yoshida, M.; Kinjo, M. Cytometry 1999, 36, 247-253.
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isotropic diffusion with D ) 1 and the step period ∆t ) 2.5 × 10-5() 10-4τ0). The particle, crossing the wall, was coming in from the opposite side of the wall. This assumption might cause an anomaly in the correlation function somewhat at long lag times. Then, the second stage was the calculation of the photocycle of each molecule. The Jablonski scheme of the molecule was assumed.11 In this stage, the molecule could emit a photon once. The total emission was detected by a 1-bit detector with the efficiency η ) 0.05 or 1. These steps were looped until the appropriate number of steps (>108 steps) or the predetermined number of the detected photons (224 photons) were obtained. Finally, the correlation function was calculated from the photon arrival time record.12 The MC simulation was carried out with the following parameters: ωz/ω0 ) 5, ktot∆t ) 1.5, φ() k21/ktot) ) 0.8, k23∆t ) 4.5 × 10-4, and k31∆t ) 0.0015, the detection efficiency η ) 0.05 or 1, and the total number of molecules N ) 200. Here, ktot ) k23 + k21 + knr is the total relaxation rate, where k23, k21, and knr are the intersystem crossing rate, the radiative relaxation rate, and the nonradiative relaxation rate, respectively. k31 is the triplet relaxation rate. The step period ∆t was presumably a small enough period compared with the diffusion time, and the number of photons in this period was judged small enough (,1). These kinetic parameters simulate the values of an R6G solution when a typical value of diffusion time, τ0 ) 50 µs, is assumed. In this case, the concentration is 0.6 nM and the average number of molecules in the excitation volume is predicted to be 0.14. The simulation step, ∆t, corresponds to 5 ns. The excitation range of simulation is σIex∆t ) 0.02-8.0 and compatible with the peak power density range from 8.5 kW/cm2 to 3.4 MW/cm2 when the cross section σ ) 1.8 × 10-16cm2 and the excitation wavelength λ ) 514.5 nm are assumed. In this calculation, the most critical difference from the experiment is the geometry of the excitation-detection field. In the experiment, the excitation field is primarily considered a Gauss-Lorentzian field and the pinhole aperture in front of the detector selects the center region of a fluorescence image to get the confocal detection. The spatial shape of the excitationdetection field is determined by the convolution of the excitation field and the aperture function; one can approximate it by employing a 3D Gaussian field, which is used in the derivation of the FCF in FCS. In contrast, the 3D Gaussian excitation field without the pinhole selection is assumed in the MC calculation. When the excitation power increases, the saturation at the center region of the emission field begins and the excitation-detection field becomes distorted. The shape of the center region of the 3D Gaussian excitation-detection field is primarily determined by the center region of the excitation field when the confocal aperture is not too small. In the usual FCS measurements, including our experiments, the aperture size is not too small because the detection efficiency should be maintained at a sufficiently high level. Therefore, the beginning of the distortion of the excitation-detection field will be approximated to the distortion of the 3D Gaussian emission field. The expansion of (11) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Plenum Press: New York, 1986. (12) Scha¨tzel, K. Single-photon correlation techniques. In Dynamic Light Scattering, The Method and Some Applications; Brown, W., Ed.; Oxford University Press: Oxford, U.K., 1993.
Figure 1. FCFs of a 4.1nM R123 in water at different excitation powers with a pinhole at 50 µm in diameter (open circles, 17 µW; solid circles, 48 µW; open squares, 169 µW; solid squares, 317 µW; open triangles, 624 µW). The inset figure is the normalized FCFs with the number of molecules to 1. The number of molecules was obtained by the fitting to eq 1. The fitting curves and three typical residuals of the fitting are plotted by solid lines.
the distortion from the center region with an increase in the excitation power causes the deviation from the 3D Gaussian field approximation of the simulation. Eventually, the simulation should agree with the experiment when the excitation is weak, although it will deviate from the actual experiment under a very high excitation region. We believe that our simulation should approximate the experimental conditions when the excitation power is not excessively high. RESULTS AND DISCUSSION Distortion of FCFs with Different Excitation Powers in Rhodamine 123 Solutions. Figure 1 shows the FCFs of a 4.1 nM R123 solution at different excitation powers from 0.017 to 0.62 mW. Each correlation curve was obtained by the average of four successive measurements, each with a 180-s accumulation. The amplitude of correlation functions was reduced strongly at higher excitation powers. In contrast, the diffusion time increases with an increase in the power as shown in the inset of Figure 1. Fast decay is also observed at a fast region of the correlation function (∼ µs) at a high excitation range, and it can be attributed to the triplet conversion. The results of a R6G solution were essentially the same as those of the R123 solution (data not shown) except for the disagreement in the absolute concentration due to the strong adsorption onto the chamber glass surface. The background due to the stray light, thermal noise, and scattering by solvent was less than 5% of the total fluorescence intensity, so it is negligible in our discussions. Therefore, our observation of the correlation amplitude with a R123 solution at different excitations does not reproduce the previous results with a R6G solution.6 The fitting to the results with eq 1 was performed with free parameters, n, τ0, q, fT, and τT. The fitting weights are evaluated from the variance of the four data points at each lag time. The fast region affected by an after-pulsing distortion (
