4 Fluorescence Correlation Spectroscopy 1
4.2 Theoretical Concept
4.2.2 Cross-correlation Analysis
An important extension of the FCS is the simultaneous detection of the two colour-channels [Eig94, Ket98, Ric89, Rig98, Sch97]. Two spectrally distinct dyes are excited at two different wavelengths. Their fluorescence light is split subsequently by dichroic mirrors and filters into the respective detection channels. With two-colour FCS, a cross-correlation analysis can be made (FCCS), where the signals of the two channels are time correlated with each other to collect information only about particles that carry both types of dyes. This method of data analysis is useful if the difference in the molecular masses, and therefore in diffusion times, of two binding partners is not large enough for autocorrelation analysis.
The cross-correlation theory can be described in analogy to the autocor-relation theory. Generalizing equation (4.6), the fluctuating signals in the two detection channels (green channel G and red channel R) are given as
δFG(t) =
with the concentrations of the single labelled speciesCi(~r, t), and the con-centration of the double-labelled species CGR(~r, t). Now the number of molecules NGR carrying both fluorophores in the volume is given by
NGR = NGNR
NCC (4.19)
where NG and NR represent the particle numbers in the two autocorrela-tion channels respectively, and NCC is the inverse amplitude of the
cross-4 Fluorescence Correlation Spectroscopy
correlation. This is only valid if there is no cross-talk, meaning the fluor-escent light of the green dye is also detected in the red detection channel.
The effective volume and the diffusion time for the cross-correlation function are given by [Sch97]:
Veff = π3/2(rG2 +rR2) (zG2 +zR2)1/2
23/2 (4.20)
τGR = (r2G+rR2)
8DGR (4.21)
Therefore, when different laser spot sizes are used, the cross-correlation curve lies between the two autocorrelation curves.
4.2.3 Artefacts
Refractive Index
An important issue when focusing a laser through a high-NA objective and detecting through the same objective is the refractive index of the sample.
Focusing and detection will be optimal only if the refractive index of the sample exactly matches that of the immersion medium of the objective.
In all experiments a water immersion objective was used, therefore the refractive index of the immersion medium is 1.33. However, when per-forming FCS in buffer solutions or at various temperatures, the refractive index of the sample may be differ from the refractive index of the immer-sion medium. The refractive index mismatch introduces slight aberrations into the laser focusing and the confocal fluorescence detection, and leads additionally to a slight offset of center of focus and center of detection along the optical axis. Both effects lead to an apparently larger detection volume and thus to an apparently smaller diffusion coefficient. The effect of the refractive index mismatch on the focus is illustrated in figure 4.3.
The calculation was done by J. Enderlein, J¨ulich. As it can be seen also from this figure, the influence on the focus, and therefore on the volume, is much smaller when setting the focus closer to the surface of the cover
Theoretical Concept
Figure 4.3: The influence of the refractive index mismatch on the focus (by J. En-derlein, J¨ulich). (A) Calculation for the refractive index of the used buffer at 10◦C (n=1.3360), the focus is placed 200µm above the surface of the cover slide. The volume is 0.64 fL. (B) Refractive index of the used buffer at 40◦C (n=1.3329), same focus distance. The volume is 0.43 fL. (C) Same as in (A), but the focus is placed 20µm above the surface of the cover slide. The volume is now 0.43 fL. (D) Same as in (B), same focus distance as in (C). The volume is now 0.42 fL.
slide. Therefore, some of the measurements in this work were performed at a distance of 50µm instead of 200µm. Measuring closer to the coverslide surface has one drawback: the diffusion of the particle may be influenced by the surface.
Foci displacement FCCS
In a two-colour setup, additional aspects have to be considered. First, in nonideal systems there can be a considerable amount of cross-talk due to the broad absorption and emission spectra of the dyes. It must be taken into account that both dyes may be excited to some extent by both lasers and emit in both spectral detection ranges. Selecting a proper dye sys-tem (although it is hardly possible to find a perfectly separable dyes) the effect of cross-talk can be neglected. Second, the size of the observation volumes is wavelength dependent. The linear dimensions are in first or-der proportional to the excitation wavelength, resulting in focal volumes differing by a factor of 2.18 for the commonly used laser lines of 488 nm
4 Fluorescence Correlation Spectroscopy
Figure 4.4:Influence of the displacement on the cross-correlation curve. (A) Cross-correlation curve, (B) normalized cross-Cross-correlation curve. The diffusion time of a particle with a concentration of ∼10 nM in a perfectly aligned setup would be 218µs, the lateral and axial distances for the blue and red channel are 160 nm, 800 nm, 208 nm, and 1040 nm, respectively. The lateral and axial displacement of the two foci of 50 nm in x- and y-direction and 100 nm in z-direction leads to a decrease of 15 % in amplitude (A), and an increase in diffusion time to 252µs (B).
and 633 nm. Third, the centres of the respective foci may show an axial or lateral displacement, due to chromatic aberrations along the optical path.
In confocal laser scanning microscopy, values between 50 and 100 nm have been reported [Ede99,Esa00], which is of the order of the beam waist and thus not negligible. The displacement produces a noticeable reduction of the amplitude and a slightly slower decay of the correlation function, cor-responding to an apparently longer diffusion time, as shown in figure 4.4 [Wei02]. The calculation was performed for a fluorophore with a concentra-tion of∼10 nM and a diffusion time in a perfectly aligned setup of 218µs.
The two lateral and axial distances r0 and z0 have a value of 160 nm and 800 nm for the blue channel, and a value of 208 nm and 1040 nm for the red channel, respectively. The centres of the foci are displaced by 50 nm in x and y-direction and by 100 nm in z-direction. This displacement leads to a decrease of 15 %, and the diffusion time is increased to 252µs.
Results
4.3 Results
4.3.1 Binding of RPA to ssDNA
In addition to the SPR experiments, the binding of RPA to the 26-mer labeled with Cy5 was also studied by FCS. The same system, including the buffer, was used for this study. For the measurements, a HeNe-laser with 5 mW power at 633 nm was focused by a water immersion Zeiss C Apochromat 40×1.2 objective. A 90µm pinhole was used in the confocal detection channel. The focus of the lens was placed 200µm above the surface of the cover slide.
Normalized autocorrelation curves for the ssDNA–RPA interaction for different RPA concentrations are shown in figure 4.5. For the determ-ination of the diffusion times of ssDNA and ssDNA–RPA complex the structure parameter S is required, which can be obtained from the meas-urement of Cy5. The diffusion times of free and protein-bound ssDNA were determined from the autocorrelation functions with no protein and excess of protein in the solution from the fit to a one component model, (equation (4.15)). From this a diffusion time τDNA = 152.1±2.4µs for the free DNA and τcomplex = 450.8±5.2µs for the complex could be ob-tained at 25◦C. The measured diffusion time of the complex can by used to determine the hydrodynamic radiusRh by the Stokes-Einstein equation
D= k T
6πηRh (4.22)
where η is the viscosity of the solution. Calculations yield a value of
∼ 7 nm, which is in good agreement with AFM ([Lys04] and cryo-TEM experiments.
The progress of complex formation as a function of RPA concentration is depicted in figure 4.6, which is obtained by a reverse titration of RPA against ssDNA at a constant concentration. In order to determine the equilibrium constantKD, the measured diffusion time has to be converted into the degree of binding. For that purpose the autocorrelation
func-4 Fluorescence Correlation Spectroscopy
1 E - 3 0 . 0 1 0 . 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0
1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0
G(τ)
τ [ m s ]
Figure 4.5: Normalized autocorrelation functions for different RPA concentrations at 25◦C. The percentage of complex was determined by a two component fit to each function. Curves for 0 % (—), 40 % (—), 81 % (—) and 100 % (—) ssDNA–
RPA complex are shown. An increase in diffusion time with increasing complex fraction can clearly be seen.
0 5 1 0 1 5 2 0
1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0
τ [µs]
c ( R P A ) [ n M ]
Figure 4.6: Progress of complex formation at 25◦C as a function of RPA concen-tration.
Results tions for the complex formation were fitted using equation (4.16) for two particles (K = 2), i.e. ssDNA molecules and ssDNA–RPA complexes.
The two diffusion times τDNA and τcomplexobtained from measurements of the free ssDNA and at saturating protein concentrations were kept fix for this fitting procedure. The degree of binding θ = φ2 = 1−φ1 remained the only unknown parameter and was determined from equation (4.16) at each titration point.
The relationship between the equilibrium constant and the degree of binding θ can be derived from the law of mass action:
KD= (1−θ)[B]([A]−θ[B])
θ[B] (4.23)
where [A] is the RPA concentration, [B] is the concentration of ssDNA, andKD is the equilibrium constant for the dissociation. Rearranging this equation and solving the quadratic formula leads to
θ = [A] + [B] +KD±p
[B]2−2[A][B] + 2[B]KD+ 2KD[A] + [B]2+KD2 2[B]
(4.24) Only the solution where θ = 1 for high [RPA] is a meaningful solution, therefore
The resulting data (degree of binding as a function of RPA concentration) and the corresponding fit using equation (4.26) is shown in figure4.7. The fit resulted in a value of 2.61±0.80×10−10M for KD at 25◦C. In order to ensure that the dissociation and formation processes are not kinetically
4 Fluorescence Correlation Spectroscopy
0 5 1 0 1 5 2 0
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
bound fractionθ
c ( R P A ) [ n M ]
Figure 4.7:Degree of binding at 25◦C as a function of RPA concentration.
hindered, the complex formation was observed also by adding protein to ssDNA, yielding the same value for the equilibrium constant. Comparing this value with the one obtained by SPR, one finds that the equilibrium constant is larger by a factor of 25. In order to explain the difference between the equilibrium constants, one has to take also a closer look at the thermodynamics of biomolecular interactions (see 4.3.3).
In the above FCS experiments the quantum efficiency of the dye was not influenced by the binding process. Previous experiments showed that the binding of DNA to RPA is not altered significantly by attaching a label to the DNA [Hey01]. It was found that the equilibrium constant remained the same within the experimental error for labelled and unla-belled DNA. Moreover, there was no interaction between RPA and free Cy5. Therefore significant deviations due to labelling can be ruled out in the experiments. Photobleaching occurred at laser attenuation above 0.7 %, therefore, the experiments were performed below this threshold value. Photobleaching also occurred at RPA concentrations above a con-centration of 50 nM, which indicates that a slowly diffusing complex with
Results two or more RPA molecules is formed. It is known that RPA forms dimers or trimers upon binding to DNA [Bla94], depending on the length of the DNA.