Relaxation Kinetics by Fluorescence Correlation Spectroscopy: Determination of Kinetic Parameters in the Presence of Fluorescent Impurities

Abstract

The use of Fluorescence Correlation Spectroscopy (FCS) in combination with Förster Resonance Energy Transfer (FRET) is gaining popularity as a tool to investigate kinetics in equilibrium conditions. The technique is based on the study of fluorescence fluctuations in small numbers of molecules, and is particularly well-suited to investigate conformational dynamics in biopolymers. In practice, its applicability is often hindered by the presence of certain impurities such as partially labeled biomolecules, excess of free fluorophore, or partially dissociated multi-subunit complexes. Here, we show that the simultaneous measurement of the fluctuations in the donor and acceptor intensities allows the determination of the kinetic relaxation time of the reaction in the presence of donor-only particles when cross-talk is negligible, or in cases where all species have the same diffusion coefficient. Theoretical predictions are supported with the results of Monte Carlo simulations, and demonstrate that the applicability of the technique is more general than previously thought.

The use of Fluorescence Correlation Spectroscopy (FCS) to investigate conformational dynamics in biopolymers has increased in popularity during the last decade.^1–6 A common approach takes advantage of the strong distance-dependence of the Förster Resonance Energy Transfer (FRET) phenomenon. The method involves attaching covalently a donor-acceptor FRET pair to the biopolymer so that the timescales of the fluctuations in donor and acceptor fluorescence intensity can be used to determine the relaxation time (τ_R) of the process under equilibrium conditions.^2,7 Briefly, the experiment involves illuminating the sample with a laser that excites selectively the donor, and measuring the fluorescence intensity of the donor and acceptor in an optically-restricted volume containing a low number of molecules in solution (fig. 1). Note that the acceptor is not excited directly by the laser, but only through energy transfer from the donor excited state. Fluorescence fluctuations are then quantified using the auto-and crosscorrelation functions (G_DD, G_AA and G_DA), which are subsequently fitted to the appropriate physical models to extract the desired kinetic information (i.e. k₁₂ and k₂₁).^2,8

Figure 1.

Schematics of the FCS experimental measurement and chemical system investigated in this work. Point molecules diffuse freely throughout a Gaussian observation volume (green surface). Species 1 and 2 exist in equilibrium, and interconvert with rates k₁₂ and k₂₁ for the forward and backward reaction respectively. The two states are distinguishable by their FRET efficiency, which determines the visibility of each state in the donor and acceptor detectors. Species 3 has constant brightness in both detectors throughout the experiment. The fluorescence intensity, which is the sum of the contributions of all molecules, is measured in two independent detectors (blue: donor, red: acceptor). Fluctuations arise due to translational diffusion (all species), and due to chemical kinetics (species 1 and 2). Fluctuations are analyzed in terms of their auto- or cross-correlation functions (G_DD, G_DA, and G_AA), defined as: G_xy(τ) = 〈δI_x (t) δI_y (t + δt)〉/[〈I_x (t)〉 〈I_y (t)〉] where δI (t) = I(t) − 〈I(t)〉 represents the deviation from the mean.

Contamination with a population of donor-only particles (species 3, fig. 1) creates a potential problem that has been long recognized. These particles contribute to the fluctuations in fluorescence intensity due to translational diffusion within the observation volume, and will in general affect the experimentally measured correlation decays. The problem is significant in biophysical research due to the difficulty of producing fluorescently labeled biopolymers with 100% yield, and because fluorescent acceptors are usually more prone to photochemical degradation (photobleaching). In addition, some biochemically-unstable systems can dissociate at the low concentrations required for the experiment (typically 100 pM–10 nM), leaving donor-only particles diffusing in solution. Although the same is true for acceptor-only particles, they do not contribute to the measured signal because they are not excited during the experiment.

These problems have been addressed recently by applying experimental excitation/emission schemes that allow the filtering of the fluorescence data according to criteria such as color, lifetime and polarization.^5,9–11 Each of these methods has their own strengths and limitations that depend on the characteristics of the system under study. For instance, the use of alternating excitation to distinguish donor-only particles⁵ requires strict single-molecule conditions which results in filtered data sets with poor signal-to-noise ratio, especially at timescales close to the diffusion time. Felekyan at al. recently proposed an elegant scheme that introduces fluorescence polarization as a variable for data filtering¹⁰, but its applicability to fluorescently-labeled biopolymers, that often exhibit two or three rotational correlation times¹², has not been demonstrated and does not appear straightforward.

In this letter, we will show that the determination of τ_R is still possible using conventional FCS in the presence of donor-only particles in many situations of interest in biophysical research. We actually consider the more general situation in which species 3 has visibility in the acceptor channel as well, provided that its relative visibility in the donor and acceptor detectors is constant throughout the experiment. A typical example of this type of situation is the unwanted detection of donor photons in the acceptor detector, known as crosstalk.

It has been well-established that the one-color autocorrelation function of a mixture of species is given by a linear combination of the individual correlation functions, g_i(τ), in which each species contributes proportionally to its number of molecules (N_i) and the square of its brightness (B_i):^8,13

$$G(\tau )=\sum _i{N}_i{B}_i^2{g}_i(\tau )/{\left(\sum _i{N}_i{B}_i\right)}^2$$ (1)

In the previous equation, the individual components g_i(τ) represent just the shapes of the decays, or in other words, they are normalized so that g_i(0) = 1.

Analogously, we can express the two-color correlation functions as:

$$G_{xy}(\tau )=\frac{{\displaystyle \sum _i}{N}_i{B}_{i,x}{B}_{i,y}{g}_{i,xy}(\tau )}{{\displaystyle \sum _i}{N}_i{B}_{i,x}{\displaystyle \sum _i}{N}_i{B}_{i,y}}={\displaystyle \sum _i}\frac{f_{i,x}{f}_{i,y}}{N_i}{g}_{i,xy}(\tau )$$ (2)

where B_i,x and B_i,y represent the brightness of species i in detectors x and y respectively and G_xy represents the total auto- (x = y) or cross- (x ≠ y) correlation function of the system. The right-hand side of the equation is expressed in terms of the fractional intensities in detectors x and y, defined as $f_{ix}=N_i{B}_{ix}/{\displaystyle \sum _i}{N}_i{B}_{ix}$ and $f_{iy}=N_i{B}_{iy}/{\displaystyle \sum _i}{N}_i{B}_{iy}$.

For species 3, fluctuations arise from translational diffusion only, and therefore the correlation functions (g_3,DD = g_3,AA = g_3,DA) can be written in terms of its diffusion coefficient (D₃) and the radial and axial semiaxis of the Gaussian observation volume (ω₁ and ω₂) as^13–14

$$T_3(\tau )={\left[\left(1+4D_3\tau /{\omega }_1^2\right){\left(1+4D_3\tau /{\omega }_2^2\right)}^{1/2}\right]}^{-1}$$ (3)

For species 1 and 2, fluctuations in fluorescence intensity arise from both diffusion and conformational dynamics. The corresponding correlation functions can be expressed as the product of a diffusion (T(τ)) and a kinetic term (Q_x,y(τ)) provided that the diffusion coefficients of the two states are identical (D₁ = D₂).^1,7 Equation 3 can be used to describe the contributions of diffusion by using the diffusion coefficients of species 1 and 2. The kinetic term Q_x,y(τ) has been derived in previous work⁷, and can be expressed in terms of the equilibrium constant of the process (N₂/N₁) and the fractional intensities of states 1 and 2 as:

$$Q_{x,y}=\frac{(f_{1x}+f_{2x})\left(f_{1y}+f_{2y}\right)-\left(f_{1y}-f_{2y}{N}_1/N_2\right)\left(f_{2x}-f_{1x}{N}_2/N_1\right)e^{-\tau /{\tau }_R}}{(N_1+N_2)\left(f_{1x}{f}_{1y}/N_1+f_{2x}{f}_{2y}/N_2\right)}$$ (4)

From eq. 2, and assuming that the diffusion coefficients of states 1 and 2 are identical, the auto- and cross-correlation decays of the system described in fig. 1 can be expressed as:

$$G_{xy}(\tau )=\frac{f_{3x}{f}_{3y}}{N_3}{T}_3(\tau )+\left(\frac{f_{1x}{f}_{1y}}{N_1}+\frac{f_{2x}{f}_{2y}}{N_2}\right)T_1(\tau )Q_{x,y}(\tau )$$ (5)

Although the presence of species 3 affects in principle all measured correlation decays, we will show that the determination of τ_R is still possible in many cases of interest even if N₃, D₃ and B_3x,y are unknown.

The assumption that the diffusion coefficients of states 1 and 2 are identical is necessary to obtain closed-form expressions for the contributions to the correlation function.¹⁴ Although this may seem too restrictive, the differences in diffusion coefficient between two conformational states of a biomolecule are often small and hard to measure by FCS.¹⁵ An increase in hydrodynamic radius of about 50% has been measured, however, in studies of denaturation of globular proteins^16–17, indicating that deviations can be expected when such dramatic changes in molecular shape are involved. Although the auto- and cross- correlation functions of a two-state system cannot be expressed analytically when D₁ ≠ D₂, they can still be calculated numerically, providing a means by which the effects of neglecting variations in frictional coefficient can be evaluated.

One interesting case of importance in biophysical research is the situation where species 3 has visibility in the donor detector only. This could be due to incomplete chemical labeling, photochemical degradation of the acceptor, or due to partial dissociation of a subunit containing the donor, and assumes that crosstalk is negligible. Partial dissociation is particularly relevant in research involving biochemical systems comprised of many subunits that might dissociate at the low concentrations used in these experiments. In this case, the donor-only particle does not necessarily have the same diffusion coefficient as the donor-acceptor particles, but interestingly, we will show that the measurement of the relaxation time of the reaction is still possible.

In this case, f_3A = 0, and therefore G_DA and G_AA are proportional to T₁(τ)Q_DA(τ) and T₁(τ)Q_AA(τ) respectively (eq. 5) as it is the case when N₃ = 0. In other words, the presence of species 3 does not affect the shapes of these two correlation decays, but just their amplitudes. This is the basis for the successful determination of τ_R even when N₃ ≠ 0. It should be noted that the amplitude of G_DA does in fact depend on N₃ (f_1D and f_2D are functions of N₃), so it is not true that the presence of donor-only particles does not affect the cross-correlation decay as it has been assumed elsewhere¹⁸. Yet, although the amplitude of the decay is affected, the relaxation time can be determined from the temporal decay of the ratio F_A(τ) = G_AA(τ)/G_DA(τ) as follows. This ratio is independent of the diffusion contributions, and can be written as [K₁ + K₂ exp(−τ/τ_R)]/[K₃ + K₄ exp(−τ/τ_R)], where the constants K_1–4 are functions of the fractional intensities, N₁ and N₂ (eq. 4). It can be shown by simple algebraic manipulation that the function

$$Y(\tau )=\frac{F_A(\tau =0)-F_A(\tau \to \infty )}{F_A(\tau )-F_A(\tau \to \infty )}=\frac{K_4}{K_3+K_4}+\frac{K_3}{K_3+K_4}{e}^{\tau /{\tau }_R}{f}_{3A}=0$$ (6)

does not depend on N₃ (see supplemental material for more details). In addition, a plot of ln Y(τ) vs τ is linear with slope 1/τ_R at long lag times (K₄ ≪ K₃exp (τ/τ_R)), allowing the determination of the relaxation time from measurements of G_AA(τ)/G_DA(τ) even in the presence of donor-only species. This procedure was tested using Monte Carlo simulations as discussed below.

Another case of interest is the situation where D₃ = D_1,2, with any value of f_3A. This represents the case where a fraction of the particles is not dynamic (i.e. the donor-acceptor distance remains constant during the experiment), or a situation of donor-only labeled particles with significant cross-talk. In this case, T₃(τ) = T_1,2(τ), and an analysis analogous to the one presented in the previous paragraph shows that the ratios F_A(τ) = G_AA(τ)/G_DA(τ) and F_D(τ) = G_DD(τ)/G_DA(τ) can both be used to construct functions that yield τ_R even in the presence of species 3 (see supplemental information for more details):

$$Y(\tau )=\frac{F_A(\tau =0)-F_A(\tau \to \infty )}{F_A(\tau )-F_A(\tau \to \infty )}=\frac{F_D(\tau =0)-F_D(\tau \to \infty )}{F_D(\tau )-F_D(\tau \to \infty )}=\frac{K_4}{K_3+K_4}+\frac{K_3}{K_3+K_4}{e}^{\tau /{\tau }_R}{D}_3=D_{1,2}$$ (7)

The procedures described above are illustrated in fig. 2, which represents the results of three Monte Carlo simulations of the behavior of the system of fig. 1. A three dimensional random walk was used to simulate translational diffusion, and the back and forth chemical reactions between species 1 and 2 were simulated as Poisson processes as described elsewhere¹⁹ (see supporting information for details). The parameters used in the simulations were N₂/N₁ = 3, N₂ + N₁ = 0.91, τ_R = 125 μs, B_1D = 2,000, B_2D = 200, B_3D = 3,000, B_1A = 400, B_2A = 2,000, ω₁ = ω₂ = 0.35 μm, and D₁ = 50 μm² s⁻¹. The remaining parameters varied as follows: case A: N₃ = 0, case B: N₃ = N₁, D₃ = 400 μm² s⁻¹, B_3A= 0 (species 3 is smaller than species 1 and 2 and has no visibility in the acceptor detector), and case C: N₃ = N₁, D₃ = D₁ and B_3A= 300 (species 3 has visibility in the acceptor detector and the same diffusion coefficient as species 1 and 2). The diffusion coefficients represent typical values for a protein (D ~ 50 μm² s⁻¹)²⁰ and a free fluorescent dye (D ~ 400 μm² s⁻¹)²¹, the latter being the extreme example of donor-only contamination.

Figure 2.

Left: Correlation decays obtained from Monte Carlo simulations for cases A–C. The black, blue and red lines represent G_DD, G_AA, and G_DA, respectively. Right: Results of Monte Carlo simulations analyzed as described in the text (eq. 7). The red, blue and green curves are constructed from the G_AA/G_DA ratio (F_A) for cases A, B and C respectively. The orange trace was constructed from the G_DD/G_DA ratio (F_D) for case C. The black straight line represents ln[K₃e^τ/τR/(K₃ + K₄)] calculated from the parameters used in the simulation. The blue dotted vertical line denotes the relaxation time of the reaction (125 μs). Inset: lnY(τ) calculated from G_DD/G_DA for case B (blue) showing poor linearity at all times. The scales of both axes are the same as in the parent plot.

The correlation decays obtained in all three simulations are in excellent agreement with the curves predicted by equations 3–5 (see supporting information). Figures 2A–C show the simulated G_DD (black), G_DA (red) and G_AA (blue) decays for the three cases described above. As expected, G_DD for case B is greatly influenced by the presence of large quantities of fast-diffusing donor-only particles. However, because in this case species 3 has no visibility in the acceptor channel, G_AA overlaps with the decay obtained for case A (supporting information fig. S5). The cross-correlation decay overlaps only in shape, but the amplitude of G_DA is lower because the large excess of donor-only particles affects the average donor intensity measured in the experiment (supporting information, fig. S6).

Importantly, despite the significant differences in the individual correlation decays, the three cases yield the same ln Y(τ) vs τ plot (fig. 2D), which is linear with a slope 1/τ_R at long times. In figure 2D, the red, blue and green lines represent the natural logarithm of Y(τ)=[F_A(τ = 0)− F_A(τ → ∞)]/[F_A(τ) − F_A(τ → ∞)] for cases τ A, B τ and C respectively, while the orange line represents the same function calculated from F_D(τ) = G_DD(τ)/G_DA(τ) for case C (f_3A ≠ 0, D₃ = D_1,2). The black straight line represents the term ln[K₃e^τ/τ_R/(K₃ + K₄)] (see eq. 6) calculated from the parameters used in the simulation, and therefore has a slope of 1/τ_R. The dotted vertical line represents the relaxation time for the simulated reaction, indicating that linearity is poor at τ ≪ τ_R. The inset in fig. 2D shows Y(τ) calculated from the donor autocorrelation function (F_D(τ) = G_DD(τ)/G_DA(τ)) for case B, showing a clear deviation from linearity. This is expected because the fast-diffusing contaminating particles greatly affect the donor-autocorrelation decay, but leave the shape of the cross-correlation decay unaltered because their visibility in the acceptor detector is negligible.

In conclusion, these results support the idea that it is possible to measure the relaxation time of certain reactions in equilibrium using FRET and FCS even in the presence of excess donor-only particles provided that cross-talk is not significant. It should be noted that the effects of crosstalk can be eliminated by performing alternating laser excitation as described in reference ⁹. Yet, if species 3 has significant visibility in the acceptor detector, the determination is possible if its diffusion coefficient is identical to that of species 1–2. In this case, the ratios F_A(τ) = G_AA(τ)/G_DA(τ) or F_D(τ) = G_DD(τ)/G_DA(τ) can be used to construct functions that yield τ_R independently of N₃ and f_3A. The results of further simulations that support these conclusions are included as supporting material.