7392
J. Phys. Chem. B 2007, 111, 7392-7400
Measuring Conformational Dynamics: A New FCS-FRET Approach Tedman Torres†,‡ and Marcia Levitus*,†,‡,§ Department of Physics, Department of Chemistry and Biochemistry, and The Biodesign Institute, Arizona State UniVersity, Tempe, Arizona 85287-5601 ReceiVed: January 25, 2007; In Final Form: April 3, 2007
Fluorescence correlation spectroscopy (FCS) has recently emerged as a powerful technique to study conformational dynamics of biomolecules, but often its applicability is limited by the difficulty of separating the contributions of kinetics from those due to diffusion. We present a new approach based on the simultaneous analysis of the auto- and cross-correlation functions of the intensities measured in two independent detectors for a donor-acceptor labeled biomolecule. Fluctuations in fluorescence intensity are a consequence of diffusion and the variations in fluorescence resonance energy transfer (FRET) efficiency due to changes in donoracceptor distance. Although the complete description of the correlation functions requires that diffusion is well-characterized, the ratio of any two correlation functions depends on kinetic parameters only. This provides a means by which kinetic information can be obtained independently of the diffusion contributions. As proof of principle, we reanalyze data obtained in previous work with nucleosomes. In contrast to our previous work, where a donor-only sample was used to characterize diffusion, we now show that the same kinetic information can be obtained from a single experiment with a double-labeled biomolecule. This eliminates not only the need of a reference, but also artifacts associated with changes in the observation volume between measurements.
Introduction Single-molecule spectroscopic techniques have emerged in the past decade with great promise for enhancing our understanding of the behavior of individual biological macromolecules.1 Their remarkable potential relies on the fact that they allow time-dependent reactions to be studied without the need for synchronizing a population of molecules. The most frequent approach involves the use of fluorescence resonance energy transfer (FRET) at the single-molecule level to study timedependent conformational changes in real time. FRET is a distance-dependent interaction between the electronic excited states of two dye molecules in which excitation is transferred from a donor molecule to an acceptor molecule without emission of a photon.2,3 Examples of single-molecule FRET applications include the investigation of the conformational dynamics of DNA and RNA four-way junctions,4 the subunit movements in the F0F1-ATP synthase system during ATP synthesis and hydrolysis,5,6 DNA-histone interactions in nucleosomes,7 the conformational distributions of polypeptides,8 and so forth. However, the extremely low signals obtained in such measurements (∼5 photons/ms) limit their application to the study of slow (∼>50 ms) kinetic processes. Fluorescence correlation spectroscopy (FCS), a technique traditionally used to study diffusion properties of biomolecules, has been recently used as an alternative to study conformational fluctuations in a wide range of time scales. The accessible time scales span almost 7 orders of magnitude, all the way from submicroseconds to seconds. In this technique, the spontaneous fluorescence intensity fluctuations of a small number of * Author to whom correspondence should be addressed. Marcia.levitus@ asu.edu. † Department of Physics. § Department of Chemistry and Biochemistry. ‡ The Biodesign Institute.
molecules are analyzed statistically in order to extract dynamic information of the system.9 Although the technique was introduced more than 30 years ago,10 its full potential became evident only after the technical advances in confocal microscopy in the late 1990s. Since then, FCS has been primarily used to study the diffusion properties of macromolecules11,12 and to characterize the photophysics and photochemistry of dyes.13-16 More recently, this technique has been applied to the investigation of conformational fluctuations of DNA molecules17,18 and nucleosomes.19 In the first of such studies, Bonnet et al. investigated the spontaneous fluctuations between the open and closed states of a molecular beacon (DNA hairpin) containing a fluorophore and a quencher covalently attached to the two ends of the DNA molecule.18 When the hairpin loop is closed, fluorophore and quencher are in close proximity, and the fluorescence is quenched. In the open conformation, the fluorophore and quencher are far apart, so the fluorescence of the dye is restored. In this way, fluctuations between the open and closed hairpin conformations create fluctuations in fluorescence. Experimentally, these fluctuations are measured in a small open volume which is created by focusing a laser beam to a diffraction-limited spot and using a confocal pinhole to limit the detection of photons to light originating from the focal plane (Figure 1). Intensity fluctuations are quantified by their autocorrelation function, G(τ) (eq 1). The typical strategy is to compare the fluorescence fluctuations of a sample labeled with the fluorophore only, with the fluctuations of a sample labeled with both the donor and the quencher. In the first case, diffusion of the particles into and out of the confocal volume is expected to be the only source of intensity fluctuations, whereas in the second case, the intensity fluctuations are due to both diffusion and intramolecular conformational dynamics. In this way, the fluorophore-only sample is used as a reference to characterize
10.1021/jp070659s CCC: $37.00 © 2007 American Chemical Society Published on Web 06/05/2007
Conformational Dynamics Studied by FCS-FRET
Figure 1. Conformational dynamics of the two-state system considered in this work. The kinetic rates of interconversion between states 1 and 2 are represented by k12 and k21. The biomolecule contains a donor (red) fluorescent probe which is excited directly by a laser. A fraction of the excitation energy is transferred to an acceptor (cyan) molecule by FRET, with a distance-dependent efficiency. A small open volume is created by focusing a laser beam to a diffraction-limited spot and using a confocal pinhole to limit the detection of photons to light originating from regions close to the focal plane (observation volume). Molecules diffuse freely in and out of the observation volume, producing fluctuations in the fluorescence intensity of both acceptor and donor. Fluctuations are due to diffusion and to changes in FRET efficiency due to the conformational dynamics of the macromolecule. Correlation analysis allows the quantification of the rates of the processes leading to these fluctuations.
the diffusion properties of the system, so the contributions of conformational dynamics can be separated from the contributions of diffusion in the double-labeled sample. More recently, we used a similar approach to investigate nucleosome dynamics.19 In this work, a donor-acceptor FRET pair was used instead of the fluorophore-quencher combination. The diffusion properties of the system were characterized with a donor-only sample, and the results were then used to isolate the kinetic contributions in the donor-acceptor nucleosome. This procedure, although seemingly simple, suffers from three major drawbacks that limit its applicability. First, it relies on the preparation of an identical sample containing only the donor. Second, since the kinetic information is obtained from the ratio of two autocorrelation decays (donor only and donor-acceptor samples), it is extremely critical that the two samples are measured under identical optical conditions. The amplitude of the autocorrelation decay depends on the shape and dimensions of the confocal volume, which are very difficult to control precisely between measurements. This issue becomes even more critical when measurements are performed inside artificial vesicles or living cells, where measuring both samples under identical optical conditions becomes a task practically impossible to achieve. Third, although the fluctuations in the acceptor intensity are measured in the same experiment, this analysis uses only the information registered in the donor detector and thus wastes the valuable information that has been acquired in the acceptor channel. In principle, one could imagine using solvent viscosity effects to separate the diffusion and conformational dynamics characteristic rates. The reciprocal of the diffusion constant scales linearly with viscosity, so one could shift the diffusion component to longer correlation times allowing a larger separation from the conformational dynamics contributions. This approach would not require a donor-only reference, but it would still require a series of successive experimental determinations, and it would be unfeasible in biological environments such as cells or vesicles. Moreover, it is likely that the rate of conformational dynamics also depends on viscosity. Kramer’s theory predicts a linear relationship between the kinetic rate of the reaction and the reciprocal of viscosity for processes that
J. Phys. Chem. B, Vol. 111, No. 25, 2007 7393 are limited by diffusion, so changes in viscosity would not help separating the diffusion and dynamics time scales. This is seen for example in folding reactions of proteins where the collapse of the polypeptide chain occurs during the rate-limiting step of the reaction.20 In this work, we investigate a new methodology of analysis that allows the study of conformational dynamics independently of the diffusional properties of the system. We derived the analytical expressions of the donor-donor, donor-acceptor, and acceptor-acceptor correlation functions of a freely diffusing two-state system containing a FRET pair, and found that the ratio of any two correlation decays isolates kinetic information from diffusion contributions. In this way, kinetic information can be extracted without the need of an auxiliary experiment with a donor-only sample to characterize the diffusion properties. Furthermore, this approach eliminates the potential artifacts associated with changes in the confocal volume between measurements. As a proof of principle, we successfully apply this new method of analysis to experimental FCS data we obtained previously using the “conventional” approach. Theoretical Basis Consider the system depicted in Figure 1. A donor-acceptor double-labeled macromolecule exists in equilibrium between two conformations characterized by two distinguishable FRET efficiencies (EFRET, defined as the quantum efficiency of energy transfer). The conversion between states is assumed to be instantaneous, so there are no other detectable states during the measurement. Molecules diffuse freely in and out of the probe volume, producing fluctuations in both the donor and acceptor intensities. Intensity fluctuations are quantified by their auto(x ) y) or cross- (x * y) correlation function, which compares the fluorescence intensity at a given time with the intensity after a certain lag time τ
Gxy(τ) )
〈Ix(t)Iy(t + τ)〉
(1)
〈Ix(t)〉〈Iy(t)〉
Here, the fluorescence intensity measured at detector x and time t is represented by Ix(t), and the angle brackets represent the following time average
〈Ix(t)Iy(t + τ)〉 ) lim
Tf∞
1 T
∫0T Ix(t)Iy(t + τ) dt
(2)
The correlation function measured in an FCS experiment over a limited data accumulation time, T, is a good approximation to the ideal correlation function if T . τ. In the biophysical literature, G(τ) is often defined in terms of the fluorescence fluctuations, δIx,y(t) ) Ix,y(t) - 〈Ix,y(t)〉
Gxy(τ) )
〈δIx(t)δIy(t + τ)〉 〈Ix(t)〉〈Iy(t)〉
)
〈Ix(t)Iy(t + τ)〉 〈Ix(t)〉〈Iy(t)〉
-1
(3)
Equation 3 represents the definition of the autocorrelation function of the fluorescence fluctuations, δIx,y, whereas eq 1 represents the autocorrelation function of the fluorescence intensities, Ix,y. In the FCS field, the term “autocorrelation function” is usually used to describe the normalized autocorrelation function of the intensity fluctuations, which is identical to the normalized autocovariance function of the intensities. We will use this definition in the remainder of the manuscript for consistency within the field, and point out that the normalized autocorrelation function of the intensities is simply obtained
7394 J. Phys. Chem. B, Vol. 111, No. 25, 2007
Torres and Levitus
from the normalized autocorrelation function of the intensity fluctuations by shifting the decay by one unit (eq 3). The number of molecules of a specified type (conformation 1 or 2) in an open region can change by diffusion or by chemical reaction (Figure 1). These processes are coupled, so that, assuming normal 3D diffusion, the rates of relaxation of concentration fluctuations are determined by the system of equations
(
) (
k21 ∂δC1(r,t)/∂t D ∇2 - k12 ) 1 2 ∂δC1(r,t)/∂t k12 D1∇ - k21
)(
δC1(r,t) δC1(r,t)
)
(4)
Here, δCi(r, t) ) Ci(r, t) - 〈Ci(r, t)〉 represents the fluctuation in concentration of species i ) 1, 2 (in number of molecules per unit volume) at position r and time t, Di is the diffusion coefficient of component i, and kij are the chemical rate constants defined in Figure 1. The fluctuations in fluorescence intensity, from which the auto- and cross-correlation functions are calculated according to eq 3, are determined by the fluctuations in concentration, the photophysical properties of the fluorophores, the FRET efficiencies in states 1 and 2 (E1,2), and the illumination and detection profiles of the experimental setup
δID1,2(t)
) g Q (1 - E1,2) D D
D
δIA1,2(t) ) gADQAE1,2
∫W(r)δC1,2(r, t) dr
∫W(r)δC1,2(r, t) dr
(5)
Here, D is the extinction coefficient of the donor at the excitation wavelength, QD,A is the fluorescence quantum yield of the donor and acceptor, respectively, and the factor g accounts for the quantum efficiency of the detector as well as geometrical and optical filtering losses inherent to the experimental arrangement. W(r) is a function that describes the spatial distribution of the detected light, and is a convolution of the spatial distribution of the focused light in the sample space and the optical transfer function of the objective-pinhole combination, which determines the spatial collection efficiency. For simplicity, we define the visibility of state i ) 1, 2 in detector x ) D, A as
ηD1,2 ) gDDQD(1 - E1,2)
(6)
ηA1,2 ) gADQAE1,2
(7)
Gxy(τ) ) T(τ)Xxy(τ)
(9)
The diffusion term, depending on W(r), is determined by many aspects of the optical design, such as whether the objective is under- or overfilled, the size of the pinhole, the shape of the laser beam, and so forth. As will be discussed below, T(τ) can be obtained as a simple analytical function only if W(r) is assumed to be Gaussian in all dimensions. However, we will not make this assumption, and we will prove that kinetic information can be obtained independently of the knowledge of W(r). It can be shown that the amplitude of T(τ) equals the reciprocal of the mean number of molecules in the observation volume, N.9 Since sample concentration affects the amplitude of the correlation decays, but not their shapes, we will assume N ) 1 in the remainder of the discussion for simplicity. The kinetic contributions, Xxy(τ), depend on the kinetic rates and the relative visibilities of each state in each detector
Xxy(τ) ) 1 +
f1f2(ηx2/ηx1 - 1)(ηy2/ηy1 - 1) (f1 + f2ηx2/ηx1)(f1 + f2ηy2/ηy1)
e-(k12+k21)τ (10)
with f1 ) k21/(k12 + k21) and f2 ) k12/(k12 + k21) representing the fraction of molecules in states 1 and 2, respectively. Equation 10 is a general expression for any two-state system containing two fluorescent tags with different visibilities in two independent detectors. In the particular case of Figure 1, where the donor is excited directly by the laser and part of the excitation energy is transferred by FRET to a fluorescent acceptor, the visibilities of each state (i, j ) 1, 2) in each detector (x, y ) D, A) are defined in eqs 6 and 7, and the kinetic components of the autoand cross-correlation functions can be expressed as
XDD(τ) ) 1 +
k12k21(E1 - E2)2 [k21(1 - E1) + k12(1 - E2)]2
XAA(τ) ) 1 +
k12k21(E1 - E2)2 (k21E1 + k12E2)2
e-(k12+k21)τ
e-(k12+k21)τ
(11a) (11b)
XDA(τ) ) 1 k12k21(E1 - E2)2 [k21(1 - E1) + k12(1 - E2)](k21E1 + k12E2)
e-(k12+k21)τ (11c)
so that eq 5 can be written in a general way as D,A δID,A 1,2 (t) ) η1,2
∫
W(r)δC1,2 (r, t) dr
(8)
The above equations assume no crosstalk (i.e., no acceptor is detected in the donor detector, and no donor is detected in the acceptor detector). In the most general case, a fraction of the acceptor intensity will reach the donor detector and vice versa, so the visibilities defined in eqs 6 and 7 will contain contributions from both donor and acceptor (see Appendix A). The system of eqs 3, 4, and 8 was solved by employing Fourier transformations and the appropriate boundary conditions following the formalism introduced by the pioneering work of Magde et al.21 A detailed derivation has been published elsewhere.22 If the diffusion coefficients of species 1 and 2 are assumed to be equal (D1 ) D2 ) D), Gxy(τ) can be expressed as the product of a term that represents the kinetic contributions to the total correlation function (Xxy(τ)) and a term that contains the diffusion contributions (T(τ)).
Whereas the analytical description of the total correlation functions Gxy requires that T(τ) is known (eq 9), the ratio of any two decays (GDD/GDA, GAA/GDA, or GDD/GAA) is independent of the diffusion contributions and thus isolates the kinetic components. Therefore, the measurement and analysis of these ratios provides a means by which rate constants can be evaluated independently of the diffusion properties of the system. This is particularly valuable when T(τ) is hard to reproduce between measurements or simply unknown. Results Features of GDD(τ),GAA(τ) and GDA(τ). As expected, the kinetic contributions to the autocorrelation functions, XDD(τ) and XAA(τ), decrease monotonically with time, while XAD(τ), the kinetic contribution to the cross-correlation function, shows the opposite behavior (see Figure 3). This is a consequence of the fact that the visibilities of states 1 and 2 in the two detectors are dictated by FRET, so an increase in donor signal causes a
Conformational Dynamics Studied by FCS-FRET
J. Phys. Chem. B, Vol. 111, No. 25, 2007 7395
Figure 2. Predictions for the cross-correlation function (GDA) using E1 ) 0.8, E2 ) 0.1, k12 ) 2 × 103 s-1, k21 ) 5 × 103 s-1, and γ ) 1/25. The decays were calculated from eqs 11c (kinetic component) and 13 (diffusion component). Equation 14 predicts the occurrence of a “bump” for tD values greater than 0.2 ms. Note that the decay for tD ) 0.1 ms is maximum at the origin and decays monotonically to zero, while the decays for tD > 0.2 show a maximum at some positive value of τ.
concomitant decrease in the acceptor, and vice versa. In this way, the factor (η2x/η1x - 1)(η2y/η1y - 1) in eq 10 is always negative for x * y and always positive for x ) y. If the visibilities of both states in each detector are identical (η2x ) η1x and η2y ) η1y), the total correlation functions are insensitive to the dynamics of the conformational change, so that Xxy(τ) ) 1 and Gxy ) T(τ) for any detector combination. Thus, the visibility of each state should be as different as possible in each detector for FCS to be useful in the measurement of conformational dynamics. Since T(τ) decreases monotonically from T(0) ) 1 to T(τ f ∞) ) 0, independently of the particular form of W(r), GDD and GAA will also decrease monotonically with time. In contrast, GDA can present a maximum at τ > 0 depending on the relative magnitudes of the kinetic and diffusion contributions. Mathematically, GDA will present a maximum at τ > 0 when
∂GDA(τ) ∂XDA(τ) ∂T(τ) ∂ ln T(τ) ) 0 ) T(τ) + XDA(τ) w + ∂τ ∂τ ∂τ ∂τ XDA(τ) - 1 (12) (k12 + k21) XDA(τ) Equation 12 can be satisfied for positive values of τ only under certain conditions. For instance, if W(r) is assumed to be Gaussian with radial and axial semiaxis r0 and z0, respectively, the diffusion component becomes
T(τ) )
1 (1 + 4Dτ/r02)(1 + 4Dτ/z02)1/2
(13)
In this case, a maximum in Gxy will occur at a positive value of τ if the following inequality holds (see Appendix B)
tD >
[
]
2 + γ k21E1(1 - E1) + k12E2(1 - E2) 2 k k (E - E )2 12 21
1
2
(14)
Figure 3. Predictions for the diffusion (T) and kinetic (XDD, XAA, XDA) contributions to the total auto- and cross-correlation functions (GDD, GAA, GDA). (1A-C) Solid green: diffusion contributions calculated from eq 13 assuming a Gaussian observation volume with γ ) 1/25. Solid black: XDD(τ). Dotted red: XAA(τ). Dashed blue: XDA(τ). The kinetic contributions were calculated from eq 11a-c using E1 ) 0.8, E2 ) 0.1, and k12 ) 2 × 103 s-1, k21 ) 5 × 103 s-1 (panels A and B) or k12 ) 2 × 102 s-1, k21 ) 5 × 102 s-1 (panel C). (2A-C) GDD (solid black), GAA (dotted red), and GDA (dashed blue) calculated from Xxy(τ) and T(τ) using eq 9.
Here, γ ) (r0/z0)2 is a geometric factor that represents the radial-to-axial ratio of the Gaussian confocal volume, and tD ) r02/(4D) is a measure of the residence time of the macromolecule in the observation volume. Equation 14 predicts a “bump” in the cross-correlation decay (i.e., a maximum in GDA at τ > 0) if the diffusion time, tD, is greater than a quantity determined by the rate constants, the FRET efficiencies, and the geometry of the observation volume (Figure 2). Otherwise, diffusion is much faster than conformational dynamics, and GDA(τ) decreases monotonically from its maximum value at τ ) 0 to zero at τ f ∞ (Figure 2, extreme left curve). Figure 2 shows an example of the expected results for a system with E1 ) 0.8, E2 ) 0.1, k12 ) 2 × 103 s-1, and k21 ) 5 × 103 s-1. All calculations have been performed by assuming a Gaussian volume with γ ) 1/25 (i.e., z0 ) 5r0), which is a reasonable value for confocal experimental setups.23 Using these parameters, eq 14 predicts a maximum in GDA for values of tD > 0.2 ms. As a reference, a macromolecule with a diffusion coefficient D ) 2 × 10-7 cm2‚s-1 (∼250 bp DNA24) would
7396 J. Phys. Chem. B, Vol. 111, No. 25, 2007 have a tD of approximately 0.7 ms if r0 is taken as 250 µm (approximately the diffraction-limited radius created by a green laser). Note that although eq 14 can be used to predict the existence of a “bump” in GDA, the position of the maximum has to be calculated numerically from eq 12. Figure 3 shows the kinetic and diffusion contributions to the total auto- and cross-correlation functions for a system with E1 ) 0.8 and E2 ) 0.1 under a variety of conditions. The kinetic components, XDD(τ), XAA(τ), and XDA(τ), were calculated from eqs 11a-11c, while the diffusion contribution was calculated using eq 13, which assumes a 3D Gaussian observation volume. In the preceding paragraphs, we have assumed a Gaussian observation volume and normal three-dimensional diffusion with the only purpose of illustrating the behavior of the different correlation functions under different conditions. If the observation volume is not Gaussian, as is often the case,25 T(τ) cannot be expressed analytically, complicating the analysis significantly. This is true even if the observation volume can be described analytically by a function different than a Gaussian (e.g., a Gaussian-Lorentzian). However, we stress the fact that this assumption will not be necessary in our final analysis. Separation of the Kinetic and Diffusion Contributions. Although the experimentally measured auto- and cross-correlation functions could be fitted in principle with the analytical expressions of GDD, GAA, and GDA to yield kinetic and diffusion information simultaneously, this procedure requires that the observation volume is correctly characterized by a threedimensional Gaussian profile and is prone to several experimental difficulties (see Discussion section). In contrast, the kinetic contributions can be investigated independently of the diffusion properties of the system by analyzing the ratios of the auto- and cross-correlation functions. Equation 9 shows that the diffusion contributions behave as a common multiplicative function for GDD, GAA, and GDA, so the ratio of any two functions depends exclusively on the rate constants that govern the dynamic process one wishes to study and the FRET efficiencies of both states. This is true under the assumption that the diffusion constant does not change significantly during the process, independently of whether the confocal volume can be characterized analytically. Thus, kinetic rates can be calculated even if the optical setup has not been properly aligned or characterized. Figure 4 shows an example of the results expected for GDD/ GDA, GDD/GAA, and GAA/GDA. These ratios depend exclusively on E1, E2, k12, and k21, so they can be used to study kinetic processes independently of diffusion. The inset in Figure 4 shows the behavior of GDD/GDA for a system with E1 ) 0.8 and E2 ) 0.1 and equilibrium constant K ) k12/k21 ) 2/5. The different curves were obtained by successively increasing both k12 and k21 by a factor of 10 from k12 ) 20 s-1 and k21 ) 50 s-1 (far right) to k12 ) 2 × 104 s-1 and k21 ) 5 × 104 s-1 (far left). Since the pre-exponential factors of the kinetic components defined in eqs 11a-11c depend on the equilibrium constant K ) k12/k21, but not on the absolute values of k12 and k21, the amplitude of the ratios shown in the inset are identical. However, GDD/GDA decays faster as the kinetic rates increase. It can be shown that GDD/GDA - 1 decays to half its initial value at τ1/2 ) [ln(XDA(τ ) 0) + 1)]/(k12 + k21), where XDA(τ ) 0) is a function of the FRET efficiencies and the equilibrium constant. Since 0 e XDA(τ ) 0) e 1, τ1/2 e (ln 2)/(k12 + k21). In practice, obtaining kinetic information from these ratios would involve fitting any two ratios (e.g., GDD/GDA and GAA/ GDA) simultaneously with four parameters: E1, E2, k12, and k21. However, if k21(1 - E1) + k12(1 - E2) ≈ k21E1 + k12E2, GDD
Torres and Levitus
Figure 4. Predicted ratios for E1 ) 0.8, E2 ) 0.1, k12 ) 2 × 103 s-1, and k21 ) 5 × 103 s-1. GDD/GDA: solid line. GAA/GDA: dotted line. GDD/GAA: dashed line. Inset: calculated GDD/GDA ratios for a system with K ) k12/k21 ) 2/5. The rates were increased by a factor of 10 from k12 ) 20 s-1 and k21 ) 50 s-1 (far right) to k12 ) 2 × 104 s-1 and k21 ) 5 × 104 s-1 (far left). All ratios are independent of τD and γ, and depend exclusively on the kinetic rates and FRET efficiencies of each state.
≈ GAA, so other independent pieces of information would be needed to obtain all four parameters. For instance, in many cases, one or both FRET efficiencies can be measured independently using standard fluorescence instrumentation. For instance, if k21/ k12 . 1 or k21/k12 , 1 (i.e., the equilibrium is shifted toward one of the two states), the FRET efficiency measured in bulk corresponds to the FRET efficiency of the most populated state. In addition, in many cases, k21/k12 can be manipulated by shifting the equilibrium with an external stimulus (e.g., ionic strength, temperature, etc.). Experimental Test. To test the proposed methodology, we reanalyzed data that we acquired in previous work with fluorescently labeled nucleosomes. In this work, we investigated the dynamics of spontaneous nucleosome conformational changes, in which a stretch of DNA transiently unwraps off the histone surface and then rewraps.19 This process was studied using double donor-acceptor labeled nucleosomes, using the “traditional” FCS approach, where the donor-only macromolecule is used as a reference to characterize the diffusion properties of the system (Greference (τ) ) T(τ)). Once diffusion was characterDD ized with the reference, the kinetic contributions due to the opening-closing process in the donor-acceptor double-labeled particle were isolated from the total donor autocorrelation decay sample sample reference (τ)). Note (Xsample DD (τ) ) GDD (τ)/T(τ) ) GDD (τ)/GDD that, in this analysis, we only used the signal recorded in the donor detector (i.e., we only analyzed GDD). Furthermore, since the kinetic information was extracted from the ratio of two independent autocorrelation decays, measured with two different samples, any change in the confocal volume between measurements would have invalidated the assumption that T(τ) is the same in both autocorrelation decays. This procedure also requires assumptions regarding the FRET efficiencies, so only two parameters (k12 and k21) are used to fit one curve. The rate constants obtained in this work were k12 ) 3.6 s-1 and k21 ) 20 s-1 (see ref 19 for all experimental details). Here, as a proof of principle, we reanalyze the results obtained in the original work using our proposed methodology. Figure 5A shows the GDD and GDA decays for the donor-acceptor labeled nucleosome sample. Note that, in contrast to our original
Conformational Dynamics Studied by FCS-FRET
J. Phys. Chem. B, Vol. 111, No. 25, 2007 7397 keeping the value of k12/k21 fixed at 0.04. The dotted lines in Figure 5B represent the expected decays for k12 ) 0.4 s-1 (right) and k12 ) 1.6 s-1 (left), which represent one-half and two times the value obtained in the fit, respectively. We note that, even if the ratio is independent of the diffusion coefficient of the macromolecule, the fact that the kinetic rates are much slower than the characteristic diffusion time implies that the time scales of interest will be at the tails of GDD and GDA, where the signal-to-noise ratio is poor. A better signalto-noise could have been obtained by increasing the diffusion time, so that GDD and GDA decay at longer lag times. In practice, this could be achieved by increasing the pinhole size, which increases the confocal volume. Webb and co-workers25 showed that deviations from a Gaussian volume are most severe when using a large detector aperture, which results in systematic artifacts in the FCS autocorrelation. While this is important when analyzing the total decay (Gxy), it is of no concern when analyzing the ratios. Discussion
Figure 5. A. Experimental GDD(τ) (black line) and GDA(τ) (red line) decays obtained with a donor-acceptor double-labeled nucleosome (see all experimental conditions in ref 18). Inset: GDD(τ) superimposed to the GDA(τ) decay multiplied by an arbitrary constant. Note that both decays seem to overlap in shape. B. Ratio of the two curves of part A. The large noise at τ > 100 ms is a consequence of the low amplitude of both GDD and GDA. The black solid line represents the fit to the ratio of eq 11a and c, from which we obtained k12 ) 0.8 s-1, k21 ) 20 s-1, E1 ) 0.89, and E2 ) 0 (see text). The dotted lines at either side represent simulated curves obtained using k12 ) 0.4 s-1 (right) and k12 ) 1.6 s-1 (left), keeping E1 ) 0.89, E2 ) 0, and k12/k21 ) 0.04 as obtained in the fit. These lines are shown as guidelines to assess the uncertainty in the determination.
analysis, here we take the ratio of two functions that have been measured simultaneously using the donor-acceptor sample. In this way, this new analysis eliminates the need of a donor-only reference sample and eliminates the experimental artifacts associated with potential changes in W(r) between measurements. This example is interesting in that the kinetic contributions to the total correlation functions are masked by the diffusion components, so that GDD and GDA appear to overlap in shape (Figure 5A inset). It is only after the ratio GDD/GDA is taken that the kinetic features are revealed (Figure 5B). We fitted the experimental GDD/GDA ratio with the ratio of eqs 11a to 11c. The results of two independent measurements overlap within experimental uncertainty. A series of constraints were used to reduce the number of fitting parameters from four to two. First, we fixed k12/k21 ) Keq to the value measured in bulk using the same sample by Li et al., Keq ) 0.04.26 Second, we constrained the FRET efficiencies to obtain results consistent with the FRET efficiency measured in bulk, Ebulk ) 0.85. The FRET efficiency measured for an ensemble of molecules in thermal equilibirium is the average of the FRET efficiencies of the individual states weighted by their mole fractions, i.e., Ebulk ) Eavg ) f2E2 + f1E1 ) E1/(1 + Keq) + E2Keq/(1 + Keq) ) 0.96E1 + 0.04E2. From this fit, we obtain k12 ) 0.8 s-1, k21 ) 20 s-1, E1 ) 0.89, and E2 ) 0. The obtained rate constants are very similar to the ones we obtained in our previous work using the donor-only reference (k12 ) 3.6 s-1, k21 ) 20 s-1), whereas the FRET efficiencies are consistent with the values expected from the position of the fluorophores in the nucleosome.26 To assess the uncertainty in the determination of the kinetic rates, we simulated the GDD/GDA decays for different k12 and k21 values,
We developed a new methodology to obtain kinetic rates from fluorescence correlation measurements that eliminates the need to characterize the diffusion properties of the system. We proved that provided that the diffusion coefficient of the diffusing particle does not change during the conformational change of interest, the ratio of any two auto- or cross-correlation decays depends exclusively on the rate constants of the process and the FRET efficiencies of the two conformations. In this way, this method of analysis does not require that the observation volume is thoroughly characterized, and can be applied even if deviations from a Gaussian profile are severe. Furthermore, since GDD, GAA, and GDA can be measured simultaneously, this methodology is not sensitive to experimental factors that can potentially affect the observation profile (W(r)) between measurements. This represents a significant improvement over methods that use a reference sample to characterize diffusion in an independent experiment, and opens up the possibility of investigating dynamic processes in complex biological environments where W(r) might not be well-defined or might be hard to reproduce. Van Orden and collaborators have devised a dual detection volume cross-correlation technique that allows the characterization of the diffusion properties of the system from the same set of fluorescence intensity measurements used to characterize the kinetic processes of the biomolecule of interest.27 In this method, fluctuations are observed as molecules flow sequentially between two detection volumes offset by a few micrometers. The flow velocity and the distance between detection volumes are selected so as to attain conditions where the conformational fluctuations make a negligible contribution to the cross-correlation function of the fluorescence from the two detection volumes. This allows the characterization of the translational diffusion properties of the biomolecule, which are subsequently used to isolate the kinetic terms in the autocorrelation analysis. Several requisites are needed to achieve this goal. First, the analysis of the dualvolume cross-correlation function relies on the assumptions that the two detection volumes are identical and Gaussian in shape and that translational diffusion is well-described by a standard 3D model. These assumptions are needed in order to obtain an analytical expression of the cross-correlation function between the two observation volumes. Second, in order to establish experimental conditions where conformational dynamics do not contribute to the cross-correlation function, the average transit time of the molecules between the two observation volumes
7398 J. Phys. Chem. B, Vol. 111, No. 25, 2007
Figure 6. Schematics of the two-color detection. The total fluorescence (red ) donor, cyan ) acceptor) is split by a dichroic filter with a wavelength-dependent transmission efficiency R(λ). Each component is directed toward an independent detector. The functions βD,A(λ) represent the transmission curve of all the optical components between the dichroic filter and each detector, whereas the function κ(λ) describes the wavelength-dependent quantum efficiency of each detector. The term crosstalk refers to the donor intensity detected in the acceptor detector, and vice versa.
has to be controlled so it is much greater than the relaxation time of the conformational fluctuations one wishes to investigate. Van Orden and collaborators showed that this is experimentally feasible for conformational fluctuations on the sub-microsecond time scale, but it might not be possible to achieve conditions that satisfy this requirement when dealing with slower kinetics. If the separation between the two foci is in the micrometer range, and the diffusion coefficient of the biomolecule is on the order of D ) 10-10 m2 s-1, the transit time due to diffusion would be on the microsecond time scale. Since the transit time cannot be shorter than that determined by diffusion, this methodology would not allow the investigation of conformational changes in the microsecond-to-second time scales (e.g., the nucleosome fluctuations described above, which show dynamics on the ∼50 ms time scale). In contrast, our new methodology can be used to investigate conformational fluctuations on a wide range of time scales, and can be put into practice using conventional FCS instrumentation. Furthermore, a key aspect of our FRET-FCS approach is that all correlation decays can be acquired using digital correlator cards, which are part of all commercial FCS instruments. This is an advantage over other methods that have been proposed in the past, which require that fluorescence intensities are measured with exceptional time resolution and compute the auto- and cross-correlation functions after the measurement. For instance, Wallace et al.17 proposed a methodology where kinetic information is obtained from the analysis of the fluctuations in the socalled proximity ratio (defined as P ) IA/(IA + ID)). In this work, the authors calculate and analyze the autocorrelation function of the proximity ratio (GP(τ) ) 〈P(t)P(t + τ)〉/〈P(t)〉2), which requires that IA(t) and ID(t) are measured with very high time resolution for long periods of time, so P(t) can be calculated point-by-point and autocorrelated after the experiment. This approach is not only computationally intensive, but more importantly, it is limited by the smallest binning time that allows the calculation of P(t) with a reasonable signal-to-noise ratio. For example, Wallace at al. used a 20 µs binning time to measure both IA and ID, so GP cannot be calculated at shorter lag times, and faster dynamic processes would then be missed.
Torres and Levitus If the binning time is reduced, it will eventually reach the point where zero or one photon per bin is registered in each detector, making the computation of P(t) impossible. In contrast, the calculation of GDD, GDA, and GAA is still possible under these conditions, making it possible to calculate these functions at shorter times. The computation of the correlation function of a vector composed of zeros and ones is possible, and yields a meaningful result, provided that the calculation is performed over a large number of photons. In this way, in contrast to GP, GDD, GDA, and GAA can be measured in the sub-microsecond time scale even if the average signal in this time scale is less than one photon. Another advantage of analyzing GDD, GDA, and GAA directly relies on the fact that correlator cards measure GDD, GAA, and GDA directly but cannot measure or generate GP, making the approach presented by Wallace et al. unfeasible with conventional FCS instrumentation. Although we have shown that the ratio of two correlation functions in the FRET-FCS method is independent of the characteristic diffusion time of the molecule (τD), the signalto-noise ratio is not. As shown in Figure 5, the signal-to-noise ratio of GDD/GDA is very poor at long lag times, where both GDD and GDA have decayed to very low values. Thus, the slowest measurable rate is determined by the macromolecule residence time in the confocal volume. The optical setup can be manipulated to some extent to increase this time, even if the shape of the confocal volume is distorted from the Gaussian profile. Although lifetimes in the subsecond scale might be measured in this way, these time scales fall in the optimum range for single-molecule experiments with immobilized molecules. In this way, the new proposed FCS-FRET methodology is complementary in time scales to single-molecule experiments, which are most useful in the 0.1-10 s time scale range. Finally, an important limitation of our new approach is given by the assumption that the diffusion coefficients of the two conformational states are the same. However, the differences in the diffusion coefficients between two folding states are generally quite small, and in fact hard to measure by FCS.28 For instance, Stellwagen and co-workers have studied the migration of DNA extensively and found that a 199 bp fragment of DNA containing a 46° bend has just a ∼2% mobility decrement in solution with respect to straight DNA of the same length.29 This suggests that our method can be safely applied to a variety of problems involving conformational changes where the hydrodynamic radius of the biomolecule does not change significantly. Yet, there are many biophysical processes of interest where the diffusion coefficient is expected to have a measurable change, with folding-unfolding of proteins being an obvious example. Chattopadhyay et al. determined the hydrodynamic radius of the folded and unfolded states of a rat intestinal fatty acid binding protein (IFABP) by FCS, and found a 50-80% diference depending on unfolding conditions. Other authors have found similar results in studies with a variety of proteins.30-32 We are currently investigating ways to take into account changes in the diffusion coefficient in our work. Conclusions We have presented a new methodology to study intramolecular conformational dynamics in biopolymers which allows the determination of kinetic rates over a large range of time scales. The new method is based on the simultaneous measurement and analysis of the fluorescence fluctuations of a donoracceptor FRET pair located within the biopolymer under study. Fluorescence fluctuations are quantified through the auto- and cross-correlation functions (Gxy; x, y ) D, A), which can be
Conformational Dynamics Studied by FCS-FRET
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measured simultaneously using two independent detectors. Although the complete description of the correlation functions requires that the diffusion contributions to the fluorescence fluctuations are well-described and included in the analysis, the ratio of any two correlation decays depends exclusively on the FRET efficiencies and interconversion rates of the two conformations in equilibrium. In this way, we have shown that the simultaneous measurement of all three correlation functions provides a means by which diffusion and kinetics can be decoupled, so the reaction rates can be measured independently of the hydrodynamic properties of the system. This eliminates the need of using a donor-only sample as a reference to characterize diffusion, and since all correlation decays can be measured simultaneously using the same sample, optical artifacts associated with changes in the observation volume between consecutive measurements are also eliminated. Our long-term goal is to study conformational dynamics in complex biological environments (e.g., membranes and cells), where characterizing the diffusion properties of the biopolymer can be particularly difficult and hard to reproduce. Acknowledgment. We thank Jonathan Widom (Northwestern University), Timmothy Newman (ASU), and John Abreu for helpful discussions, and Prof. Thomas Jovin (Max Plank Institute for Biophysical Chemistry) for critical comments on the manuscript.
AD ) (E1 - E2)(cADQA - cDDQD) cADQA(E1k21 + E2k12) + cDDQD[k21(1 - E1) + k12(1 - E2)] AA ) (E1 - E2)(cAAQA - cDAQD) cAAQA(E1k21 + E2k12) + cDAQD[k21(1 - E1) + k12(1 - E2)] (A5) Due to the asymmetry in the spectral band of most fluorophores, which show a steeper increase before the peak compared with the slower decrease at longer wavelengths, it is often possible to minimize the amount of acceptor intensity detected in the donor channel (i.e., cAD ≈ 0). However, the fact that the donor emission usually has a slowly decaying tail that overlaps with the acceptor emission makes cDA harder to neglect. Thus, although it is usually possible to find conditions in which corrections to eq 6 are unimportant, corrections to eq 7 are usually required. In this case
AD )
(E1 - E2)(cAAQA - cDAQD)
If a fraction of the donor intensity is recorded in the acceptor detector and vice versa, the visibilities defined in eqs 6 and 7 should be reformulated as
ηD1,2 + cDDDQD(1 - E1,2) + cADDQAE1,2
(A1)
ηA1,2 ) cDADQD(1 - E1,2) + cAADQAE1,2
(A2)
where the coefficients cxy represent a measure of the fraction of the intensity of donor and acceptor that is measured in each detector. They can be expressed in terms of the normalized spectral distribution of the fluorescence emission of donor and acceptor (FD(λ) and FA(λ)), the transmission properties of the dichroic filter (R(λ)), the transmission efficiency of the bandpass filter and other optical components in front of each detector (βD,A(λ)), and the wavelength-dependent efficiency of the detectors (κ(λ))
∫0∞ Fx(λ)κ(λ)βD(λ)[1 - R(λ)]
cxA )
∫0∞ Fx(λ)κ(λ)βA(λ)R(λ)
x ) D, A
x ) D, A
(A3)
Here, we assume that the dichroic filter is nonabsorbing (so the reflectance and the transmittance add up to one), and that the two detectors have the same spectral response. Plugging eqs A1 and A2 into eq 10, we obtain the following general expression for the kinetic contributions to the total correlation functions:
Xxy ) 1 + k12k21AxAy e-(k12+k21)τ where
[k21(1 - E1) + k12(1 - E2)]
AA )
Appendix A
cxD )
-(E1 - E2)
(A4)
cAAQA(E1k21 + E2k12) + cDAQD[k21(1 - E1) + k12(1 - E2)] (A6) so in order to extract the kinetic rates from the analysis of the ratios of Gxy, the quantity (cAAQA/cDAQD) has to be measured independently. A possible way to achieve this is by measuring the signal in the acceptor detector with a donor-only sample (I0) and with a donor-acceptor sample of known FRET efficiency and same absorbance at the excitation wavelength (IE). From eq A2, the ratio of these signals equals
IE ) E[cAAQA/(cDAQD) - 1] I0
(A7)
It is worth noting that the measurements necessary to characterize crosstalk do not need to be carried out with the same sample one wishes to investigate. For instance, the fraction of the donor intensity recorded in the acceptor detector, which is usually the most important contribution to crosstalk, can be determined using a solution of the dye used as the donor. Alternatively, crosstalk can be characterized from the spectral properties of the optical components used in the setup from eq A3. Appendix B From eqs 12 and 13, a maximum in GDA will occur at a time τm > 0 if a positive value of τ exists so that
-
[
]
XDA(τm) - 1 γ/2 1 + ) (k12 + k21) τm + tD γτm + tD XDA(τm)
(B1)
For simplicity, we rewrite eq 11c as XDA(τ) ) 1 - A exp(aτ), so eq B1 becomes
7400 J. Phys. Chem. B, Vol. 111, No. 25, 2007
Torres and Levitus
1 γ/2 aA + ) τm + tD γτm + tD eaτm - A
(B2)
lengthy, and beyond the scope of this paper, so we will not present it here. References and Notes
Consider the function
F(τ) )
[
1 γ/2 aA + τ + t γτ + tD e -A D aτ
]
(B3)
so that F(τm) ) 0. The goal is to prove that eq 14 is a necessary condition for F(τ) ) 0 to be true at some positive value of τ. Equation 14 can be manipulated to yield
aAtD + A > 1
(B4)
so, if the inequality in eq 14 holds, the functions y1 ) exp(at) and y2 ) aAt + (aAtD + A) will cross at some t > 0. This is true, since y2 is a straight line with intercept greater than 1, and y1 can grow indefinitely (a > 0). Thus, there exists a positive τ (which we call τ2) so that
aAtD + A + aAτ2 ) eaτ2 w
aA 1 ) eaτ2 - A τ2 + tD
(B5)
Therefore, F(0) > 0 (eq 14), and F(τ2) ) -γ/[2(γτ2 + tD)] < 0. Since F is continuous on the closed interval [0, τ2], there is at least one positive number τm < τ2 such that F(τm) ) 0 (intermediate value theorem). Therefore, if eq 14 holds, there will be a maximum in GDA at a time 0 < τm < τ2. We note that, as long as eq 14 holds (i.e., tD > (2 + γ)(1 A)/(2aA)), τm decreases with decreasing tD (Figure 2). When tD ) (2 + γ)(1 - A)/(2aA), F(τ) is 0 at the origin, so τm ) 0 and the maximum in GDA occurs at τ ) 0. Therefore, the position of the maximum in GDA moves to shorter times as the difference between tD and (2 + γ)(1 - A)/(2aA) decreases, and disappears when this difference equals zero. We demonstrated above that the inequality of eq 14 predicts the existence of a “bump” in GDA. However, a complete proof requires that we show that there is no “bump” when tD < (2 + γ)(1 - A)/(2aA). We proved this to be true for γ e 1, which represents the most physically relevant situation23 where z0 g r0 (i.e., the confocal volume is longer in the axial direction). However, we note that a maximum in GDA is mathematically possible when eq 14 does not hold, but this requires γ . 1, which represent physically unrealistic situations. This proof is
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