Letter pubs.acs.org/JPCL
Amplitude of Relaxations in Fluorescence Correlation Spectroscopy for Fluorophores That Diffuse Together Yandong Yin, Rongfeng Yuan, and Xin Sheng Zhao* Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Department of Chemical Biology, College of Chemistry and Molecular Engineering, and Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing, 100871, China S Supporting Information *
ABSTRACT: The amplitude of chemical relaxations in fluorescence correlation spectroscopy (FCS) is an important parameter that directly relates to not only the equilibrium constant of the relaxations but also the number of individual fluorophores that diffuse together. In this Letter we answer the question how exactly the amplitude of the relaxations in FCS changes with respect to the number of identical fluorophores on one cargo. We anchored tetramethylrhodamine molecules onto each arm of a DNA Holliday junction molecule so that the codiffusing dyes were capable of performing independent fluorescent fluctuations. We found that the amplitudes of the relaxations were inversely proportional to the number of the dyes on each cargo molecule, well agreeing with the theoretical prediction derived in this Letter. The result provides a guideline for the FCS data analysis and points out a simple way to determine the number of molecules that a cargo carries. SECTION: Kinetics and Dynamics
H
−1 ⎛ ⟨I(t )I(t + τ )⟩ τ ⎞ G (τ ) = = G(0)⎜1 + ⎟ τD ⎠ ⟨I(t )⟩2 ⎝
ow many proteins, nucleic acids, or other biomolecules that a cargo system (e.g., exosome) carries is a key feature for understanding the mechanism of some cargo assisted signal transductions or other biological processes.1−4 To directly and accurately count the molecules taken by each cargo, however, is still difficult for most current techniques on an ensemble level. Single-molecule assays, such as the surface-immobilized singlemolecule5−8 and single-molecule trapping9−11 approaches, can succeed to obtain the number information by directly counting the photobleaching steps of fluorescent probes tagged to the on-cargo molecules. Other single molecular methods, such as photon counting histogram (PCH),12−14 can also address this issue through the brightness analysis of each cargo molecule that carries multiple fluorophores. Akin to single molecule assays, fluorescence correlation spectroscopy (FCS)15−27 has been developed to monitor the thermodynamic fluctuations that allows people to simultaneously obtain multiple kinetic and thermodynamic parameters,28 including the relaxation time of conformational or chemical fluctuation (τR), diffusion coefficient (D), equilibrium constant of the physical and chemical processes (K),29,30 and so on. However, we have not yet found a discussion about how FCS deciphers the number of fluorophores that diffuse together with their cargo molecule. (The issue here is not about how the number of independently diffusing fluorophores is related to the FCS functions.). A classical expression about the FCS correlation function, G(τ), for a system of the single-fluorophore-labeled molecules (one-cargo-one-fluorophore) reads28 © 2013 American Chemical Society
−1/2 ⎛ ⎛ ⎛ τ ⎞⎞ τ ⎞ ⎜⎜1 + αR exp⎜ − ⎟⎟⎟ + 1 × ⎜1 + 2 ⎟ w τD ⎠ ⎝ τR ⎠⎠ ⎝ ⎝
(1)
where I(t) is the instantaneous fluorescent intensity, τD is the characteristic time for the molecule diffusing through the optical region, w = wz/wr in which wz and wr are the geometry dimensions of the confocal volume in the FCS measurement, G(0) = 1/N with N being the average number of independently diffusing molecules in the confocal volume, and αR is the preexponential amplitude of a relaxation and is a function of the equilibrium constant of the process.29,30 G(τ) can be split into G(τ ) = G(0)G D(τ )G R (τ ) + 1
(2)
where −1/2 −1 ⎛ τ ⎞ ⎛ τ ⎞ G D(τ ) = ⎜1 + ⎟ ⎜1 + 2 ⎟ τD ⎠ ⎝ w τD ⎠ ⎝
⎛ ⎛ τ ⎞⎞ G R (τ ) = ⎜⎜1 + αR exp⎜ − ⎟⎟⎟ ⎝ τR ⎠⎠ ⎝
(3)
Received: November 16, 2012 Accepted: January 2, 2013 Published: January 2, 2013 304
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both the equilibrium constant and the number of the fluorophores are encoded in the pre-exponential amplitude. To verify the theoretical prediction, we took a DNA Holliday junction31,32 as a cargo molecule. The DNA Holliday junction is formed by hybridization through specific base pair recognition of four predesigned different single-stranded DNA (ssDNA, Supporting Information, Tables S1 and S2). In this way, we can easily put a precise number of TMR molecules onto the DNA Holliday junction by labeling the dye onto desired single strands. First, we anchored different numbers of TMR dyes on the arms of each DNA Holliday junction composed of the TMR-HJ series (Figure 2a; Supporting Information, Table S1). In this case, no chemical quenching is available, and we monitored the relaxation due to the singlet−triplet transition of the TMR molecule.33,34 Indeed, in the FCS curves of this cargo system shown in Figure 2b the amplitude significantly decreased with the increasing number of probes. Through a global fit of each FCS curve (Supporting Information, Figure S1 and Table S4) we obtained a series of the amplitude α (Methods) and found that the amplitude was linearly against 1/n (Figure 2c). Notably, the intercept on the axis of amplitude, which theoretically should be 0 according to eq 5, given by the linear fit was 0.030 ± 0.003. This could be due to the incomplete hybridization, which resulted in the existence of lower order labeling of TMR on the DNA molecules. As shown in the PAGE image of the DNA Holliday junction in Figure 3, some DNA strands did not form the full size DNA Holliday junction because the stoichiometry of different DNA strands could hardly be controlled to be exactly the same. Other possibilities, such as incomplete labeling during DNA synthesis and invalid labeling caused by photobleaching, could also bring in lower order labeling molecules in all one-cargo-multifluorophore cases. These less labeled molecules mixed within the one-cargo-multifluorophore system would lead to a nonzero intercept in the extrapolation. Because the DNA Holliday junction is relatively massive, we could see the correlation coming from the rotational diffusion35,36 in the short time range (Figure 2b, Supporting Information, Figure S1 and Table S4). Interestingly, as the number of probes varied from 1 to 4, the rotational amplitude also exhibited a linear dependence on 1/n with a nonzero intercept (Figure 2c, bottom). The rotational fluctuation from different TMR molecules would be completely correlated and exhibit identical correlation if the whole DNA Holliday junction system (including the DNAs, the linkers, and the dye molecules) was a rigid body. The almost 1/n dependence indicated that the rotational coherence was mostly destroyed between different dyes, suggesting that the system has wobbling and vibrations that are faster than or comparable to the time scale of the rotational relaxation. In fact, neither the DNA Holliday junction, nor the linker between DNA and TMR, nor the TMR itself can be regarded as a perfectly rigid body.37,38 The mostly uncorrelated rotational relaxation is therefore conceivable. Similarly, the fitting line in Figure 2c (bottom) does not go through the origin of the coordinate. It should partially represent the fact that different dyes on the same cargo molecule do have some rotational correlation because of the partial rigidity of the system and should partially be due to the incompleteness in labeling and hybridization. We then anchored TMR−guanine pairs to the arms of the DNA Holliday junction composed of the TMR-G-HJ series to form a chemical relaxation system (Figure 4a; Supporting Information,
which are contributions from the translational diffusion and physical or chemical relaxation, respectively. Compared with the FCS curve of the one-cargo-one-fluorophore system (Figure 1a,b), we observed a smaller amplitude in the correlation for relaxations in the one-cargo-multifluorophore system, in which each cargo carries a certain number of fluorophores or fluorescently labeled molecules (Figure 1c,d). In the one-cargo-multifluorophore system, the fluorophores on the same cargo diffuse together but undergo physical and chemical fluctuations individually. It is therefore interesting to understand how exactly the correlation function of such onecargo-multifluorophore system is related to that of the onecargo-one-fluorophore system (eq 1). In this Letter, we derived a theoretical expression for the FCS function of a freely diffusing cargo, on which n individual fluorescent probes are employed, and experimentally verified this theoretical prediction by applying DNA Holliday junction molecules as the cargo, on which one or more tetramethylrhodamine (TMR) dye molecules were labeled. Considering that n identical fluorescent probes are anchored on a freely diffusing cargo so that their diffusional motions are completely correlated but their physical and chemical relaxations occur independently (i.e., totally uncorrelated). The fluorescence intensity from this cargo can be written as I(t) = I1(t) + I2(t) + ... + In(t), where Ii(t) is the fluorescence intensity from the ith probe. The time-averaged fluorescence intensity from each probe should be the same and is written as . The fluorescence autocorrelation function on one cargo is therefore written as n
n
⟨∑i = 1 Ii(t ) ∑i = 1 Ii(t + τ )⟩ ⟨I(t )I(t + τ )⟩ = G (τ ) = n 2 ⟨I(t )⟩ ⟨∑i = 1 Ii(t )⟩2 n
= =
n
∑i = 1 ⟨Ii(t )Ii(t + τ )⟩ + ∑i ≠ j ⟨Ii(t )Ij(t + τ )⟩ n2⟨I0⟩2 1 n−1 Gii(τ ) + Gij(τ ) n n (4)
where Gii(τ) is the single fluorophore autocorrelation function and Gij(τ) is the cross-correlation function between different fluorophores. For simplicity, we consider only one relaxation process on a probe with a relaxation time τR. The autocorrelation function of the ith probe is just that of the onecargo-one-fluorophore system, Gii(τ) =GD(τ)GR(τ)+1, while the cross-correlation between the ith and jth probe should be Gij(τ) =GD(τ)+1 because the two probes perform chemistry independently but diffuse together with the cargo molecule. Now that the number of the cargos in the confocal volume fluctuates around the average number N, a factor of G(0) = 1/N will be derived as in the one-cargo-one-fluorophore system.28 Combining all these factors together the general correlation function for the one-cargo-multifluorophore system reads −1/2 −1 ⎛ ⎛ τ ⎞⎞ ⎛ α τ ⎞ ⎛ τ ⎞ ⎜⎜1 + R exp⎜ − ⎟⎟⎟ + 1 G(τ ) = G(0)⎜1 + ⎟ ⎜1 + 2 ⎟ n τ w τ ⎝ τR ⎠⎠ ⎝ ⎝ D⎠ ⎝ D⎠
(5)
It is seen that the only difference between the onecargo-multifluorophore system and the one-cargo-one-fluorophore system is that now the pre-exponential amplitude is replaced by α = αR/n, where αR is the amplitude when n = 1. In other words, 305
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Figure 1. Schematic illustration of the one-cargo-one-fluorophore and one-cargo-multifluorophore systems. (a) In the one-cargo-one-fluorophore system both chemical reaction and diffusion of different fluorophores are uncorrelated in an FCS experiment. (b) An example of the FCS curve of the system illustrated in panel a by using the DNA Holliday junction with 1 TMR labeled on each DNA Holliday junction molecule (TMR-G-HJ-1, Supporting Information, Table S2). (c) In the one-cargo-multifluorophore system, both chemical reaction and diffusion of fluorophores on different cargoes are uncorrelated, but the different fluorophores on the same cargo are completely correlated in diffusion yet uncorrelated in chemical reactions in an FCS experiment. (d) An example of the FCS curve of the system described in panel by using the DNA Holliday junction with 4 TMR labeled on each DNA Holliday junction molecule (TMR-G-HJ-4, Supporting Information, Table S2). The dashed lines in both panels b and d represent the FCS curves when only the translational diffusion was considered.
and fitted the FCS curves for the sample of TMR-HJ-2 at different concentrations. Indeed, both N (so that the concentration of the cargo) and the pre-exponential amplitude (so that the number of the fluorophores on the cargo) can be separately fitted with sufficient accuracy in one FCS curve (Supporting Information, Figure S3 and Table S6). FCS is a powerful technique that is easy to implement and sensitive in probing dynamic processes. It will find more and more applications in biological problems in vitro and in vivo. Multiple labeling on biomolecules or particles could often be encountered, especially when studies are carried out in living cells and on molecular complexes where controlling the number of fluorophores is very hard if not impossible. The accurate relation between the pre-exponential amplitude in the FCS curve and the number of codiffusing fluorophores anchored on the same cargo is an important fact that has mostly been left out the consideration previously in the FCS analysis. Most importantly, without knowing the relationship discussed in this Letter, derivations on equilibrium constants and corresponding rate constants of the tested fluctuation by using the FCS technique would be wrong for a multiple labeling system according to eq 5. When multiple labeling is implemented on a cargo one has to be careful about calculating the equilibrium constant by using the pre-exponential amplitude. In such a situation, the classical expression derived for one-cargo-one-fluorophore system to calculate the equilibrium constant from the amplitude29,30 must be revised accordingly. The number information coded in the preexponential amplitude can be used to count the on-cargo molecules that diffuse together. It is often a key issue to know how many molecules a molecular complex carries in a biological process. Single-molecule assays are capable of providing the information on the number of fluorophores labeled on a cargo. Counting steps in photobleaching,42 analyzing photon-anti-
Table S2). When TMR is terminally labeled on a dsDNA, it can stack on the end of the dsDNA as if an additional base39,40 and its fluorescence is quenched by the adjacent guanine due to photoinduced electron transfer (PET).30,41 The chemical relaxation between the stacked state and the unstacked state of the TMR therefore gives rise to individual fluorescence fluctuations on each TMR−guanine labeled arm of the Holliday junction. Figure 4b shows the FCS curves of the DNA Holliday junction carrying 1 to 4 TMR-guanine pairs. Through a global fit of the FCS curves (Supporting Information, Figure S2 and Table S5) we observed much more prominent amplitude caused by the PET induced process, and the amplitude of the chemical relaxation is, again, a linear function of 1/n (Figure 4c, top). In this case, the extrapolation of 1/n relationship became more ideal, probably because PET only occurred within the dsDNA and the dyes so that the ssDNA did not make contributions to the amplitude of fluorescence quenching. A minor component due to the singlet−triplet transition (Figure 4b, Supporting Information, Figure S2 and Table S5) was also observed, and its amplitude was linearly against 1/n, too, as aforementioned (Figure 4c, bottom). Similarly, a minor contribution from the rotational diffusion was observed. Because the time scales of the rotational relaxation and the PET quenching were very close, they could not be determined just by fitting. In the data analysis, the rotational contribution in Figure 4b was taken to be that in Figure 2b (Methods). According to eq 5, N and n seem to be coupled together. One may wonder to what extent they can be determined confidently. The fact is that their coupling is very weak as long as the time scale of the chemical relaxation is faster than and well-separated from that of the translational diffusion. In such a case, the fit to the translational diffusion provides a good and consistent internal scaling reference so that N and n can be determined independently. To demonstrate this, we recorded 306
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Figure 3. Polyacrylamidegel electrophoresis (PAGE) image of the DNA Holliday junction molecules. TMR-HJ-1, TMR-HJ-2, TMRHJ-3, and TMR-HJ-4 are the Holliday junctions with 1 to 4 TMRs labeled on each DNA Holliday junction molecule, respectively, which do not have the PET quenching (Supporting Information, Table S1). The TMR-G-HJ-1, TMR-G-HJ-2, TMR-G-HJ-3, and TMR-G-HJ-4 are the Holliday junctions with 1 to 4 TMR-Guanine pairs anchored on each molecule, respectively, which have the PET quenching (Supporting Information, Table S2). The ssDNA is one of the four strands of the Holliday junction molecules, and the dsDNA is the hybridization of the ssDNA with its complementary ssDNA (Supporting Information, Table S3). The partially hybridized DNA and unhybridized DNA mixed within the DNA Holliday junction system were seen.
transition-induced fluctuation could be such a relaxation process internally existing in most of chromophores that are widely applied in FCS and single-molecule techniques. Furthermore, the amplitude of the singlet−triplet fluctuation can be easily tuned and controlled by changing the illumination power. One can artificially introduce such kind of chromophores and predetermine αR for the singlet−triplet transition so that n can be obtained according to eq 5 by monitoring the singlet−triplet fluctuation. Other fluctuations, such as the PETinduced fluctuation, could also be considered. An interesting example can be seen in the recent experiment of the diffusion-decelerated fluorescence correlation spectroscopy (ddFCS), where a polystyrene microsphere was used as the cargo to slow down the diffusion motion.23 With a diameter of 1.87 μm, the streptavidin-coated microsphere could carry as much as thousands of fluorescent molecules on its surface. However, because the size of the microsphere was much bigger than the confocal volume, only a small part of the microsphere was under the laser illumination when it slowly diffused through the optical region. Hence, the number of dye molecules contributing to the amplitude as the factor in α = αR/n is the average number of the molecules in the laser focus other than that of all dyes a microsphere carries. In the calculation of the equilibrium constant one should put the effective number of dyes in the place instead of the number of dyes on the whole microsphere. In summary, for a system that fluorophores diffuse together but individually carry out physical and chemical relaxations, the amplitude of the correlation function from the relaxations in the FCS curve is proportional to the inverse of the number of the fluorophores. This property has the potential application to easily determine the number of the fluorophores on one cargo. This finding also has very important consequence on the calculation of equilibrium constant and rate constant when the amplitude of the relaxation is used as one of the original sources. Our study implies that the FCS technique is hopeful to
Figure 2. Amplitudes of the correlation decay caused by singlet− triplet transition and rotation. (a) Schematics of the DNA Holliday junctions that carry 1 to 4 TMR (green) as fluctuating probes. (b) Normalized FCS curves of the DNA Holliday junctions (solid line) with different numbers of probes as described in panel a. The dashed line represents the FCS curves when only the translational diffusion was considered. (c) Pre-exponential amplitude (mean ± s.d.; n = 3) for the singlet−triplet transition (top) and rotational diffusion (bottom) given by a global fit (Methods, Supporting Information, Table S4) of the FCS curves in panel b. The solid lines are the linear fit against 1/number of probes, respectively.
bunching,43 and performing photon statistics14 (by differing the brightness of the one-cargo-multifluorophore system from that of the one-cargo-one-fluorophore system) are some successful examples. We offered here an alternative way to acquire the information with the FCS technique. By knowing the relationship provided herein the number of fluorophores on a cargo can be determined simultaneously with other properties in one FCS experiment. To apply the technique, there must exist in the chromophore at least one clear fluorescence fluctuation process faster than the translational diffusion. The relaxation processes among the on-cargo fluorophores have to be independent, and the pre-exponential amplitude in a one-cargo-one-fluorophore case (αR) has to be sorted out. It is not difficult to find such a process that αR can be predetermined and is uncorrelated among different fluorophores. The singlet−triplet 307
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Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
This work was supported by NKBRSF (2012CB917304 and 2010CB912302) and NSFC (21233002 and 20973015).
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Figure 4. Amplitudes of the correlation decay caused by PET quenching and singlet−triplet transition. (a) Schematics of the DNA Holliday junctions that carry one to four TMR-guanine pairs as fluctuating fluorescent probes. The pink bar stands for guanine, whereas the ball stands for TMR. TMR is at the fluorescent bright state (green) when it is open and at the dark state (gray) when it stacks to the end of the double-stranded arm and is quenched by the adjacent guanine. (b) FCS curves of the DNA Holliday junctions with different numbers of probes as described in panel a. The dashed line represents the FCS curve when only the translational diffusion was considered. (c) Pre-exponential amplitudes (mean ± s.d.; n = 3) for the PET quenching (top) and singlet−triplet transition (bottom) given by a global fit (Methods; Supporting Information, Figure S2 and Table S5) on the curves in panel b. The solid lines are the linear fit against 1/number of probes.
simultaneously count the on-cargo signal molecules and monitor other thermodynamic, kinetic, and dynamic information in a complex system.
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ASSOCIATED CONTENT
S Supporting Information *
Methods and Figures S1−S3 and Tables S1−S6. This material is available free of charge via the Internet at http://pubs.acs.org
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. 308
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