ARTICLE pubs.acs.org/ac
Information Content in Fluorescence Correlation Spectroscopy: Binary Mixtures and Detection Volume Distortion Jonathan D. Lam,† Michael J. Culbertson,† Nathan P. Skinner,† Zachary J. Barton,† and Daniel L. Burden*,† †
Chemistry Department, Wheaton College, Wheaton, Illinois 60187, United States
bS Supporting Information ABSTRACT: When properly implemented, fluorescence correlation spectroscopy (FCS) reveals numerous static and dynamic properties of molecules in solution. However, complications arise whenever the measurement scenario is complex. Specific limitations occur when the detection region does not match the ideal Gaussian geometry ubiquitously assumed by FCS theory, or when properties of multiple fluorescent species are assessed simultaneously. A simple binary solution of diffusers, where both mole fraction and diffusion constants are sought, can face interpretive difficulty. In order to better understand the limits of FCS, this study systematically explores the relationship between detectionvolume distortion, diffusion constants, species mole fraction, and fitting methodology in analyses that utilize a two-component autocorrelation model. FCS measurements from solution mixtures of dye-labeled protein and free dye are compared to simulations, which predict the performance of FCS under a variety of experimental circumstances. The results reveal a range of conditions necessary for performing accurate measurements and describe experimental scenarios that should be avoided. The findings also provide guidelines for obtaining meaningful measurements when grossly distorted detection volumes are utilized and generally assess the latent information contained in FCS datasets.
’ INTRODUCTION Fluorescence correlation spectroscopy (FCS) is widely used for monitoring translational diffusion, spectroscopic properties, molecular structural dynamics, and reaction kinetics in numerous analytical scenarios.111 Examples include the measurement of synthetic and biological polymers,12,13 nanoscale particles,14,15 environmental toxins,16 living cells,17,18 and the early detection of disease.19,20 The technique commonly employs a diffractionlimited (or nearly diffraction-limited) three-dimensional (3D) confocal detection volume (DV) to monitor fluorescent fluctuations of molecules moving in and out of this probed region. Fluctuating fluorescence signals are collected and correlated to produce a curve that can be interpreted through mathematical fitting. The general utility of FCS has been widely recognized for decades. Both the older and more-recent FCS literature contains considerable discussion regarding the reliability and interpretation of FCS information.2138 Although these studies explore various aspects of the measurement technique or data analysis individually, the principal factors contributing to systematic bias in FCS are inherently inter-related. These factors include (1) a lack of correspondence between the 3D geometry of the experimental DV and the geometry assumed in the mathematical model; (2) a lack of correspondence between the presumed molecular dynamics and the dynamics actually occurring in the sample; (3) the effectiveness of the mathematical fit and its tolerance to local minima for a given number of model parameters; and (4) the signal-to-noise ratio present in the correlation data. In this study, we explore the relatedness of these factors in a uniquely holistic fashion, whereby the impact of all four factors is probed simultaneously with a comprehensive numerical r 2011 American Chemical Society
simulation (Single Molecule Diffusion Simulator, SMDS).The simulator closely mimics the performance of an FCS microscope under various experimental conditions. More specifically, we focus our investigation on the impact of factors 1, 3 and 4, while factor 2 is tightly controlled within the simulation. In this study, the shape of the detection volume is systematically altered. In addition, the molecular dynamics of the sample and the model are perfectly matched, while the impact of the fitting method is explored by using different free and fixed variables. Lastly, interaction of the signal-to-noise ratio with these factors is revealed by controlling species brightness and simulated instrument noise. The simulation algorithm utilized for this work has been described previously.39 It stands distinct from other FCS simulators by making use of a networked cluster of computer processing units (CPUs), where each CPU executes a random walk of point particles through a 3D DV. Collectively, the nodes of the cluster allow the simulation of long measurement times, while enabling data acquisition conditions to be varied rapidly. In addition, the SMDS uses a 3D spatial map of the DV, allowing the evaluation of arbitrary DV shapes. Output is interpreted by fitting and the fitted parameters are compared to the simulation input values. This permits direct assessment of the accuracy afforded by a specific FCS apparatus. We also directly compare the predictions of the SMDS to measurements of aqueous standards. The results verify the integrity of the simulator, permitting the latent information within an FCS dataset to be evaluated under a variety of conditions. Received: March 16, 2011 Accepted: May 23, 2011 Published: May 23, 2011 5268
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Analytical Chemistry Numerous questions can be addressed using this approach. For example, given a specific level of DV distortion, how reliable are the extracted values for particle concentration or mole fraction? When does poor signal-to-noise ratio lead to data misinterpretation? How much foreknowledge of the sample and the instrument is required to extract a meaningful representation of solution properties for a given type of sample? In the context of protein analysis, small organic dyes with aqueous diffusion constants of ∼3 106 cm2/s are ubiquitously used as labels.4043 Conjugation of the dye to the protein often lowers the diffusion constant of the dye by a factor of 310, depending on the size of the protein. Because an excess of free dye is commonly used in labeling protocols, and because the reaction yield is often ω2), which differs radically from standard dimensions. Although the representative radii are used in eqs 1 and 2 for fitting purposes, discrete DV maps (Figures 1AC) are used to generate autocorrelation curves in the SMDS. Concentration calculations are based on the representative Gaussian volumes, in a manner analogous to experimental FCS measurements. Simulation Scenarios. Two-component simulations performed by the SMDS use a fast component (Df), set at 2.9 106 cm2/s. Slow components (Ds) are chosen to be either 2.9 107 cm2/s (0.1 Df) or 9.7 107 cm2/s (0.33 Df). The value chosen for Df matches the actual diffusion constants of RB and the values chosen for Ds define a range around the diffusion constant of R-BSA (i.e., 4.5 107 cm2/s, according to measurements performed on our microscope). In addition, because the 0.1 scenario produces greater graphical separation between the two components in the autocorrelation curve (see Figure S3 in the Supporting Information), the effectiveness of the fitting algorithm can be tested by comparing results from the 0.33 and 0.1 scenarios. Theoretical mole fractions of the fast component (af,th) are simulated at 0.05, 0.3, 0.5, 0.7, and 0.95. We define brightness relative to the fast component and set the relative brightness ratio of the slow component (Rs) to 0.25, 1, and 4. This mimics analysis scenarios ranging from a dim slow component (Rs = 0.25), as in situations where the covalently joined tag undergoes fluorescent quenching (or is off resonance with the instrument optics), to a bright slow component (Rs = 4),
as in situations where multiple fluorescent tags are bound to one protein molecule. Autocorrelation functions from mixtures of these two species produce different decay profiles, but distinctions between the curves grow subtle at extreme mole fraction or brightness ratios, where information content is generally lost to noise. The signal-to-noise ratio of simulated curves is adjusted to approximately match that from RB:R-BSA solutions (see the Supporting Information). Data Analysis. Autocorrelated datasets are fit using eqs 1 and 2. For the RB:R-BSA solutions, we use a brightness ratio (Rs) of 1.13, as determined via measurement of each solution individually (see the Supporting Information). Two methods of fitting are employed in an attempt to quantify the latent information residing in the correlation curves. In both cases, the shape factor (γ), average intensity ÆIæ, background ÆBæ, relative brightness of the slow component (Rs), and K are set as fixed parameters, while DC, the average number of molecules for the fast component (ÆNfæ), the average number of molecules for the slow component (ÆNs)æ, and the diffusion constant of the slow component (Ds) freely vary. The “fixed” fitting method holds the value of the fast component (Df) constant. The “free” approach simply allows both Df and Ds to vary freely. We also explored scenarios where Ds, Df, and Rs were all allowed to vary freely; however, this combination yields nonreproducible results, which indicates overparametrization of the model. Results Matrix. Results track error in the fitting parameters that arise from changes in (1) species mole fraction, (2) the number of fitting parameters, (3) DV distortions, (4) relative brightness of the fluorescent species, and (5) the difference in magnitude between diffusion constants. We evaluated 180 specific combinations of these five factors, which are tabularized in matrix form (see Table S2 in the Supporting Information). The number of entries in the matrix arises from two different fitting schemes, three different DV profiles, two different diffusion constant ratios, three values of relative brightness, and five mole fractions (2 3 2 3 5 = 180). Each of the 180 possible combinations was performed in triplicate, resulting in a total of 540 simulations.
’ RESULTS AND DISCUSSION The SMDS produces a fluorescence record of particles diffusing through a user-supplied 3D DV map. Figure 1 shows three DV profiles that range from the ideal theoretical Gaussian (Figure 1A) to a highly distorted DV produced by crossing two laser beams (of the same wavelength) in the sample plane (Figure 1C). Figure 1B shows a DV for an aligned FCS microscope and closely matches the Gaussian approximation, although not entirely. Deviations from the ideal are particularly notable in the outer edge, where evidence of Airy rings from the focused laser can be seen. Figure 1B represents conditions that are commonly employed by FCS experimentalists. Although DV distortions for experiments using only one excitation wavelength are generally not as extreme as those displayed in Figure 1C, beam overlap for dual-color crosscorrelation measurements can be difficult to achieve.49 In this study, the distortions present in Figure 1C serve as an illustrative benchmark for characterizing worst-case analytical scenarios. For brevity, we highlight the representative trends in the 180element results matrix that appear to be most significant. Figure 2 gives one perspective of the relationship between diffusion constants (Ds, Df), mole fraction (af,th), relative brightness (Rs), the number of variable fitting parameters, and DV 5270
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Figure 2. Accuracy of diffusion parameters at various theoretical mole fractions of the fast component (af,th). In all panels, the horizontal black line is the expected diffusion constant. In panels AF, Ds/Df = 0.33. Data contained in panels AC arise from the theoretical Gaussian, while data in panels DF were generated using the distorted beam profile. Relative brightness of the slow component is represented as Rs = 0.25 (solid red square), Rs = 1 (solid green circle), and Rs = 4 (solid blue triangle). In panels GI (gold diamond, RB:R-BSA measurement), relative brightness was measured to be 1.13, making these panels most comparable to the green markers in panels AC. Panels A, D, and G show results from the fixed fitting scenario. Panels B, C, E, F, H and I show results from the free fitting scenario. Error bars are the SEM analysis results for three independent simulations. Inset in panel C shows the SEM magnification for each data point in panel C.
distortion. As expected, accurate diffusion constants are obtained from the Gaussian DV. Figures 2AC show diffusion results over the full range of af,th with deviations arising at the extreme low and high values of Rs. Specifically, in Figure 2A, error in the estimated diffusion constant increases when the slow component is dim (Rs = 0.25, red) and the slow component makes up a small fraction of the mixture (i.e., af,th is large). This systematic change begins when the signal generated by the slow component no longer dominates the correlation function and, instead, is overtaken with signal from the fast component and inherent shot noise. At large mole fractions (af,th = 0.95, in Figures 2A and 2B), the deviations from the expected diffusion constant (Ds) are substantial. Accompanying the increased error is a parallel loss of precision, as evidenced by the increased size of the error bars in data points at large mole fraction (af,th) and small relative brightness (Rs). In Figure 2B, a larger relative error and variance in Ds is observed (compared to Figure 2A), presumably because of the increased number of variable fitting parameters. The monotonic rise in diffusion coefficient for Ds is also coupled with a drop in af (see Table S2 in the Supporting Information) as the fitting routine attempts to compensate for the inflated diffusion constant. In the case of Figure 2B (at Rs = 0.25, red), the average
value of Ds actually exceeds the expected value of the fast component (i.e., 2.9 106 cm2/s), due to the extra fitting parameter in the model. The systematic deviation of the fast diffusion component (Df) in Figure 2C remains relatively small throughout the range of mole fractions and relative brightness evaluated. This suggests that measurement of the fast component possesses a larger tolerance to low signal conditions. However, the inset in Figure 2C does show a systematic loss of precision as the mole fraction of the fast component (af,th) grows small. This indicates that the limit of reliability for this parameter is being approached at small af,th. Trends in Figures 2AC demonstrate a fundamental quantitative limit inherent to FCS, even with idealized optical alignment. In these panels, information is apparently lost to noise when deviations from the expected values become notable. The incurrence of error in an FCS experiment will, of course, be dependent upon the particular signal, noise, and acquisition time for a given measurement. Panels GI show results from the RB:R-BSA standard measurements, which were acquired for the same period of time (600 s) and the optical conditions were adjusted so that the signal-to-noise ratio approximately matched that of the simulated curves. Here, the trends observed experimentally 5271
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Figure 3. Comparative impact of detection volume distortion (theoretical Gaussian (hollow red square); aligned (hollow green circle); distorted (solid blue triangle)) on diffusion constants. The expected value is shown with the horizontal black line. Only small differences from the expected values are seen under fixed fitting conditions (A), while larger deviations are observed under free fitting conditions (B and C) across the range of theoretical mole fractions (af,th). In all three panels, Ds /Df = 0.33 with Rs = 1 (equal brightness).
generally match the trends defined by the green lines (Rs = 1) from Figures 2AC. The notable exception is the deviation seen in the 0.95 data point from Figure 2I, which can be explained by a lack of ideal BSA homogeneity. More generally, the trends observed here, as well as the trends noted in Figure 5 (shown later in this paper), indicate that simulations performed over a broad range of detection scenarios provide meaningful insight into the expected errors and limits of FCS. Figures 2DF show data generated using the highly distorted DV. Interestingly, despite the distortion severity, usable diffusion results can be attained under some circumstances. The singlecomponent calibration procedure, which is analogous to DV calibration methods performed in all quantitative FCS measurements, creates a Gaussian approximation of the distorted DV. Even though the dimensions of the substituted profile are not similar to the distorted DV, accurate results can be obtained across the full range of mole fraction (af,th) when sufficient foreknowledge of the system is available (i.e., Df is fixed) and the brightness of the slow diffuser is moderate or high (Rs g 1). However, the data also suggest that it is not possible to acquire accurate diffusion information when there is insufficient foreknowledge of the system (Df and Ds are free). Figures 2E and 2F show large systematic deviations and relatively large SEMs (standard error of the mean) across the full range of mole fraction values. These figures demonstrate how DV distortions can, under certain circumstances, place significant limits on the usability of FCS. Under most laboratory conditions, DV distortions are not as extreme as the distortion represented in Figure 1C. More typically, DV profiles are similar to Figure 1B, which is closely approximated by a Gaussian, deviating only in its outermost regions. Figure 3 compares the impact of progressive DV distortions on the diffusion parameters for the situation where each species is equally bright (Rs = 1). As can be seen, accurate diffusion results are obtained over a reasonably large range of mole fraction (af,th), with exceptions occurring at extreme values. When the fast diffusion constant is held fixed (Figure 3A), a small error in the slow diffusion constant (Ds) is present only for mixtures dominated by the fast component (i.e., large af,th). However, when Df is also allowed to vary freely (see Figures 3B and 3C), a systematic bias is revealed. These results highlight the impact of misaligned DV profiles (even when only slightly misaligned) in the determination of the diffusion constant. As is also evidenced in Figure 2, the general susceptibility to error in Ds grows large as af,th increases (Figure 3B), while the susceptibility of Df to error increases as af,th decreases (Figure 3C).
Figure 4. Comparative impact of DV distortion on extracted concentration (CT) for Ds/Df = 0.33 (hollow red circle) and Ds/Df = 0.1 (hollow blue inverted triangle). The horizontal black line indicates the expected (theoretical) value (CT,th = 0.7 molecules/μm3 or 1 nM): (A) ideal Gaussian, (B), aligned, and (C) distorted. Each data point is averaged from 15 trials (five mole fractions, af,th = 0.050.95, three trials each). For all determinations of CT, Df and Ds were allowed to vary freely.
In addition to diffusion constants, the total concentration and mole fraction information can be extracted from eq 1. Both parameters are calculated from the average number of molecules for each species (ÆNfæ and ÆNsæ) and the representative volume of the beam profile (VDV). Figure 4 compares the impact of DV distortions on the total concentration, CT = (ÆNfæ þ ÆNsæ)/VDV. 5272
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Figure 5. Percent error in the measured mixture composition (af) compared to the theoretical mole fraction (af,th) from (A) fixed and (B) free fitting scenarios. Ds/Df = 0.33. Increasing distortion in the DV profile generally increases error (Gaussian (solid red square), aligned (solid green circle);, distorted (solid blue triangle); RB:R-BSA measurement (solid orange diamond)). Points are reported as averages (N = 3) (for (solid orange diamond) (N = 2)). In both panels A and B, a brighter slow component (Rs = 4) produces the largest overall error, most notably when af,th is small.
Simulated concentrations were chosen to produce a mixed solution with a total theoretical concentration (CT,th) of 1 nM (0.7 molecule/μm3) and the resulting autocorrelation data sets were fit to determine a measured concentration (CT). Data are reported as the average of all CT for a given brightness ratio. Consistent with previous observations, increasing DV distortion produces greater deviations from CT,th (see Figures 4AC). Although CT is generally accurate across the range of brightness ratios (Rs) for the theoretical Gaussian profile (Figure 4A), a minor systematic offset can be seen when volume distortions are present (Figure 4C). In Figure 4A the minimal errors are most likely a result of averaging over the full range of brightness ratios (Rs), which amounts to 15 measurements. It is particularly noteworthy that even a severely distorted DV (Figure 4C) incurs errors that are only moderately larger than those seen with the aligned and ideal DVs. This suggests representative Gaussian dimensions enable reasonably accurate absolute concentration measurement, without undue concern for DV alignment. This is potentially nonintuitive, given that the representative Gaussian dimensions in no way reflect the actual contours of the distorted DV profile (i.e., in this case, ω1 > ω2). Figures 5A and 5B (fixed and free fitting scenarios, respectively) show results from the determination of mixture composition, where the percent error in mole fraction of the fast component (af) is plotted as a function of theoretical mole fraction (af,th) and relative species brightness (Rs). Although the trends shown in Figure 5 represent the 0.33 scenario only, similar trends are found in the 0.1 dataset. As can be seen, the largest errors in af are found at the smallest mole fraction (af,th) and largest relative brightness (Rs). This occurs because signal generated by dim, fast-moving species at low concentrations is masked by the relatively large signal from the brighter, slowmoving species at high concentration. This trend parallels the loss of precision shown in Figure 2C and the increased error evident in Figure 2F. In addition, when the relative brightness of the slow component is large (Rs = 4) and the theoretical mole fraction of the fast component increases, errors drop to negligible values, even for the distorted beam profile. A similar trend is also seen for diffusers of equal brightness (Rs = 1). With a dim slow component (Rs = 0.25) errors remain relatively constant across the range of tested mixtures (af,th) for both fixed and free fitting scenarios. Note also the trend agreement between simulation results and the experiments performed on RB:R-BSA standards, where Rs = 1.13.
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The major difference between fixed and free fitting scenarios is the lower bound of error that is achievable. In Figure 5A (fixed scenario), errors are reduced to ,1% for large af,th values. In Figure 5B (free scenario), errors generally shrink no further than 5%10%. The overall trend identified in Figure 5 indicates that the mixture composition plays a predominant role in the accuracy of mole fraction determinations (i.e., a large negative slope occurs along the af,th axis). As can be seen, the best measurements of the ratio occur at relatively high concentrations of the fast component.
’ CONCLUSION Detection volume distortions can be a major confounding factor in complex fluorescence correlation spectroscopy (FCS) measurements. Thus, it is important to fully understand the errors that can arise if the detection volume (DV) is not ideal, which is the case for all FCS instruments. This investigation explored a wide range of potential analytical scenarios. Although some combinations of parameters produce large systematic errors, accurate results can be achieved using any DV profile, as long as appropriate calibration procedures have been performed. For these investigations, signal levels were kept relatively high, which creates adequate interplay between the simulated species, the DV, and the fitting model, so that the run-to-run reproducibility is high and underlying systematic biases are not unduly obscured by noise. In general, as signal levels drop, a lack of reproducibility obscures the trends reported here. In addition, factors not included in this study, such as fluorophore photobleaching, triplet state effects, and the correspondence between the mathematical model and the solution dynamics, can be evaluated using the same approach. Such studies are ongoing and relevant for commonly encountered FCS measurement applications in the physical, biological, and engineering sciences. ’ ASSOCIATED CONTENT
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Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*Tel.: (630) 752-5065. Fax: (630) 752-5996. E-mail address:
[email protected].
’ ACKNOWLEDGMENT Support for this work has been provided by the American Chemical Society Petroleum Research Fund (ACS-PRF), the Camille and Henry Dreyfus Foundation Teacher-Scholar Award, and the National Science Foundation (Nos. 0957197, 0958697). J.D.L., N.P.S., and Z.J.B. gratefully acknowledge the Wheaton College Alumni Association and the Nelson Research Endowment for financial support. ’ REFERENCES (1) Boukari, H.; Sackett, D. L. Methods Cell Biol. 2008, 84, 659–678. (2) Cherdhirankorn, T.; Best, A.; Koynov, K.; Peneva, K.; Muellen, K.; Fytas, G. J. Phys. Chem. B 2009, 113, 3355–3359. (3) De Santo, I.; Causa, F.; Netti, P. A. Anal. Chem. 2010, 82, 997–1005. 5273
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