Lisa J. Carlson Department of Chemistry, University of Rochester, Rochester, NY 14627

Transcription

1 Lisa J. Carlson Department of Chemistry, University of Rochester, Rochester, NY I. INTRODUCTION All processes in nature tend to progress toward a state of equilibrium, and the temporal quantification of microscopic nonequilibrium processes that play a role in disrupting this pattern can be used to predict the macroscopic behavior of a system as it approaches equilibrium. 1 In order to achieve this goal, fluorescence correlation spectroscopy (FCS) and autocorrelation statistical analysis are used to study molecular diffusion and the kinetics of chemical equilibria by monitoring concentration fluctuations in a small ensemble of molecules. The historical development of this analytical technique, its principles, and its employment as a means of determining the volume and mass of an unknown protein and DNA sample are presented. Fluorescence correlation spectroscopy is used to study both the Brownian diffusion and the Figure 1: The confocal Gaussian detection volume. 1 equilibrium kinetics of a chemical system. 2 As molecules diffuse through a small, open detection volume, the number of particles in this space varies around its equilibrium value (Figure 1). Two major results can be extracted by tracking the emission from these particles: the temporal autocorrelation defines the time scale of diffusion and its variance provides the average number of fluorescent molecules in the detection volume. 2 Hence, when changes in the fluorescence are measured over time, these data provide information that can be used to determine diffusion coefficients, rate constants, and sample concentrations; aggregation and dynamics linked to rotation and translation are also of interest and can be studied in this manner. 3 Historically, fluorescence correlation spectroscopy is the mathematical descendant of quasi-elastic light scattering (QELS) spectroscopy. 2 While both FCS and QELS use a small sample volume to noninvasively probe concentration fluctuations, it is the enhanced sensitivity of fluorescence to conformational, environmental, and chemical changes in a system that allows FCS to be more useful in these scenarios than tracking scattered light. The roots of fluorescence correlation spectroscopy trace back to 1903, when Smoluchowski first outlined the relationship between Brownian movement and autocorrelation, which is used in the statistical analysis of FCS data. 4 However, it wasn t until 1972 that Magde, Elson, and Webb applied these ideas to the design of FCS spectrometers at Cornell University in order to study the kinetics of the reversible binding between ethidium bromide (a fluorescent nucleic acid synthesis inhibitor) and DNA. 4 was introduced by Koppel et al. in 1976; 2 The notion of incorporating a confocal microscope into a typical FCS setup since that time, advances in detection, identification, and characterization of single molecules in dilute solution have catalyzed a Renaissance in fluorescence correlation studies. 1

2 December 2005 Carlson - 2 II. THEORY The primary observable in FCS is fluorescence, and changes in fluorescence intensity reflect the concentration fluctuations of a molecular system (Figure 2a). 1 In FCS, the autocorrelation of this fluorescence variation is used to evaluate the temporal progression of a system around its equilibrium state. The autocorrelation is the cross-correlation of a signal with itself and is obtained by comparing a measured value at a time t with that at a later time (delayed by τ), as shown in Figure 2b. In this sense, one would expect two signals taken at nearly the same time to have a high correlation value and those taken farther apart to result in a lower correlation value (Figure 2c). The amplitude of the autocorrelation function is influenced by the number of molecules in the detection volume. The relative effect of one particular molecule on the total measured fluorescence decreases as the number of molecules increases, and the normalized amplitude of the autocorrelation function declines accordingly. 2 It is for this reason that extremely dilute concentrations are used for FCS studies, such that approximately five molecules are desired in the detection volume at one time. The normalized autocorrelation function can be expressed as the product of the fluorescence fluctuation at a given time t and a later time t + τ, normalized by the square of the average fluorescence: 1,2 (1) G(τ) = <δf(t)δf(t+τ)>/<f(t)> 2, where δf(t) is the difference between the fluorescence intensity at time t and its average value. More specifically, if chemical kinetics are neglected and only one species is being detected, then the autocorrelation function for the confocal observation volume that takes the shape of a prolate ellipsoid is given by 1,2 (2) G D (τ) = [N(1+τ/τ D )(1+τ/(ω 2 τ D )) 1/2 ] -1. Here, N is the average number of fluorescent molecules, τ D is the typical time that a molecule spends in the observation volume, and ω is the axial to lateral ratio of this space. Finally, this idea can be extended to account for more than one diffusing species in the observation volume when chemical kinetics are ignored by taking a weighted linear combination of the correlations for each non-interacting species: 2 m 2 (3) G D (τ) =! Fi Gi ) /(! m (" F ). i= 1 i= 1 2 i Figure 2: (a) A typical fluorescence signal measured in time. (b) A binned portion of the fluorescence in (a) indicating the average fluorescence. (c) The autocorrelation of fluorescence, which characterizes fluctuations caused by diffusion. 2

3 December 2005 Carlson - 3 The autocorrelation function reveals the time required for a molecule to diffuse through an open observation volume. Diffusion is the macroscopic result of random thermal motions that occur on a molecular level, and the collisions between solute particles and solvent molecules experiencing thermal movement are responsible for this phenomenon. 5 Fick s first and second laws of diffusion describe the rate of diffusion of a solute across an area in three and one dimensions, respectively. However, the work of Brown, Guoy, Einstein, and Smoluchowski is responsible for the development of the random walk equation, which uses a diffusion coefficient, D, to relate the average time required for a molecule to travel an average distance, <x>, in time t. 5,6 (4) <x> 2 = 2Dt Once the diffusion coefficient is known, the Einstein relation bridges this value and the drag coefficient, γ: 5,6 (5) D = k B T/γ. For a sphere with radius r, the drag coefficient is calculated using the Stokes equation 5,6 (6) r = γ/6πη, where η is the viscosity of the solvent. In this manner, the radius, mass, and volume of a diffusing particle may be determined. III. EXPERIMENT These ideas were implemented in a fluorescence correlation spectroscopy experiment designed to target the volume and mass of two samples: an unknown protein antibody and DNA. A block diagram of the Nikon inverted confocal microscope that was used as the foundation for this experimental setup is depicted in Figure 3. An Nd:YAG laser emitting at λ = nm was passed through a telescope in order to overfill the back aperture of an NA x oil immersion objective (η = 1.515). The fluorescence emitted by the excited sample was collected using the same objective and was passed through a notch filter and pinhole before being detected by an avalanche photodiode. A preliminary estimate of the concentration required to detect up to ten single molecules, as well as the probability of detecting one such particle in a 2.31 x L detection volume is displayed in Tables 1 and 2. 2,7 The concentration of an 81 nm stock solution of unknown protein was adjusted to 81 nm by a two step dilution with a buffer Figure 3: Experimental setup. 2 composed of 50 nm K 3 PO 4, 0.1% OG buffer, and 0.2% bovine serum albumin in water (prepared June 1, 2005). One hundred microliters of this sample was injected into the sample holder and the focus of the laser (P = 230 µw) was raised into the liquid. Data were collected for 15 seconds at a 10 µs time resolution and were binned every 2 milliseconds (Table 4). Three initial trials were carried out for the protein, corresponding to the following temperatures (listed in sequential order):

4 December 2005 Carlson C, 46.1 C, and 26.2 C. The temperature was adjusted in this order because light was not directed to the detector during the first trial for the intermediate temperature. The unknown nm DNA sample was prepared from a 2.15 µm stock solution and an aqueous tris chloride/edta buffer by following the two-step dilution process used for the unknown protein. A series of six trials were carried out to study the effect of increasing temperature (-3.9 C, 10.0 C, 20.4 C, 29.7 C, 40.6 C, and 59.2 C) on the diffusion rate and the average number of molecules in the observation volume. The laser power was 200 µw and the data were collected by using 25 second collection times at 10 µs time resolution and then binning data every 200 milliseconds (Table 4). A similar procedure was followed to collect data for the blank tris chloride/edta buffer at -3.9 C. The construction of autocorrelation curves for these trials was completed in MATLAB and the curves were fitted with equation (2) in Origin. IV. ANALYSIS The mass and volume of the unknown protein and DNA were calculated using information available from a plot of the autocorrelation function versus time for each species. Plots were constructed for the background (T = -3.9; Plot 1), the protein at three temperatures (T = -2.4 C, 26.2 C, 44.8 C; Plots 2-4), and DNA at six temperatures (T = -3.9 C, 10.0 C, 20.4 C, 29.7 C, 40.6 C, 59.2 C; Plots 5-10). The MATLAB software package was used to calculate the autocorrelation of the temporal data collected during the experiment; the graphing software package Origin was then used to fit the data to the normalized autocorrelation function, equation (2). During the fitting procedure, the value of ω was held constant (ω = 3.54 for all trials; Table 1), while τ D and N were allowed to vary. An initial approximation of the average number of fluorescent molecules, N, in the observation volume was obtained by taking the reciprocal of the limit of G(τ) as τ approached zero: G D (0) = [N(1+(0)/τ D )(1+(0)/(ω 2 τ D )) 1/2 ] -1 = N -1. Typically, one hundred Levenberg-Marquardt iterations were carried out for both N and τ D until their values remained unchanged between iterations. The average diffusion time was then obtained from the fitted value of τ D, equal to the time at the inflection point of the fitted curve. Temporal emission data for the non-fluorescent DNA buffer was collected as a point of comparison between samples with and without Plot 1: TrisCl/EDTA Buffer (T = -3.9 C) labeled molecules (Plot 1). As expected, the autocorrelation function has no amplitude for this trial. Oppositely, the autocorrelation function provides a good fit for the protein and DNA data, and illustrates the behavior of these molecules around their equilibrium states (Plots 2-10). Since the amplitude of the autocorrelation function is directly linked to the number of fluorescent molecules within the detection volume, it is noted that the amplitude of

5 December 2005 Carlson - 5 the autocorrelation function for the DNA samples is quite low, especially for trials DNA 2 and 6, whose average number of fluorescent molecules is approximately 988 and 281, respectively. concentrations of DNA and protein would be used if the experiment were to be repeated. Lower The diffusion times obtained through this fitting procedure are summarized in Tables 5 and 6. By implementing equations (4), (5), and (6), the average diffusion time of 457 microseconds yielded the average radius and volume of the unknown protein as pm and pm 3, respectively. The mass of the protein was determined to be about x kd; this molecule was an antibody whose mass is known to be near 150 kd. As the temperature increased, the diffusion time for one molecule decreased, which agrees with the idea that molecules move faster when more thermal energy is available. The average number of fluorescent molecules in the detection volume was 20, which is slightly high for single molecule detection. One consideration in this determination is that the possibility of protein denaturation was introduced for the Protein 2 and 3 trials, since some metastable proteins denature in this temperature regime. The temperature was first ramped to 44.8 C before being reduced to 26.2 C for the Protein 2 trial, and so some uncertainty was introduced due to this nonlinear temperature variation. The calculated radius of the DNA sample was pm (V = pm 3 ) and its mass was x 10-9 kd. The average diffusion time was found to be 703 microseconds and the mean number of fluorescent molecules in the detection volume was 77. The magnitude of N is cause for some concern because as N increases, the amplitude of the autocorrelation curve decreases and makes accurate curve fitting more difficult. In contrast to the protein experiment, the diffusion time of the DNA and temperature were directly related here. The trials DNA 2 and DNA 6 were excluded from the radius and mass determinations because the amplitude of the autocorrelation function was such that a signal could not be reliably distinguished. Because the results of this study are not realistic for the unknown protein or DNA, it is worthwhile to account for these deviations from expected results. Sample preparation, data collection, and theoretical modeling are three areas that could be adapted to improve the validity of these results. The samples were typically too concentrated for single molecule detection since the average number of molecules in the observation volume was 20 for the protein and 77 for the DNA. If these values were limited to less than ten molecules, then both the amplitude of the autocorrelation function and the signal to noise ratio would be much higher. Also related to sample preparation, the buffer used in the protein trials included OG, which is a detergent. The viscosity of the unknown solutions was assumed to be equal to that of water (Table 1). However, detergents typically decrease the viscosity of a solution and enable faster diffusion of a molecule throughout that solvent. This trend was displayed for both solutions, and viscosity may have played a role in both the protein and DNA trials. The experimental setup and data collection techniques could be adjusted to improve the quality of the data. Glass cover slips exhibit autofluorescence, and while this constant background does not strongly affect the autocorrelation fits, it does lead to a decreased signal to noise ratio. As a substitute, quartz cover slips could be used to eliminate this additional background noise. The setup also incorporated an Nd:YAG laser operating at a power near 200 µw (Table 4). It is possible that the dyes used to label the protein and

6 December 2005 Carlson - 6 DNA experienced photobleaching at this excitation intensity. In order to confirm or dismiss this supposition, the experiment could be carried out as a function of varying laser intensity in order to determine the rate at which the dyes bleach. In addition, an examination of the time resolution used in data collection indicates that although we could achieve only ten-microsecond time resolution, the calculated diffusion times of 457 µs (protein) and 703 µs (DNA) were well above the threshold limits for this value. As such, improving the time resolution would not largely influence the calculated diffusion times. Oppositely, if the experiment were to be repeated, the number of data points collected during a time scan would be increased in order to improve the SNR. Finally, the fitting of autocorrelation curves may have been responsible for some error in the results of this study. Inaccuracy related to estimating the number of fluorescent molecules in the detection volume may have contributed to some variation in the determination of N and τ D from the autocorrelation curve fits. Modifying the sample preparation and data collection as described above would increase the signal to noise ratio, making curve fitting more accurate and reliable. V. CONCLUSIONS Fluorescence correlation spectroscopy is an optical technique that makes use of diffusion and kinetics to aid in the understanding of equilibrium processes. As an example of this idea, FCS was used to determine the mass and radius of an unknown protein (m = x kd; r = pm) and DNA (m = x 10-9 kd; r = pm). The approximate mass of the protein is known to be near 150 kd, and so the calculated results for the unknown protein and DNA are not valid. Nevertheless, meaningful information can still be extracted from this study by accounting for improvements that could be made in the sample preparation, experimental setup, data collection, and curve fitting. As advances in experimental techniques are made, it is likely that fluorescence correlation spectroscopy coupled with autocorrelation analysis will provide a meaningful approximation of the mass and size of these diffusing particles.

7 December 2005 Carlson - 7 Appendix Table 1: Constants and Associated Symbols Quantity Symbol Value Units Laser Wavelength λ nm Magnification M 100 x Numerical Aperture NA Refractive Index n Beam Waist Δx nm Depth of Focus Δz nm Confocal Detection Volume V det E-20 L Axial:Lateral Ratio ω Avogadro's Number N A 6.022E+23 molecules/mole Number of Fluorescent Molecules N - - Number of Molecules of Interest x 1 - Poisson Probability of Detection P x - - Typical Protein Density ρ 1,380 kg/m3 Viscosity of Water η 1.000E-03 Pa*s Boltzmann Constant k B 1.381E-23 J/K Temperature T 298 K Radius of a Globular Protein r - nm Drag Coefficient γ N*s/m Diffusion Coefficient D - m 2 /s Drift Time t 1D - s Root Mean Square Velocity v rms - m/s Sample Calculations Δx = ((0.6098)(λ))/(NA) = ((0.6098)( nm))/(1.4) = nm Δz = ((2)(n)(λ))/(NA) 2 = ((2)(1.515)( nm))/(1.4) 2 = nm V det = ((4/3)(π)(Δx/2) 2 (Δz/2))(1 m/1 x 10 9 nm) 3 = ((4/3)(π)(23,172.4 nm/2) 2 (8,224, nm/2))(1 m/1 x 10 9 nm) 3 (1 L/1 m 3 ) = x L ω = Δz/Δx = nm/ nm = 3.549

8 December 2005 Carlson - 8 Table 2: Protein concentration and probability of detection as a function of N N x [Protein] (M) [Protein] (nm) Probability E E E E E E E E E E Sample Calculations N = 1 [Protein] (M) = N/(N A )(Vdet)) = (1 molecule)/((6.022 x molecules/mole)( x L)) = x M [Protein] (nm) = ([Protein] (M))(1 x 10 9 nm/1 M)) = ( x M)( 1 x 10 9 nm/1 M)) = nm P x = ((N x )(e -N ))/(x!) = ((1 1 )(e -1 ))/(1!) = Parallel calculations were carried out for N = 2-10.

9 December 2005 Carlson - 9 Table 3: Drift time and v rms as a function of protein size Protein Mass (kd) r (nm) γ (N*s/m) D (m 2 /s) t D (s) v rms (m/s) a E E b E E c E E d E E e E E f E E g E E Sample Calculations m = 10 kd r = ((m/n A )/((4/3)(π)(ρ))) 1/3 (1 x 10 9 nm/1 m) = (((10 kg/mol)/( x molecules/mole))/((4/3)(π)(1.380 x 10 3 kg/m 3 )) 1/3 (1 x 10 9 nm/1 m) = nm γ = 6πηr = (6)(π)(1 x 10-3 Pa*s)( nm)((1 kg/ms 2 )/1 Pa)(1 N/(1 kgm/s 2 ))(1 m/1 x 10 9 nm) = x N*s/m D = k B T/γ =((1.381 x J/K)((1 kgm 2 /s 2 )/1 J)(298 K))/(( x N*s/m)((1 kgm/s 2 )/1 N)) = x m 2 /s t D = (Δx) 2 /(2D) = ((23,172.4 nm)(1 m/1 x 10 9 nm)) 2 /((2)( x m 2 /s)) = s v rms = ((3k B T)/(m/N A )) 1/2 = (((3)(1.381 x J/K)((1 kgm 2 /s 2 )/1 J)(298 K))/(10 kg/mol/(6.022 x molecules/mol))) 1/2 = m/s Parallel calculations were carried out for all other masses of protein.

10 December 2005 Carlson - 10 Table 4: Data Collection Parameters Sample Concentration Temperature Power Collection Time Binning (nm) ( C) (µw) Time (s) Resolution (µs) (ms) Protein Protein Protein Tris/Cl EDTA Buffer DNA DNA DNA DNA DNA DNA

11 December 2005 Carlson - 11 Table 5: Summary of Protein Results Protein Average T ( C) T (K) N t D (s) 3.200E E E E-04 t D (ms) D (m 2 /s) 8.390E E E E-07 γ (Ns/m) 4.457E E E E-15 r (m) 2.364E E E E-13 r (pm) Mass (kd) 4.601E E E E-10 V (pm 3 ) Plot 2: Protein 1 in Buffer (T = -2.4 C) Data: Protein1 O r i g i n D e m o O r i g i n D e m o e m O r i g i n D Chi^2/DoF o = 001 R^2 = N ±0 O r i g i n D e m o O r i g i n D e m o r i g i O td n D e m o 032 ±6.2742E-6 w 3.54 ± Plot 3: Protein 2 in Buffer (T = 26.2 C) Data: Protein2 O r i g i n D e m o O r i g i n D e m o O r o i g i n D e m Chi^2/DoF = 001 R^2 = N ±0 O r i g i n D e m o O r i g i n D e m o O r i n D i g td e m o 067 ±001 w 3.54 ± Plot 4: Protein 3 in Buffer (T = 46.1 C) Data: Protein Chi^2/DoF r i g i n D e m = 001 O r i g i n D e m o O r i g i n D e m o O o R^2 = N ± td r i g D038 e m o ±0 O r i g i n D e m o O r i g i n D e m o O i n w 3.54 ±0 r i D e m O r i g i n D e m o O r i g i n D e m o O g i n o -

12 December 2005 Carlson - 12 Table 6: Summary of DNA Results DNA Average Average (Excluding 2 & 6) T ( C) T (K) N t D (s) 3.200E E E E E E E E-04 t D (ms) D (m 2 /s) 8.390E E E E E E E E-07 γ (Ns/m) 4.432E E E E E E E E-14 r (m) 2.351E E E E E E E E-13 r (pm) Mass (kd) 4.524E E E E E E E E-09 V (pm 3 ) Plot 5: DNA 1 in Buffer (T = -3.9 C) Plot 8: DNA 4 in Buffer (T = 29.7 C) O r i g i n D e m o O r i g i n D e m o O rdata: i g i n DNA1 D e m o Data:DNA4 Chi^2/DoF = E-6 D e m O r i g i n D e m o O r i g i n D e m o O o r i g i n R^2 = N ± td 027 ±0 O r i g i n D e m o O r i g i n D e m o O i n D r i g w e m o 3.54 ±0 Chi^2/DoF i g i n D e m = 001 O r i g i n D e m o O r i g i n D e m o O r o R^2 = N ± itd g i e13 m o ±0 O r i g i n D e m o O r i g i n D e m o O r n D w 3.54 ±0 Plot 6: DNA 2 in Buffer (T = 10.0 C) Plot 9: DNA 5 in Buffer (T = 40.6 C) Data: DNA2 Chi^2/DoF O r i g i n = o E-6 O r i g i n D e m o O r i g i n D e m o D e m R^2 = 065 N ± tdo 075 n D e m o ±0 O r i g i n D e m o O r i g i n D e m o r i g i w 3.54 ±0 O r i g i n D e m o O r i g i n D e m o O rdata: i g i n DNA5 D e m o Chi^2/DoF = 001 rr^2 i g i n D e m = O r i g i n D e m o O r i g i n D e m o O o N ±0 td 089 ±004 O r i g i n D e m o O r i g i n D e m o O i n r i g w D e m o 3.54 ±0 r i g D e m o O r i g i n D e m o O r i g i n D e m o O i n Plot 7: DNA 3 in Buffer (T = 20.4 C) Plot 10: DNA 6 in Buffer (T = 59.2 C) O r i g i n D e m o O r i g i n D e m o O rdata: i g i n DNA3 D e m o Data: DNA_DNA6 Chi^2/DoF = E-6 rr^2 i g i n D e m = 307 O r i g i n D e m o O r i g i n D e m o O o N ±0 td 03 ±003 r i g w e m o 3.54 ±0 O r i g i n D e m o O r i g i n D e m o O i n D OChi^2/DoF r i g i n D e = E-6 O r i g i n D e m o O r i g i n D e m o m o R^2 = 615 N ± OtD r 06 D e m o ±0 O r i g i n D e m o O r i g i n D e m o i g i n w 3.54 ±0

13 December 2005 Carlson - 13 Acknowledgements The author is indebted to John Lesione for his assistance in planning, collecting, and analyzing data for this project. She also thanks Katie Leach for helpful revisions and discussions related to this work. References (1) Maiti, S.; Haupts, U.; Webb, W. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, (2) Hess, S. T.; Hunag, S.; Heikal, A. A.; Webb, W. W. Biochemistry 2002, 41, (3) Aragón, S. R.; Pecora, F. J. Chem. Phys. 1976, 64, (4) Microscopy from Carl Zeiss. D /$file/40-535_e.pdf (accessed November 2005). (5) Laidler, K. J.; Meiser, J. H.; Sanctuary, B. C. Physical Chemistry; Fourth ed.; Houghton Mifflin Company: Boston, MA, (6) Howard, J. Mechanics of Motor Proteins and the Cytoskeleton; Sinauer Associates, Inc.: Sunderland, MA, (7) Földes-Papp, Z.; Demel, U.; Tilz, G. P. Proc. Natl. Acad. Sci. U.S.A. 2001, 98,