Single-Molecule Fluorescence Trajectories for Investigating Molecular

fit to a two-dimensional Gaussian function to determine their x,y positions with subpixel resolution. Scatter plots and histograms of molecular step s...
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Anal. Chem. 2003, 75, 4351-4359

Single-Molecule Fluorescence Trajectories for Investigating Molecular Transport in Thin Silica Sol-Gel Films Karla S. McCain, David C. Hanley, and Joel M. Harris*

Department of Chemistry, University of Utah, 315 South 1400 East, Salt Lake City, Utah 84112-0850

Single-molecule fluorescence tracking has been used to examine diffusion of small molecules in sol-gel films in order to identify spatial heterogeneity in the structure and molecular diffusivities for different regions of the film. Fluorescence intensity profiles from single molecules are fit to a two-dimensional Gaussian function to determine their x,y positions with subpixel resolution. Scatter plots and histograms of molecular step sizes indicate that the trajectories conform to the predictions of a two-dimensional random walk. The mean-square step size is shown to be an unbiased estimate of the variance of the stepsize probability distribution and a valid statistic for determining the diffusion coefficient from a molecular trajectory. The diffusion coefficients measured for different molecules are subjected to an F test, which showed that the sol-gel film exhibits spatial variation in the diffusion coefficient on a micrometer-length scale. The spatial variation in diffusivities is a measure of structural heterogeneity of these films.

Sol-gels are a family of high-surface-area materials that are produced from the hydrolysis of metal alkoxides.1,2 Sol-gels prepared from silicon alkoxides to produce porous silica are an important class of structures that are applied in chromatographic separations,3,4 in fuel cells,5 and as supports for optical sensors.6-8 The successful application of sol-gel materials in these applications relies on efficient transport of molecules within the porous structure. For example, the response of a sensor depends on how quickly analyte molecules can diffuse into the sol-gel to immobilized reagent molecules that produce the sensor response. The reversible cycle time of such a sensor depends on how quickly those molecules can diffuse in to and out of the structure. * Corresponding author. E-mail: [email protected]. (1) Brinker, C. J.; Scherer, G. W. Sol-Gel Science: The Physics and Chemistry of Sol-Gel Processing; Academic Press: Boston, 1990. (2) Iler, R. K. The Chemistry of Silica; John Wiley and Sons: New York, 1979. (3) Hayes, J. D.; Malik, A. Anal. Chem. 2000, 72, 4090-4099. (4) Nakanishi, K.; Shikata, H.; Ishizuka, N.; Koheiya, N.; Soga, N. J. High Resolut. Chromatogr. 2000, 23, 106-110. (5) Anderson, M. L.; Stroud, R. M.; Rolison, D. R. Nano Lett. 2002, 2, 235240. (6) Han, L.; Niemczyk, T. M.; Lu, Y.; Lopez, G. P. Appl. Spectrosc. 1998, 52, 119-122. (7) Wang, C.; Li, C.; Lin, Y.; Chau, L. Appl. Spectrosc. 2000, 54, 15-19. (8) Makote, R.; Collinson, M. M. Anal. Chim. Acta. 1999, 394, 195-200. 10.1021/ac0345289 CCC: $25.00 Published on Web 08/01/2003

© 2003 American Chemical Society

Diffusion of molecules within sol-gel materials is not a simple process since these materials can be structurally and chemically heterogeneous, characterized by a size range of porosities.1 Molecular diffusion in sol-gel materials has been studied in a variety of means including leaching experiments,9 electrochemistry,10-12 and attenuated total reflectance Fourier transform infrared spectroscopy.13 Recently, molecular diffusion in thin sol-gel films has been characterized on a fast time scale by total internal reflection fluorescence correlation spectroscopy.14 In all of these experiments, however, the measured diffusion represents that of a molecular ensemble that averages, over a large area of the sample, differences in the rates of transport that might arise from the heterogeneity of the sol-gel materials. Single-molecule fluorescence detection brings the ability to observe the behavior of individual molecules instead of measuring properties of a large ensemble. This is an important tool for investigating heterogeneous systems because it allows for spatial variation in the sample, as it impacts the local behavior of molecules, to be examined and characterized. Single-molecule spectroscopy has been exploited in order to examine the diversity of adsorption sites sampled by dye molecules on a chromatographic surface15-19 and the variation in molecular environments for small molecules within polymer and sol-gel films.20-26 To examine molecular diffusion within sol-gel films, single-molecule (9) Watson, J.; Zerda, T. W. Appl. Spectrosc. 1991, 45, 1360-1365. (10) Collinson, M. M.; Rausch, C. G.; Voigt, A. Langmuir 1997, 13, 72457251. (11) Collinson, M. M.; Zambrano, P. J.; Wang, H.; Taussig, J. S. Langmuir 1999, 15, 662-668. (12) Howells, A. R.; Zambrano, P. J.; Collinson, M. M. Anal. Chem. 2000, 72, 5265-5271. (13) Rivera, D.; Harris, J. M. Anal. Chem. 2001, 73, 411-423. (14) McCain, K. S.; Harris, J. M. Anal. Chem. 2003, 75, 3284-3292. (15) Wirth, M. J.; Ludes, M. D.; Swinton, D. J. Anal. Chem. 1999, 71, 39113917. (16) Swinton, D. J.; Wirth, M. J. J. Phys. Chem. B 2001, 105, 8679-8684. (17) Wirth, M. J.; Swinton, D. J. Anal. Chem. 1998, 70, 5264-5271. (18) Wirth, M. J.; Swinton, D. J. J. Phys. Chem. B 2001, 105, 1472-1477. (19) Ludes, M. D.; Wirth, M. J. Anal. Chem. 2002, 74, 386-393. (20) Deschenes, L. A.; Vanden Bout, D. A. J. Phys. Chem. B 2002, 106, 1143811445. (21) Deschenes, L. A.; Vanden Bout, D. A. J. Phys. Chem. B 2001, 105, 1197811985. (22) Deschenes, L. A.; Vanden Bout, D. A. Science 2001, 292, 255-258. (23) Wang, H.; Bardo, A. M.; Collinson, M. M.; Higgins, D. A. J. Phys. Chem. B 1998, 102, 7231. (24) Mei, E.; Bardo, A. M.; Collinson, M. M.; Higgins, D. A. J. Phys. Chem. B 2000, 104, 9973. (25) Hou, Y.; Bardo, A. M.; Martinez, C.; Higgins, D. A. J. Phys. Chem. B 2000, 104, 212.

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tracking can be employed, where fluorescence images are taken as a function of time and the position of individual molecules is followed to produce molecular trajectories. Single-molecule tracking has been applied to a wide range of biological systems27-43 including studying diffusion in phospholipid bilayers and within living cells. In this work, we apply single-molecule tracking to examine the diffusion of small dye molecules in sol-gel films, to examine the spatial heterogeneity of sol-gel materials. Previous attenuated total internal reflection infrared (ATR-FT-IR) measurements of the exchange of molecules in sol-gel films, following a concentration change in solution, reveal a slow diffusion component that is independent of film thickness, attributed to slow transport within intraparticle micropores.13 Total internal reflection fluorescence correlation spectroscopy (TIR-FCS) has shown14 that the more rapid diffusion of small dye molecules in the interparticle pore volumes is sensitive to the organization of particles within the film structure; more ordered films exhibited slower diffusion of molecules within smaller interparticle pores. In both of these experiments, molecular diffusion was characterized over large (mm2) areas of the sample. By examining single-molecule trajectories in small (µm2) regions of sol-gel films, it is proposed that spatial heterogeneity in the film structure can be identified by observing differences in diffusivities for different regions in the film. Statistical analysis of molecular trajectories within the film is developed so that differences in diffusivities can be interpreted quantitatively. EXPERIMENTAL SECTION Silica Particle and Film Preparation. Silica particles were prepared by a modified Sto¨ber synthesis.44 About 7.5 mL of tetraethoxysilane (Aldrich) was added to 150 mL of absolute ethanol (Omnisolve) followed by the addition of 7 mL of a 25% ammonia solution in water. This mixture was stirred for 36 h and then allowed to stand for at least 2 weeks to ensure that all ethoxy (26) Bardo, A. M.; Collinson, M. M.; Higgins, D. A. Chem. Mater. 2001, 13, 2713-2721. (27) Kues, T.; Peters, R.; Kubitscheck, U. Biophys. J. 2001, 80, 2954-2967. (28) Schu ¨ tz, G. J.; Schindler, H.; Schmidt, T. Biophys. J. 1997, 73, 1073-1080. (29) Ke, P. C.; Naumann, C. A. Langmuir 2001, 17, 5076-5081. (30) Seisenberger, G.; Ried, M. U.; Endreb, T.; Bu ¨ ning, H.; Hallek, M.; Bra¨uchle, C. Science 2001, 294, 1929-1932. (31) Schmidt, T.; Schu ¨ tz, G. J.; Baumgartner, W.; Gruber, H. J.; Schindler, H. J. Phys. Chem. 1995, 99, 17662-17668. (32) Schmidt, T.; Schu ¨ tz, G. J.; Baumgartner, W.; Gruber, H. J.; Schindler, H. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 2926-2929. (33) Ruiter, A. G. T.; Veerman, J. A.; Garcia-Parajo, M. F.; van Hulst, N. F. J. Phys Chem. A. 1997, 101, 7318-7323. (34) Dickson, R. M.; Norris, D. J.; Tzeng, Y. L.; Moerner, W. E. Science 1996, 274, 966-969. (35) Bopp, M. A.; Tarrach, G.; Lieb, M. A.; Meixner, A. J. J. Vac. Sci. Technol., A 1997, 15, 1423-1426. (36) Sonnleitner, A.; Schu ¨ tz, G. J.; Schmidt, T. Biophys. J. 1999, 77, 26382642. (37) Smith, P. R.; Morrison, I. E. G.; Wilson, K. M.; Ferna´ndez, N.; Cherry, R. J. Biophys. J. 1999, 76, 3331-3344. (38) Kubitscheck, U.; Ku ¨ ckmann, O.; Kues, T.; Peters, R. Biophys. J. 2000, 78, 2170-2179. (39) Goulian, M.; Simon, S. M. Biophys. J. 2000, 79, 2188-2198. (40) Cheezum, M. K.; Walker, W. F.; Guilford, W. H. Biophys. J. 2001, 81, 23782388. (41) Ke, P. C.; Naumann, C. A. Langmuir 2001, 17, 3727-3733. (42) Simson, R.; Sheets, E. D.; Jacobson, K. Biophys. J. 1995, 69, 989-993. (43) Jorg, E. Single Mol. 2000, 1, 225-230. (44) Coenen, S.; DeKruif, C. G. J. Colloid Interface Sci. 1988, 124, 104-110.

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Figure 1. Instrumental layout used for single-molecule tracking experiments: PB , Pellin-Broca prism; L1-4, lenses; A1-2, apertures; F1-2, filters; DM, dichroic mirror; and RG, roughened glass disk. See text for details.

groups were hydrolyzed. Then 50-mL aliquots of this suspension were diluted to 200 mL with absolute ethanol, which resulted in a particle concentration of ∼0.3% (w/v). Glass coverslips (Esco) were dip-coated with this silica suspension using a motorized stage to control the withdrawal rate at 1.0 cm/s. A 20-dip film corresponded to a thickness of 450 nm.14 This thickness was chosen so that the entire film would be within the depth of field of the microscope objective. The coverslip with the sol-gel film was placed in a flow cell with a 20 pM solution of rhodamine 6G in 90/10 v/v methanol/water containing 50 mM NaCl; these solution conditions were chosen to avoid adsorption of the dye to the silica surface.14 Fluorescence Microscopy. A diagram of the fluorescence microscope is shown in Figure 1 and has been described previously.45 The 514.5-nm line from an argon ion laser (SpectraPhysics) is used as the excitation source and is passed through a Pellin-Broca prism and apertured to remove plasma lines. The beam is shuttered with computer control by an acoustooptic deflector (AOM, Crystal Technology) to allow image transfer with no light on the detector. To accomplish this, the beam is lightly focused (L1) onto the acoustooptic crystal. When an rf signal is applied to the acoustooptic crystal, the beam is deflected and passed by an aperture (A2). The beam is collimated (L2) and then passed on a roughened glass disk, rotating at several hundred rotations per minute in order to average out the speckle pattern on the time scale of the experiment. Passing the beam through this roughened, rotating glass creates an incoherent spot source for excitation. This spot is reimaged with a 55-mm focal length, f/1.2 camera lens (L3) (Canon) into the back of the microscope (Nikon) in order to overfill the collection cone of the objective, creating a nearly uniform intensity profile over the observation area. The laser power captured by the objective is less than 1 mW. (45) Hanley, D. C.; Harris, J. M. Anal. Chem. 2001, 73, 5030-5037.

Within the microscope, the excitation beam is passed through a 514.5-nm pass-band filter (F1) with a 10-nm bandwidth and reflected by a 525-nm dichroic mirror (DM), which are contained in the microscope filter cube (Chroma Technology Corp.). The reflected light then impinges on the back of a 100×, 1.3 NA, infinity-corrected, 0.20-mm working distance, oil immersion microscope objective. The excitation light converges at the sample in a 190-µm spot. Fluorescence from the sample is collected by this same objective and passes through the dichroic mirror and a 530 high-pass filter (F2) also contained in the filter cube. The fluorescence is then directed to the microscope side port and is reimaged at 1:1 magnification by a relay lens (L4) onto the detector. The CCD detector (Andor) contains an antireflectioncoated, 1024 × 1024 pixels, back-thinned EEV chip with a 13-µm square pixels. The area of the CCD chip (1.77 cm2) corresponds to a detection area at the sample surface of 1.77 × 104 µm2. Images from a central area 256 × 256 pixels square were collected with a 0.2-s integration time and 0.579-s read-out time giving 0.779 s between frames. Each series of images was 300 frames long, corresponding to a total observation time of ∼234 s. Fluorescence images were processed in Matlab (Mathworks). Images were first mean centered and Fourier filtered with a Gaussian filter to remove the highest frequency (single pixel) spatial noise from the data. The center position of a fluorescence spot molecule was determined with subpixel resolution by fitting its intensity profile to a two-dimensional Gaussian function

I(x,y) ) A exp

{

}

[-(x - x0)2 - (y - y0)2] 2σ2

+B

(1)

where x0 and y0 are the x and y positions, A is the amplitude, σ is the standard deviation, and B is the offset. It was necessary to include both intensity and size parameters because the molecules also experienced diffusion along the z axis, which affected their intensity and apparent size. Rigorously, the point spread function for the microscope should be an Airy function, assuming that the back of the objective acts as a circular mask with sharp edges. However, the spatial autocorrelation of an image of stationary molecules adsorbed at a glass/air interface shows that the point spread function of the optics is Gaussian.45 This is likely due to a smooth falloff in the collection efficiency at large angles at the edges of the objective. Simulations. Numerical simulations were performed in order to determine the positional uncertainty and correspondingly slowest measurable diffusion coefficient that derive from the S/N ratio of the data. A data set of 100 images was produced by adding noise to a Gaussian spot placed at a fixed point. The noise was produced using a Gaussian-distributed random number generator programmed in Matlab. It was necessary to Fourier filter this noise with the same filter used for the real data, since the filtering slightly lowered the spatial frequency of the noise. The amplitude of the noise was adjusted so that when added to the Gaussian function simulating the signal from a fluorescent molecule, the same average S/N ratio of a molecule was produced. These simulated images were then analyzed identically to real data. Other Analytical Techniques. Scanning electron microscopy (SEM) images of the sol-gel films were acquired using a Hitachi S3000N microscope. Samples were dip-coated onto glass micro-

scope slides and then overcoated with 10 nm of gold. The diffusion of molecules in these films was also characterized by TIR-FCS. The instrumental setup and procedures for these TIR-FCS experiments have been described in detail elsewhere.14 RESULTS AND DISCUSSION Image Processing. Because of the slightly scattering nature of the sol-gel films and the time scale of molecular diffusion, acquiring images with high S/N was challenging. To improve the S/N of the acquired images to recognize moving molecules in the images, the data were Fourier filtered in two dimensions. The 2D Gaussian filter had a standard deviation of 0.83 pixel in both x and y dimensions; convoluting with this filter in both dimensions removed noise that varies on the distance scale of ∼1 pixel. The raw intensity data single molecules in these images were spots averaging 3.2 pixels across (2σ), and their size only increased ∼15% upon filtering. Because of the two-dimensional averaging that occurs even with such modest filtering, the S/N was improved considerably as shown in a comparison of an unfiltered image, the Gaussian filter, and a filtered image in Figure 2. The quantitative effect of filtering on the S/N was investigated by a 2D autocorrelation of a 50 × 50 pixel image of a film containing no molecules to compare the variance of the noise46 before and after filtering. The ratio of the noise variance after filtering to the variance before filtering was found to be 0.17, or an improvement in the S/N by a factor of 2.4. This increased the S/N of the images to a degree that single molecules could be easily observed above the noisy background. Because the convolution step used to filter the image is a linear operation, the improvement in S/N can be predicted by propagation of errors from the coefficients of the filter.47 This predicted improvement was calculated using a truncated 11 × 11 coefficient matrix of the Gaussian filter function; see Figure 2b. Five points away from the center, the Gaussian function has decayed by over 10 orders of magnitude from its maximum value, so all appreciable effects of the filter could be accounted for using this truncated function. The theoretical reduction in the noise variance by filtering was predicted to be 0.12, corresponding to an increase in S/N by a factor 2.9. The theoretical improvement in S/N is 20% higher than the factor of 2.4 that was achieved experimentally, which is probably due to lower spatial frequency noise perhaps from scattering that was not adequately averaged by this very narrow Gaussian filter. Autocorrelation analysis also allows for the determination of the contribution of different frequency noise on the overall variance. In the raw data before filtering, 93% of the signal standard deviation corresponded to single pixel-to-pixel variation, that is, where the noise arises from fluctuations that occur on the interval of individual pixels; in the filtered data, only 47% of the rms noise corresponds to the single-pixel variation. This reduction in high-frequency noise makes the intensity spots corresponding to single molecules easier to identify and then analyze. It is possible that this filtering procedure could occasionally produce spots from high-frequency random noise that could be misinterpreted as arising from single molecules. However, these spots would show no persistent correlation from one frame to the next and would, therefore, not produce a molecular trajectory. (46) Hansen, R. L.; Harris, J. M. Anal. Chem. 1998, 70, 2565-2575. (47) Phillips, G. R.; Harris, J. M. Anal. Chem. 1990, 62, 2749-2752.

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Figure 3. (a) Intensity image of a single fluorescent molecule. (b) Best-fit two-dimensional Gaussian function for the molecule in (a). (c) Residuals of the fit to the two-dimensional Gaussian.

Figure 2. (a) Unfiltered fluorescence image of single rhodamine 6G molecules in sol-gel film. (b) Gaussian Fourier filter used to filter images. (c) Filtered fluorescence image. All images are 256 × 256 pixels, corresponding to 35 × 35 µm.

In the filtered images, a small population of moving fluorescence spots could be observed, along with a larger number of spots whose motion could not be detected on the time scale of the experiment. The stationary peaks could be due to adsorption of the fluorescent probe to strong adsorption sites15-19 or the trapping of molecules in small, intraparticle pores.13 Fluorescent spots were also observed to disappear in a single frame and never reappear near the same position. These events are likely due to either single-molecule photobleaching or molecular diffusion out of the film and into the overlaying free solution. Thus, it was a 4354 Analytical Chemistry, Vol. 75, No. 17, September 1, 2003

relatively rare event for a molecule to remain in the film and exhibit observable diffusion for many sequential frames, so that its trajectory in the film could be mapped. The position of diffusing molecules was determined with subpixel resolution by fitting their fluorescence intensity distributions to a Gaussian function. This approach has been shown to be a precise method of determining the position of a single molecule from a fluorescence spot of modest S/N ratio.40 In Figure 3, a plot of a typical fluorescence intensity peak from a single diffusing molecule in the film, its Gaussian fit, and residuals are shown. The residuals show no structure, indicating the goodness of the fit. It was found that the full width at half-maximum (fwhm) of the fitted spots varied substantially, from as small as 2 pixels (0.27 µm) to more than 7 pixels (1.0 µm), with most around 3.5 pixels or 0.5 µm. This variation is likely due to scattering of

Figure 4. Molecular trajectories. Note that molecules A and B are very close to each other and are shown in the same panel. Molecule A’s trajectory is indicated by circles (b), while molecule B’s trajectory is indicated by squares (9). Molecules A, B, and C are from the same data set. Molecules D and E come from the same sample but from a second data set. For the second data set, the microscope stage was translated several hundreds of micrometers away from the location for the first data set.

fluorescence of molecules from different depths within the film and also some motion of the molecules during the 0.2-s exposure time of each frame. Interpreting Single-Molecule Trajectories. Overall, the trajectories of five molecules were collected and analyzed, corresponding to four different spatial regions in the film. In some of the trajectories, the molecule would diffuse out of the depth of focus for a few frames and then diffuse back in. Figure 4 shows the trajectories for each molecule. The molecules A and B are from the same region of the film. Molecules A, B, and C are from the same data set, while molecules D and E are from the same sample, but from a different data set collected several hundred micrometers away from the first set. Molecules A and B are ∼10 µm away from molecule C, and molecule D is ∼15 µm away from molecule E. It is certain that molecules A and B are different molecules because the molecules appear simultaneously in different locations in several frames. In general, the trajectories appear to be random walks, with no directional bias. Note how molecule A remains stationary for several seconds (near coordinates 18,23 µm) before diffusing again. This could be due to the adsorption to a strong site or diffusion within the internal pores of a single silica sol particle. Because the particles are 27 nm and below the diffraction limit, only interparticle diffusion can be followed. The longest uninterrupted trajectory measured was 44 steps, which corresponds to tracking the same molecule for a total time of 35 s. In all, the five trajectories that were collected and analyzed corresponded to the measurement of 115 molecular steps. To check for directional bias in the motions of molecules due to solution flow, vibrations, mechanical drift, or anisotropy in the dip-coated films, all of the molecular steps were plotted with each

Figure 5. Scatter plots of molecular steps centered at the origin. (a) Scatter plot for all 115 measured molecular steps. (b) Scatter plot for molecule B with 44 molecular steps.

Figure 6. Plot of cumulative squared displacement versus time and its linear fit to eq 1 for molecule B.

step starting at the origin, to produce a scatter plot of molecular step sizes and directions. The scatter plots for all of the molecular steps observed and for the longest trajectory measured are shown in Figure 5. The plots shows no bias in any particular direction and appear to be consistent with the two-dimensional Gaussian probability distribution for the location of a molecule experiencing a random walk.48 A common method to determine diffusion coefficients from single-molecule tracking data is to plot the time dependence of the cumulative square displacement from the initial location.29,32,35,36,38,39,41 According to the Einstein equation for diffusion in two dimensions,48

〈r2〉 ) 4Dt

(2)

this plot should be linear with a slope proportional to the diffusion coefficient. An example of this type of plot made for the longest measured trajectory is shown in Figure 6. From the slope of plot, the molecular diffusion coefficient was estimated to be 3 ( 2 × 10-10 cm2/s. (48) Berg, H. C. Random Walks in Biology; Princeton University Press: Princeton, NJ, 1983.

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This approach to analyzing molecular trajectory data has a number of limitations. First, the uncertainty in the reported diffusion coefficient is very large due to the deviations from a straight line; for this example, the relative error in determining D was 73%. Second, the residuals are not random since the distance of the molecule relative to its initial location is highly correlated from one step and the next; while the individual steps are random, the square displacement from the initial position evolves slowly. Third, the molecule has a large probability of randomly walking back toward its initial location. This motion is fully consistent with the model of a random walk, because the highest probability, even after long times, is for the molecule to be near its initial location. As a result, the point at which the trajectory ends has an enormous effect on the reported diffusion coefficient. For example, if this molecule had photobleached after 15 s of observation, the resulting diffusion coefficient measured would be much larger than if the molecule was observed for 22 s. The points at the end of the observation have the greatest leverage on the slope because they are the farthest away from the origin. When this method has been employed in the literature,29,31,32,35,36,38,39,41 the average cumulative square distance for many molecules is plotted versus time. This averaging significantly improves the precision of this method if many molecules can be tracked for similar periods of time. Diffusion of single molecules in phospholipid bilayers29,31,32,36,41 has been well studied and has generally relied on the averaging of many molecular trajectories comprising 12 or less images due to the speed of the diffusion and the fast camera read rates that are required. To achieve a fast framing rate, small subimages are taken and stored on the camera chip until the entire series is acquired. Our data differ substantially from this situation for two reasons. First, our trajectories vary significantly in length, which is a result of both our ability to monitor molecules for a longer period of time and the three-dimensionality of this system where molecules may diffuse to the interface out of the film. Because of the differences in trajectory length, averaging would work only for the number of points in the shortest trajectory. Second, and more importantly, diffusion in sol-gel films has the potential to be heterogeneous on micrometer distance scales, and the ability to resolve this heterogeneity is lost if trajectories from different parts of the film are averaged together. Averaging step sizes from different molecules measures ensemble properties and prohibits determining local properties sensed by individual molecules, which is a chief advantage of single-molecule detection techniques. Another approach to analyze molecular trajectories is to plot histograms of the frequency of occurrence of different step sizes. This method avoids the correlated history of a mean-square displacement analysis since each step is considered an independent trial starting at the previous location of the molecule. The expected probability density of step sizes, r, for a random walk is expected to be Gaussian with a variance that depends on the product of time and the diffusion coefficient36,37,48 multiplied by a factor 2πr that accounts for the larger area as the size of the r increases:

P(r) ) (1/(4πDt))e-r /4Dt2πr 2

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(3)

Analytical Chemistry, Vol. 75, No. 17, September 1, 2003

Figure 7. Histograms of molecular step size for molecules A-E. Fits are to eq 4. The outlier in trajectory D was not included in the fit of the distribution.

where D is the diffusion coefficient, t is time between observations, and r is the step size. Histograms of the frequency of step sizes, F, are generated by counting the number of steps observed within a given interval ∆r:

F(∆r) ) NP(r)∆r

(4)

where N is the total number of steps in the trajectory. The histograms are fitted to eq 4 to determine the variance of the distribution 4Dt (see below). Figure 7 shows histograms of step sizes for all of molecules observed; the experimental histograms agree with eq 4 even for small numbers of observations. Molecule

Table 1. Single-Molecule Diffusion Coefficients Calculated from Histograms and Average Square Step Sizes molecule

Dhisto (cm2/s)a

A B C D E

10-10

D〈r2〉 (cm2/s)b,c

N

10-10

(3 ( 1) × (7 ( 2) × 10-10 (5 ( 1) × 10-10 (7 ( 3) × 10-11 (3 ( 1) × 10-10

3× 6 × 10-10 5 × 10-10 1 × 10-10 3 × 10-10

36 44 17 5d 12

a Diffusion constant calculated from fit of displacement histogram to eq 4. b Diffusion constant calculated from average squared step size using eq 5. c Uncertainties (standard deviations) are derived from the variance of variances:49 V(〈r2〉) ∼2〈r 4〉/(N - 1). d The original trajectory had an additional point that was deleted from the analysis, based on an F test at the 99.9% confidence level. See text for discussion.

A appears to exhibit some deviation from eq 4, in part because it remains nearly stationary for several frames during its trajectory. While these deviations are comparable to the counting statistics for small numbers of occurrences, with more data points, this behavior might be fit by two distributions, one for fast motion and one for much slower motion. Similar behavior may be influencing trajectory D, which exhibits one large step within a short trajectory that is otherwise dominated by very slow motion. This single step was left out of the fit to eq 4 (see below) in order to describe the slower diffusion component that dominates this trajectory. It is not entirely possible to rule out that trajectory D is actually the result of two different molecules whose appearance and disappearance in the film are coincidental. The diffusion coefficients determined from fitting the histograms are listed in Table 1; however, the uncertainties in these values are not reported. Uncertainties could be estimated from the deviations from fitting expected histogram shape, but the observed deviations are sensitive to the size of the bins, ∆r, used to accumulate the occurrences, especially when the number of steps in the trajectory is small. This sensitivity to binning makes a quantitative comparison of the apparent diffusion coefficients, derived from histograms, difficult to interpret. Histograms of step lengths are, nevertheless, useful for examining the distributions of molecular motions to whether they follow a simple model. For trajectories that follow the random walk model of eq 3, there is a simple method to determine the diffusion coefficient based on calculating the square step size for each point in the trajectory and averaging, 〈r2〉. This approach corresponds to estimating the second moment of the probability density for the step size, P(r):

〈r2〉 ) Lim(r2/n) ) nf∞

∞ 2

0

) [e-r /4Dt(-4Dt - r2)]|∞0 ) 4Dt 2

3

r e ∫ r P(r) dr ) ∫ (2Dt) ∞

0

-r2/4Dt

dr (5)

The average square step size, therefore, represents an unbiased estimate of the distribution variance, 4Dt, which shows the correspondence between the Einstein equation and the second moment of the step-size probability distribution, P(r). This method has the advantage over calculating the cumulative mean-square displacement by considering each step as an independent sample

of the molecular motion, and thus, it does not fail as a molecule moves toward or away from its position in the first frame. Unlike fitting a histogram of the step sizes, calculating 〈r2〉 provides an estimate of the variance of the step distribution that does not depend on a bin size, ∆r. The statistics for estimating the uncertainty (variance) of the variance49 can be used to specify uncertainties of reported diffusion coefficients. Since the scatter plots (Figure 5) and histograms (Figure 7) of the molecular trajectories in the film conform to the general predictions of a random walk, we determine the variance of step lengths from the average square step size, 〈r2〉, according to eq 5. The resulting diffusion coefficients are listed in Table 1 along with their uncertainties.49 The precision to which the diffusion coefficient is estimated by this method is greatly improved compared to the plots of the cumulative mean-square displacement versus time, with an relative uncertainty averaging ∼35%. This uncertainty is a factor of 2 smaller than the uncertainty in slope of a linear fit of the cumulative displacement plots (Figure 6) and does not depend on when the trajectory begins and ends. The values for the diffusion coefficients agree with those determined by fitting the histograms of step sizes within the uncertainty determined from the statistics of the sampled variance.49 The largest differences were found for molecule A, which was observed to remain nearly motionless for several frames, and the diffusion coefficient determined from histogram fitting was smaller than that estimated from the variance probably due to the greater weighting of large step sizes in the 〈r2〉 determination. As noted above, one large step was deleted from both the histogram and 〈r2〉 analysis of trajectory D, based on an F test (see below) at a confidence level of 99.9%. This unique step was left out of the analysis in order to describe the slower motion component that dominates this trajectory. In the fluorescence images, many molecular spots were observed to be motionless over the observation time of the experiment. Because of the complex structure of these sol-gel films, it is possible for molecules to be diffusing in confined spaces that are small compared to the pixel size (135 nm) or are diffusing over small distances (27 nm) within intraparticle pores.13 This kind of motion would not be distinguishable from a truly stationary molecule due to the optical resolution of the microscope and intensity noise in the data. To determine whether these apparently “motionless” molecules were truly stationary within our ability to measure their displacements, numerical simulations were performed to model the effect of the intensity noise on the positional uncertainty of a fluorescence spot. The intensity noise was estimated from the reproducibility of the raw data in regions of the film where no molecules were detected. Stationary Gaussian intensity spots, of the same average size and intensity as the experimental single-molecule data, were modeled within an area of synthetic noise that varied from frame to frame. These images were then filtered and fit in the same manner as the experimental images, and the apparent movement of the intensity profile was characterized by a distribution of steps of size, r. The resulting histograms of step sizes for the “stationary” molecules in the film and those of a simulated stationary molecule are shown in Figure 8. The histograms for the measured and (49) Barlow, R. J. Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences; Wiley: Chichester, 1989; Chapters 5 and 8.

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Table 2. Results of F Tests To Determine Differences in Diffusion Coefficients between Trajectoriesa molecule

A

A B C D E

pass fail pass fail

B pass fail pass pass

C fail fail pass fail

D

E

pass pass pass

fail pass fail pass

pass

a The ratio of measured coefficients corresponds to the ratio of sampled variances, 〈r2〉, which is compared to the tabulated F ratio at 90% confidence,49,50

Figure 8. (a) Histogram of step sizes for an apparently stationary molecule, fit to eq 4. (b) Histogram of step sizes for simulated images with the same S/N ratio and fit to eq 4.

simulated data are nearly equivalent in width, and the diffusion coefficients calculated from the mean-square step sizes are indistinguishable from each other at even 90% confidence according to an F test. Thus, if these “stationary” molecules are indeed moving, their motion is not distinguishable from the root-meansquare positional uncertainty (〈r2〉 ) 150 nm) that is expected solely from the intensity noise in the image. The diffusion coefficient associated with the apparent mean square step size of a stationary molecule puts a lower bound on the diffusion coefficients that could be measured at this S/N ratio, which is ∼7 × 10-11 cm2/s. Given this result, it is very likely that molecule D is not moving, except for its single long step that deleted from the analysis based on an F test. At the limit of detecting diffusion in the film, the trajectory of this molecule appears to be a single fast event that carries it from one dense region of the film to another. The undetectable motion of molecule D and other “stationary” molecules in the film is consistent with the slow rates of accumulation of molecules into the intraparticle micropores of sol-gel films measured by ATR-FT-IR spectroscopy.13 For small molecules exhibiting no detectable adsorption to silica surfaces, diffusion rates within the intraparticle micropores have been estimated to be e10-13 cm2/s, which is more than 1 order of magnitude below the current limit of detection for molecular motion by the single-molecule tracking experiment. The ability of the tracking experiment to detect molecular motion could be improved by increasing the S/N of the collected images. The noise in the present experiments is about twice the photon shot noise predicted for the background fluorescence of the film, where the additional noise is probably due to scattering by the film. The S/N ratio could be improved for measurements made on less scattering, lower background samples. More signal intensity could be gathered from single molecules with a framing camera that is able to integrate signal for longer periods of time between image transfers; the duty cycle of the unmasked CCD camera used in these experiments is only 25%. If the S/N ratio were only 5 times greater than the fluorescence images acquired 4358

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in this work, the lower bound for measurable diffusion coefficients would be D ∼ 3 × 10-12 cm2/s. Such a result may not be achievable in practice since the corresponding rms position uncertainty from the intensity noise is only 30 nm, and the mechanical and optical stability of the microscope might not be sufficient to reach this limit. Experiments are currently underway to test the fundamental and practical limitations for detecting molecular motions in lower background samples imaged with a higher throughput frame-transfer CCD camera. Spatial Heterogeneity in Sol-Gel Films. A goal of this research was to determine whether inhomogeneity in molecular diffusion can be detected within different regions of the sol-gel film by these experiments. To examine this question, we utilize the statistical validity of the mean square step sizes, 〈r2〉, as an unbiased sample of the population variance (eq 4). Fisher’s F test can be used to compare sample variances49 and thus determine whether two measured diffusion coefficients (D ) 〈r2〉/4t) differ by a significant fraction. The ratio of measured diffusion coefficients corresponds to the ratio of sampled variances, 〈r2〉, which is compared to the tabulated F ratio at 90% confidence.49,50 Table 2 shows the results of F testing the observed diffusion coefficients. Of the 10 pairwise tests, 6 pass an F test at 90% confidence, indicating a measurable difference in diffusion coefficient exists between the two regions, and 4 pairs do not show a significant difference at 90% confidence. Somewhat surprisingly, molecules A and B had measurably different diffusion coefficients, even though they were diffusing in the film in regions quite near to one another. Their difference results from several frames in which molecule A appears to stop moving, at least on a distance scale that can be measured optically. It is possible that the two neighboring molecules showed differences in diffusion coefficients because they were diffusing at different depths in the film, indicating that heterogeneity exists not only in the x and y dimensions but also in the z dimension. Although each molecular trajectory was not distinguishable from every other trajectory, significant differences were observed. This is consistent with a continuous distribution of diffusion coefficients with lower bounds established for molecules that are motionless on the time scale of experiment, as discussed above. The diffusion coefficients for each molecular trajectory are plotted in Figure 9 in order from the lowest to highest values, together with error bars corresponding to the uncertainty of a sampled variance.49 These results produce a continuous variation of (50) Beyer, W. H., Ed. Handbook of Tables for Probability and Statistics, 2nd ed.; CRC Press: Boca Raton, FL, 1966.

Figure 9. Plot of the diffusion coefficient determined from the mean square step sizes, 〈r2〉, for each molecular trajectory. Results are arranged from the smallest diffusion coefficient to the largest diffusion coefficient.

diffusion coefficients, with only two trajectories, A and E, having equivalent values. In general, pairs that are nearest neighbors in this plot fail to indicate a difference in diffusion coefficient according to an F test, while all other pairs show significant differences. Because it has been previously shown that the diffusion of small molecules is sensitive to the structure and particle packing in these films,14 it is likely that the variation in diffusion coefficients corresponds to structural heterogeneity in the film on a length scale of several micrometers. Previous SEM images of similar films have shown structural inhomogeneity on a micrometerlength scale, especially for films derived from a smaller number (