Anal. Chem. 1998, 70, 2964-2971
Simulation of Single-Molecule Photocount Statistics in Microdroplets Steven C. Hill
Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783 Michael D. Barnes,* Noah Lermer, William B. Whitten, and J. Michael Ramsey
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
We present results of detailed calculations of photocount statistics for single molecules in microdroplets. A Monte Carlo approach is used to simulate effects of molecular occupancy, photobleaching, and fluorophor spatial diffusion within the droplet. The excitation rate and fraction of fluorescence collected are both position-dependent, and are computed using classical and semiclassical formalisms. Differences in the single-molecule photocount distributions (and consequently molecular detection efficiencies) are predicted for cases where the RMS diffusion length is either small or large compared to the droplet diameter on the measurement time scale. These calculations provide semiquantitative estimates of molecular detection efficiencies and illustrate some of the unique optical features of a microdroplet approach to single molecule detection. Fluorescence analysis of solutions at the single-molecule level has become an important tool in liquid-phase chemical analysis.1-20 Some applications of single-molecule probes to bioanalytical (1) Barnes, M. D.; Whitten, W. B.; Ramsey, J. M. Anal. Chem. 1995, 67, 418A423A. (2) Li, L.-Q.; Davis, L. M. Appl. Opt. 1995, 34, 3208-3217. (3) Soper, S. A.; Shera, E. B.; Martin, J. C.; Jett, J. H.; Hahn, J. H.; Nutter, H. L.; Keller, R. A. Anal. Chem. 1991, 63, 432-437. (4) Nie, S.; Chiu, D. T.; Zare, R. N. Anal. Chem. 1995, 67, 2849-2857. (5) Soper, S. A.; Mattingly, Q. L.; Vegunta, P. Anal. Chem. 1993, 65, 740-747. (6) Castro, A.; Shera, E. B. Appl. Opt. 1995, 34, 3218-3222. (7) Enderlein, J.; Robbins, D. L.; Ambrose, W. P.; Goodwin, P. M.; Keller, R. A. J. Phys. Chem. B 1997, 101, 3626-3632. (8) Keller, R. A.; Ambrose, W. P.; Goodwin, P. M.; Jett, J. H.; Martin, J. C.; Wu, M. Appl. Spectrosc. 1996, 50, A12-A32, and references therein. (9) Wilkerson, C. W.; Goodwin, P. M.; Ambrose, W. P.; Martin, J. C.; Keller, R. A. Appl. Phys. Lett. 1993, 62, 2030-2032. (10) Soper, S. A.; Nutter, H. L.; Keller, R. A.; Davis, L. M.; Shera, E. B. Photochem. Photobiol. 1993, 57, 972-977. (11) Nie, S.; Chiu, D. T.; Zare, R. N. Science 1994, 266, 1018-1021. (12) Chiu, D. T.; Zare, R. N. J. Am. Chem. Soc. 1996, 118, 6512-6513. (13) Ambrose, W. P.; Goodwin, P. M.; Enderlein, J.; Semin, D. J.; Martin, J. C.; Keller, R. A. Chem. Phys. Lett. 1997, 269, 365-370. (14) Moerner, W. E. Science 1994, 265, 46-53, and references therein. (15) Dickson, R. M.; Norris, D. J.; Tzeng, Y.-L.; Moerner, W. E. Science 1996, 274, 966-969. (16) Eigen, M.; Rigler, R. Proc. Natl. Acad. Sci. U.S.A. 1994, 91, 5740-5747; Rigler, R. J. Biotechnol. 1995, 41, 177-186. (17) Chen, D.-Y.; Dovichi, N. J. Anal. Chem. 1996, 68, 690-696. (18) Mathies, R. A.; Peck, K.; Stryer, L. Anal. Chem. 1990, 62, 1786-1791. (19) Xu, X.; Yeung, E. S. Science 1997, 275, 1106.
2964 Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
problems include DNA fragment analysis,6 single-molecule identification,7,10 and molecular manipulation.12,16,21 However, a complicating feature of single-molecule detection experiments in which the observation zone is defined by the laser beam (or by the laser beam in combination with a sheath flow) is the dependence of the fluorescence signal amplitude on the specific path that the molecule takes as it traverses the illumination region.16 Analysis of the distribution of fluorescence signal amplitudes in general becomes nontrivial since a detailed knowledge of molecular diffusion characteristics and spatial properties of the excitation field are required for quantitative modeling. While a microdroplet approach offers some advantages with respect to this problem since the molecules being probed are confined to a sphere,1,22-26 there still remain optical issues such as position-dependent fluorescence collection efficiency27-29 and local excitation intensity variations30,31 within the sphere. While simulations of single-molecule photocount statistics of molecules in liquids (stationary11 or flowing2,3,7,18) have been developed, and a Monte Carlo approach has been employed by several groups,2,3,7 simulations of single-molecule fluorescence in microdroplets must also include an analysis of the positiondependent and frequency-dependent variations in the excitation intensity, collection efficiency, and fluorescence emission rates, along with the features required for analyses in flowing liquids (photobleaching, diffusion, etc.). Because the emission can be strongly enhanced near optical resonances of the droplet, and inhibited in certain positions at nonresonant frequencies, the emission rate and collection efficiency must be integrated over a (20) Schmidt, Th.; Schultz, G. J.; Baumgartner, W.; Gruber, H. T.; Schindler, H. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 2926-2929. (21) Kung, C.-Y.; Barnes, M. D.; Lermer, N.; Whitten, W. B.; Ramsey, J. M. Anal. Chem. 1998, 658-661. (22) Whitten, W. B.; Ramsey, J. M. Anal. Chem. 1991, 63, 1027-1031. (23) Whitten, W. B.; Ramsey, J. M. Appl. Spectrosc. 1992, 46, 1587-1589. (24) Ng, K. C.; Whitten, W. B.; Arnold, S.; Ramsey, J. M. Anal. Chem. 1992, 64, 2914-2919. (25) Barnes, M. D.; Whitten, W. B.; Ramsey, J. M. Anal. Chem. 1993, 65, 23602365. (26) Barnes, M. D.; Kung, C.-Y.; Whitten, W. B.; Ramsey, J. M.; Arnold, S. In Optical Processes in Microcavities; Chang, R. K., Campillo, A. J., Eds.; World Scientific: Singapore, 1996; 135-165. (27) Chew, H.; McNulty, P. J.; Kerker, M. Phys. Rev. A. 1976, 13, 396-404. (28) Hill, S. C.; Saleheen, H. I.; Barnes, M. D.; Whitten, W. B.; Ramsey, J. M. Appl. Opt. 1996, 35, 6278-6288. (29) Hill, S. C.; Barnes, M. D.; Whitten, W. B.; Ramsey, J. M. Appl. Opt. 1997, 36, 4425-4437. S0003-2700(97)01319-X CCC: $15.00
© 1998 American Chemical Society Published on Web 06/04/1998
suitable range of emission frequencies. We have recently analyzed the physical optics problems (e.g., time average internal intensity generated by single or counterpropagating waves, and frequency integrated fraction of fluorescence collected)28,29 required for analyses of single-molecule detection in microdroplets, and are now in a position to combine these models with a Monte Carlo method which will allow simulations of the statistical nature of single-molecule detection. In this paper, we describe the integration of our semiclassical model of absorption and emission of radiation by a single molecule in a microdroplet, with a Monte Carlo approach used to simulate stochastic quantities such as molecular translational diffusion, rotational diffusion of the droplet, and photobleaching. We present results of detailed calculation of single-molecule photocount statistics. Our model of fluorescence collection from single molecules inside droplets28,29 is based on a classical calculation of the excitation intensity inside a sphere illuminated with counterpropagating waves and on a semiclassical formalism of inelastic light emission from molecules inside a sphere.27 By computing the far-zone output field generated from a single molecule (considered as a point-dipole), we are able to determine the fluorescent power radiated into a specified solid angle as a function of position within the sphere. The integrated fluorescence signal depends on the extent of spatial averaging of the output field (over the numerical aperture of the collection lens), and on the illumination geometry29 (which causes local excitationintensity variations30,31 due to interference within the droplet and reflection at the sphere-air interface). The fields we obtain could be used in calculating fluorescent images of droplets containing single molecules (similar to fluorescence imaging of single molecules15,19,20). However, the classical nature of the model precludes recovery of single-molecule fluorescence intensity correlation effects such as photon antibunching.13,32 Effects of photobleaching and translational and rotational diffusion are treated by Monte Carlo sampling of appropriate probability distributions. These calculations are compared with recent single-molecule experiments in droplet streams,21,33,34 and illustrate optimum excitation intensities with respect to molecular detection efficiencies. MODEL AND METHODS In the single-molecule detection experiments we model, a molecule at position r inside a droplet is illuminated with counterpropagating waves. The lifetime of the excited molecule depends on both its lifetime in a bulk medium and upon the position of the molecule in the microsphere.35 A lens at 90° from the laser-propagation direction collects the light. Figure 1 illustrates the problem in two dimensions in the equatorial plane. The false color image shows intensity variations in the excitation field (for a counterpropagating beam geometry). The incident waves are reflected and refracted at the droplet interface, form a (30) Lock, J. A.; Hovenac, E. A. J. Opt. Soc. Am. A 1991, 8, 1541-1549. (31) Chowdhury, D. Q.; Barber, P. W.; Hill, S. C. Appl. Opt. 1992, 31, 35183523. (32) Basche, T. H.; Moerner, W. E.; Orrit, M.; Talon, H. Phys. Rev. Lett. 1992, 61, 1516. (33) Lermer, N.; Barnes, M. D.; Kung, C.-Y.; Whitten, W. B.; Ramsey, J. M. Anal. Chem. 1997, 69, 2115-2121. (34) Barnes, M. D.; Lermer, N.; Kung, C.-Y.; Whitten, W. B.; Ramsey, J. M.; Hill, S. C. Opt. Lett. 1997, 22, 1265-1267. (35) Chew, H. J. Chem. Phys. 1987, 87, 1355-1360.
complex interference pattern inside the droplet, and excite the molecule. The pattern is a sensitive function of particle size, refractive index, and wavelength. The fraction of the emitted photons that are collected by the lens is also a function of the position of the molecule in the droplet.28 The contours show the relative fluorescence collection efficiency as a function of position in the sphere for a collection lens with 0.42 NA. Emission from molecules in some positions travels preferentially toward the collection aperture, while emission from molecules in other positions travels preferentially away from the aperture. That the fluorescence from molecules along the line of sight of the detector is collected as much as 4 times more efficiently than other positions has been confirmed experimentally.36 Perhaps the most physically realistic way to simulate singlemolecule photocount statistics in microdroplets would be to do a Monte Carlo calculation in which (1) an excitation time is computed from the excitation intensity at the molecule; (2) the emission frequency is obtained by sampling a probability distribution which includes both the molecule’s Lorentzian line shape function and the cavity’s excitation and inhibition of rates; (3) an average lifetime is computed and the emission time distribution is sampled for the time of emission; and (4) the probability that the lens collects a photon emitted from a molecule at r is calculated and, after sampling a uniform distribution, is used to determine if the emitted photon is collected by the lens. Other aspects of the problem (e.g., noise, photodetector quantum efficiencies) would be similar to those in previous simulations of single-molecule photocount statistics. Unfortunately, for simulations in which large numbers of molecules/droplets are employed, this type of approach is too intensive computationally, especially when the simulations are to be done repeatedly with different parameters (diffusion lengths, photobleaching, quantum yields, incident intensities, droplet diameters, etc.). The key problem is in the requirement for repeated evaluations of special functions (Bessel, Hankle, and associated Legendre). For each position of the molecule in the droplet and for each potential emission frequency (enough frequencies to sample the cavity resonances adequately), a series of Bessel and Hankle functions must be evaluated to calculate the average emission rate. Also, for each position inside the droplet the emission pattern must be integrated over the collection aperture, a computation that requires the associated Legendre functions in addition to the Bessel and Hankle functions. To avoid the computational requirements of a brute force solution, we use frequency-integrated fluorescence emission and collection rates. These integrated rates should provide accurate values when each molecule emits many photons, but underestimate the variations in the fluorescence collected from molecules that emit a small number of photons. Specifically, we calculate a frequency-integrated fluorescence collection rate at each position on a two-dimensional grid inside the droplet, store these values, and use symmetry and interpolation to obtain the rate at any position. With respect to the position-dependent total spontaneous emission rates we make an additional approximation: we assume that the droplet is large enough and the molecule’s emission line width is large enough that QED effects on the spontaneous (36) Barnes, M. D.; Lermer, N.; Kung, C.-Y.; Whitten, W. B.; Ramsey, J. M.; Hill, S. C. Opt. Lett., in press.
Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
2965
Figure 1. Color contour plot of the fluorescence collection ratio, F(r), (red is high, yellow is low) from a dipole inside a spherical droplet in the equatorial plane [integrated over the numerical aperture of the collection lens (NA ) 0.42), the emission frequencies (molecular line width 100 cm-1), and the dipole orientation], overlaid on a false color plot of the electric field energy density generated inside the droplet by counterpropagating plane waves (incident electric field vector polarized in the plane of the plots). The droplet diameter is 6.5 µm, the excitation wavelength is 0.5145 µm, and the center emission wavelength is 0.6 µm. The average illumination electric field energy density is 1. The color scale is for the electric field energy density.
emission rate can be ignored. This approximation is adequate for the droplets and molecules studied here, but would make the analysis inapplicable to certain combinations of droplet sizes and line widths (smaller droplets and/or narrower line widths). Figure 2 shows a flowchart illustrating the main steps in our method of single-molecule photocount simulations in droplet streams. Some details and discussion of assumptions are as follows. Illumination Intensity and Time in Beam. The illuminating beam(s) propagate in the ( z direction. As the droplet falls along the y axis through the beam, the illumination intensity at the droplet is a Gaussian function I(y) ) 2Iinc exp(-2(y/ya)2), where Iinc is the total power carried by the beam divided by πya2, and ya is the distance from the beam center to the 1/e2 intensity point. The fluorescence counting time, tbeam, is chosen to exceed the time it takes for a droplet to move between the 2Iinc/e3 intensity points. Stochastic droplet position fluctuations37 in the plane 2966 Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
perpendicular to the y axis are ignored. For the experimental cases we consider, Brownian position fluctuations (less than ∼1 µm) and the droplet diameters (1-12 µm) are small compared to the laser beam diameter (∼90 µm or larger), so this assumption is valid in the limit where the beam width is much larger than the droplet diameter. We let the molecule diffuse and emit photons for a number of time intervals, td ) tbeam/nstep, where nstep is some number of steps. Rate of Photon Emission. The average time for a molecule to be excited is
τex )
pω 2mpω ) σI(r,t) σcE2(r,t)
(1)
where pω is the energy per photon, σ is the absorption cross (37) Arnold, S.; Folan, L. M.; Korn, A. J. Appl. Phys. 1993, 74, 4291-4297.
Figure 2. Schematic of the Monte Carlo procedure for modeling the collection of fluorescence from molecules in a series of droplets.
section of the molecule, I(r,t) is the excitation intensity at position r inside the droplet at time t, m is the refractive index, c is the speed of light in a vacuum, and E2(r,t)/2 is the electric energy density. The average rate of photon emission is approximated as
1 τex + τfT
(2)
where τfT is the excited-state lifetime determined from an average of the fluorescence lifetime, triplet lifetime, and nonradiative decay time, weighted by the fraction of excitation events that result in triplet shelving. Although the droplet cavity can make τfT positiondependent, in this paper we have chosen combinations of droplet sizes and molecular emission line widths for which the approximation of constant τfT is adequate.28 Average Number of Fluorescence Photons Detected. The number of photons collected by the lens in the time interval td centered at ti is estimated as
Ne,i )
∫
tdF(ri)Ω/4π F(r)Ω/4π dt = pω pω τfT + τfT + σI(r,t) σI(ri,ti)
ti+td/2
ti-td/2
(3)
where Ω is the solid angle subtended by the collection optics, F(r)Ω/4π is the frequency-integrated fluorescent power collected from a dipole (averaged over all orientations) at r, and normalized
by the total frequency-integrated fluorescence power (integrated over 4π sr) emitted from the dipole at r in the sphere, again averaged over all orientations (the position-dependence of the normalization power is relatively unimportant for the droplet sizes and emission linewidths studied here; in our calculations we assume it is uniform).38 That is, F(r)Ω/4π is the probability that a photon emitted at r will be collected by the lens, when the dipole can take a random orientation. F(r) specifies how the collection probability varies from the collection in a bulk liquid. (In bulk, F(r) ) 1). In the integrations over frequency the power is weighted by the molecule’s Lorentzian line shape function, and by the cavity-modified relative emission rate. The position of the molecule at the center of the time interval is ri. The numbers of photons detected are obtained by scaling these values by an overall photon detection efficiency (which includes the filter transmission and the detector quantum efficiency), and then by sampling a Poisson distribution having that average number of detected photons. Photobleach-Limited Number of Photons Emitted by the Molecule. The photobleach-limited number of emission events for each molecule, Nm, follows a decaying exponential distribution40 which is sampled according to Nm ) Ne ln(X), where Ne is the average number of emission events before photobleaching, and X is a random number with a uniform distribution between 0 and 1. At each time step we compare the value of Nm with the total number of photons that have been emitted from the molecule. Length and Direction of Diffusion during the Time Step. For each time period the molecule diffuses some random length in some random direction. The average of the squared distances the molecules diffuse in time td is ∆r2 ) 6Dtd, where D is the diffusion coefficient. Translational diffusion of the molecule in three dimensions within the droplet is modeled as a random walk. The probability that the molecule moves a distance r in time td is given by a Gaussian error distribution.39 We calculate a random deviate with a Gaussian (normal) distribution40 and then multiply by (6Dtd)1/2 to obtain the random distance r. We then calculate a random direction for the diffusion of the molecule. Length and Direction of the Rotational Diffusion during the Time Step. Rotational diffusion partially randomizes the position of the molecule (and hence, affects the molecules detectability). For each time period the droplet rotates some random angle in some random direction. The average of the squared angle the droplet rotates about a single axis in time td is41 ∆θ2 ) kTtd/4πηa3, where k is Boltzman’s constant, T is the (38) The F(r) used here differs from that of refs 28 and 29 only in that the normalization is by the total fluorescence power from a dipole at r inside the sphere instead of by the fluorescence collected from a dipole in a homogeneous medium. For this paper the difference is more conceptual than practical. Note also that eq 3 assumes that the integration over dipole orientation can be brought inside the integral over frequency; although this is a reasonable approximation for the droplet sizes and lens numerical apertures used here (and essential for computational efficiency), it may not be adequate for smaller NA collection optics. (39) Moore, W. J. Physical Chemistry, 3rd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1962; pp 234 and 343. (40) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran, The Art of Scientific Computing, 2nd ed.; Cambridge: Cambridge, 1992. Uniform deviates for random positions, directions, etc., are obtained using ran2 (pp 277-280). Exponential, Gaussian, and Poisson deviates are obtained using expdev (p 278), gasdev (pp 279-280), and poidev (pp 283-285), respectively, along with ran2.
Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
2967
temperature, η is the viscosity of the air, and a is the radius of the droplet. The distance the molecule moves upon rotation is |r|p(n)(2(∆θ2))1/2, where p(n) is a random deviate with a Gaussian (normal) distribution, and where the 2 is to account for rotation about two axes (the third axis is chosen to pass through the molecule, and so the distance the molecule moves because of rotation about that axis is zero). We then calculate a random direction for the rotation and the new position of the molecule. We assume that the initial rotation of the droplet is zero, and assume that the rotational diffusion can be modeled as a discrete Markov chain (i.e., that td is long enough that the momentum from the previous step can be ignored). Number of Background (Nonfluorescence) Detection Events. The droplet-independent part of the background (the photons detected when no droplet is in the sample region) includes contributions from Raman scattering (from gases in the cell), stray light that generates fluorescence or Raman when it illuminates some surface, etc. The average number of dropletindependent photons detected is
N h nf-nd ) d
Iinc Λt + Kdarktbeam pω beam
(4)
where the superscript nf-nd indicates “nonfluorescent with no droplet,” the parameter Λ is determined empirically for each experimental arrangement, and Kdark is a rate constant for dark counts. The droplet-dependent average number of nonfluorescent photons detected, N h nf-d , where the superscript nf-d indicates d “nonfluorescent with droplet,” is primarily attributable to Raman scattering by the solvent and includes contributions from the fluorescence generated in the optics by light that is scattered by the droplet. It is approximated as
N h nf-d ) d
Iinc-avg 3 Iinc βd tbeam = 2xπ βd3 pω pω
(5)
where the diameter of the droplet is d, and the empirically determined parameter β depends on the collection optics, composition of the droplet, etc., Iinc-avg is the average of the incident intensity seen by the droplet over the time tbeam, and where the approximate equality is within 2% because tbeam is chosen sufficiently large. The diameter d is determined by sampling a Gaussian distribution. The number of nonfluorescent, droplet-dependent photons are obtained by sampling a Poisson distributions with average numbers N h nf-nd and N h nf-d . d d Width of the Background Distribution. The width of the background photocount distribution, σN, is needed to estimate the signal-to-noise ratio and the fraction of the molecules with signals 3σN greater than the average background. To obtain σN, we first find σ1, the variation in the number of average background counts (N h nf d ). We assume that σ1 is attributable to two terms: the standard deviation in the intensity σI (the variation in σI is much greater than I1/2), and the standard deviation of the droplet (41) Fuchs, N. A. The Mechanics of Aerosols; Pergamon: Oxford, 1964; p. 245. See also: Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1963; pp 432-437, and compare that analysis with Moore (39).
2968 Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
diameter, σd. The parameters are related with a propagation of random errors analysis as
( ) ( )( ) ( ) ( ) σ1
N h nf d
2
)
∂N h nf d ∂Iinc
2
σI
N h nf d
2
+
∂N h nf d ∂d
2
σd
N h nf d
2
(6)
For any particular value of average background counts, N h nf d, there is an additional shot noise, which we assume results from one cascaded Poisson-statistics process. We assume that this Poisson distribution can be approximated by a Gaussian, having 1/2 a width σ2 ) (N h nf d ) . Then the final distribution has a width σN ) (σ12 + σ22)1/2. RESULTS AND DISCUSSION The results shown have been selected to model previously reported1,21,25,33,34 and ongoing experiments, illustrate variations in some of the parameters in ways that would be extremely timeconsuming experimentally, help in understanding the effects of photon-droplet interactions on the detection probability, and suggest experimental variations in which the detection efficiencies may be increased. In attempts to validate a model by comparing experimental data with modeled results, the validation is often limited by uncertainties in the required physical and experimental parameters. In our droplet work there is considerable uncertainty in the values of τfT and Ne of R6G in 85% glycerol in water, as well as uncertainties in the experimentally determined detection efficiencies. We can say (and show example results below) that it is possible to choose parameters, from within the ranges of uncertainty for each parameter, so that the calculations fit the data quite well. Because of our goals of using the model to develop understanding and suggest further experiments, we chose, for the illustrations here, parameter values (within their uncertainty ranges) that make our results most similar to the experimentally measured curves. The average time in an excited state, τfT, the weighted average of the fluorescence lifetime (∼4 ns) and the triplet state lifetime (∼4 µs), is 12 ns which is consistent with measurements in ethanol4 (1 of ∼500 excitations results in intersystem crossing to the triplet state). The main parameters are given in Table 1. Figure 3 illustrates results of model calculations of the number of photons collected per droplet from a sequence of 1000 droplets composed of 85% glycerol in water (the raw data). The lower curve shows counts from the blank (no R6G) droplets. The average number of R6G molecules per droplet is 0.024. The line width assumed for the R6G emission spectrum (100 cm-1) is the homogeneous line width,44 which is large enough that several droplet resonances are included in the integration over emission frequencies with the 6.5 µm diameter droplets studied here; therefore, increasing the line width would have a small or negligible effect on the results.28 These calculated photocounts illustrate the same qualitative features of the measurements of (42) Lawrie, J. W. Glycerol and the Glycols; Chemical Catalogue Co.: New York, 1928; pp 204-205. (43) Snavely, B. B. In Dye Lasers; Schafer, F. P., Ed.; Springer-Verlag: New York, 1990; pp 93-95. (44) Barnes, M. D.; Whitten, W. B.; Ramsey, J. M. J. Opt. Soc. Am. B 1994, 1297-1304.
Table 1. Parameters Used in Calculations composition typical droplet diameters refractive index43 excitation wavelength beam diameter (1/e2 full width) numerical aperture of collection lens β Λ Kdark diffusion coefficient (assumed proportional to viscosity42) R6G absorption cross section at 515.4 nm43 τfT (see ref 4 for data in ethanol) Ne, average number of emission events before photobleaching R6G emission center frequency and linewidth
glycerol 85%/water 15% 6.5 and 7.5 µ m 1.454 0.5145 µm 95 µm 0.42 4.8 × 10-13 cm-1 7.9 × 10-21 cm-2 10 Hz 2 × 10-8 cm2/s 1.5 × 10-16 cm-2 12 ns 6 × 105 16666.6 cm-1, 100 cm-1
Figure 4. Calculated single molecule photocount distributions from 6.5 µm diameter glycerol/water droplets. The incident intensity was chosen high enough (4 × 106 W/cm2) so that essentially all the molecules photobleach (the fraction surviving is less than 3 × 10-5). The background counts are not included (β and Λ are negligibly small). The diffusion coefficient is 2 × 10-8 cm2/s for the thin line, and 2 × 10-4 cm2/s for the thick line. The inset shows the distribution of the fluorescence collection ratio F(r). In a bulk liquid the fluorecence collection ratio is 1 for molecules near the focal point of the collection lens.
Figure 3. Simulation of photons collected per droplet from a sequence of 1000 droplets (6.5 µm diameter, 85% glycerol in water v/v). The average number of R6G molecules per droplet is 0.024. The lower curve shows counts from the blank (no R6G) droplets. The upper curve has been offset to allow comparison. The illumination intensity for a counterpropagating beam geometry is 70000 W/cm2, and the illumination/collection time is 140 ms. The emission line width of the R6G is 100 cm-1 [see ref 28 for discussion of effects of line width on calculation of F(r)]. The squares indicate the number of R6G molecules per droplet. Other parameters are given in Table 1. The main parameters in the calculation were chosen to compare with recent experimental observations.33 The dashed lines are 3σN above background. The standard deviation of the droplet diameter is σd ) 0.1d.
Lermer et al.,33 such as the frequency of events which exceed the threshold and the distribution of peak heights. Figure 4 shows the probability versus the photocounts per droplet from a sequence of 106 droplets. Each droplet contains one R6G molecule. The intensity (4 × 106 W/cm2) and time in the beam (140 ms) are large enough to cause photodegradation of essentially all the molecules. No background counts are included. The refractive index of all droplets is that of 85% glycerol in water (v/v). The diffusion coefficient is that of 85% glycerol in water for the thin line, and is set artificially high (2 × 10-4 cm2/ s) for the thick line. With rapid diffusion the probability versus
photocounts varies almost exponentially, even though both the rate of excitation which depends on the internal intensity, I(r), and the fraction of the emitted light that is collected, F(r), vary with position inside the droplet. The probability distribution for the slow diffusion does not decay exponentially with photocounts. The nonexponential decay cannot be caused by variations in I(r), because in these calculations essentially all the molecules photobleach. The variation is caused by the variation in F(r) (the probability distribution of F(r) is illustrated in the inset), and by slow diffusion, which allows each molecule to only sample a small part of the F(r) distribution. The average time to photobleach in Figure 4 is ∼27 ms. In the droplets with D ) 2 × 10-4 cm2/s the rms diffusion distance in 27 ms is (6Dt)1/2 ∼ 57 µm (∼9 droplet diameters). The molecule diffuses fast enough that effects of inhomogeneities in F(r) are almost eliminated by averaging. In the droplets with D ) 2 ×10-8 the rms diffusion distance is (6Dt)1/2 ∼ 57 µm (∼0.09 droplet diameters). The molecule does not diffuse fast enough to sample more than a small region of the droplet (in this case the rotational diffusion of the droplet is more significant than the diffusion of the molecule). The photocount distribution is modified by the position dependence of the fluorescence collected by a lens. Figure 5 shows calculated and measured photocount distributions for droplets containing RhB. The parameters are as given in the table for R6G, except that the droplet diameter is 9 µm. The F(r) used in the calculation was for a 6.5 µm diameter droplet (because our computer programs have problems with large glycerol droplets and the normalized F(r) should be similar). The comparison between theory and experiment is good in the range where there are many molecules. It appears less good for the Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
2969
Figure 5. Calculated (solid line) and measured (diamonds) photocount distributions from droplets containing RhB. The parameters are as given in the table for R6G, except that the droplet diameter is 9 µm. In the calculations the F(r) for 6.5 µm diameter droplets was used (with the position vector r normalized by the ratio of the droplet diameters) because of computational difficulties in the integrations required for F(r) for 9 µm diameter glycerol/water droplets. Because F(r) is obtained by integrating over frequency, collection solid angle, and dipole orientation, its overall features do not vary rapidly with droplet size. The excitation distribution is that of a 9 µm diameter droplet. The diamonds at the lowest level (2 × 10-5) are actually zeros (they indicate bins with no molecules), but were given an arbitrary nonzero value so they could be indicated.
bins with very few (either 1 or 2) molecules per bin, where the numbers are so small that the statistical significance is not clear. A primary figure of merit for a single molecule detection scheme is the molecular detection efficiency, i.e., the probability that the photons detected from a single molecule exceeds some threshold. A typical value for the threshold, the value of the threshold used here, is the sum of the background (“blank”) signal and three times the standard deviation of background, σN. Figure 6 shows the molecular detection efficiency, signal-to-noise ratio, and fraction that photobleach, all as a function of the incident intensity. The droplets are 85% glycerol in water which defines the diffusion coefficient of 2 × 10-8. The incident intensities were chosen to illustrate the detection efficiencies within the range of intensities typically used in experiments, because they are in a near-optimal range (see ref 18 for optimization of single molecule S/N ratios in flowing liquids). In Figure 6a the parameters are as in Table 1. In Figure 6b the collection efficiency has been halved while keeping the background counts the same. Because we use the frequency-integrated fraction of photons collected, the calculated detection efficiency at the lowest intensity levels (say
