Anal. Chem. 1996, 68, 690-696
Single-Molecule Detection in Capillary Electrophoresis: Molecular Shot Noise as a Fundamental Limit to Chemical Analysis DaYong Chen† and Norman J. Dovichi*
Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada
Capillary electrophoresis is coupled with a single molecule detector based on laser-induced fluorescence. Individual molecules migrating from the capillary are detected and counted with 50% efficiency. Injection of 30 000 analyte molecules generates a reproducible peak consisting of at least five components. However, injection of 3000 or fewer molecules leads to a noisy and irreproducible peak. Monte Carlo simulation demonstrates that this irreproducibility results from molecular shot noise or stochastic fluctuations in the number of injected molecules. The model predicts that the relative standard deviations of peak area, peak center, and peak width are inversely proportional to the square root of the number of injected molecules. At least 104 analyte molecules (17 zmol) are required to define peak area and width with 1% relative precision. Fluctuation in the number of molecules taken for chemical analysis is a fundamental and irreducible source of uncertainty. Chemical analysis is traditionally divided into several steps: sampling, preliminary operations, measurement, calculation, and evaluation of results.1 The first step, sampling, refers to the generation of a representative sample of an inhomogeneous object.2 Although sampling is usually considered in the context of complex, macroscopic objects such as a tissue sample or an ore body, all solutions are inhomogeneous at the molecular level. This inhomogeneity presents a fundamental limit to chemical analysis. As Tomas Hirschfeld pointed out, “fluctuations arising from the stochastic variations in the number of sample particles within the sampling volume” are a basic, fundamental limit to quantitative analysis.3 By analogy with shot noise in photon counting experiments,4 the uncertainty in the amount of analyte detected is equal to the square root of the number of molecules assayed;5 we call this uncertainty molecular shot noise. Molecular shot noise becomes important when very small volumes of dilute solutions are prepared for analysis. For example, to obtain a relative precision of 1%, it is necessary to analyze at least 10 000 analyte molecules, or 17 zmol of analyte. † Current address: Department of Chemistry, University of British Columbia, Vancouver, BC V6T 1Y6, Canada. (1) Harris, W. E.; Kratochvil, B. An Introduction to Chemical Analysis; Saunders College Publishing: New York, 1981; pp 4-6. (2) Kratochvil, B.; Taylor, J. K. Anal. Chem. 1981, 53, 924A-938A. (3) Hirschfeld, T. Anal. Chem. 1976, 48, 16A-31A. (4) Malmstadt, H. V.; Franklin, M. L.; Horlick, G. Anal. Chem. 1972, 44, 63A76A. (5) Hungerford, J. M.; Christian, G. D. Anal. Chem. 1986, 58, 2567-2568.
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The incorporation of high sensitivity detectors in microcolumn separation science allows analysis of minute amounts of sample. For example, yoctomole detection limits have been reported for the capillary electrophoretic analysis of sugars, amino acids, nucleic acids, and neat dyes.6-33 Molecular shot noise should dominate the precision of these experiments. In this paper, we demonstrate the effects of molecular shot noise on the precision of peak area and shape in electrophoretic separations. (6) Wu, S.; Dovichi, N. J. J. Chromatogr. 1989, 480, 141-155. (7) Swerdlow, H.; Zhang, J. Z.; Chen, D. Y.; Harke, H. R.; Grey, R.; Wu, S.; Fuller, C.; Dovichi, N. J. Anal. Chem. 1991, 63, 2835-2841. (8) Zhao, J. Y.; Dovichi, N. J.; Hindsgaul, O.; Gosselin, S.; Palcic, M. M. Glycobiology 1994, 4, 239-242. (9) Chen, D. Y.; Dovichi, N. J. J. Chromatogr. 1994, 657, 265-269. (10) Figeys, D.; Arriaga, E.; Renborg, A.; Dovichi, N. J. J. Chromatogr. 1994, 669, 205-216. (11) Zhao, J. Y.; Diedrich, P.; Zhang, Y.; Hindsgaul, O.; Dovichi, N. J. J. Chromatogr. 1994, 657, 307-313. (12) Chen, D. Y.; Adelhelm, K.; Cheng, X. L.; Dovichi, N. J. Analyst 1994, 119, 349-352. (13) Le, X. C.; Scaman, C.; Zhang, Y.; Zhang, J. Z.; Dovichi, N. J.; Hindsgaul, O.; Palcic, M. M. J. Chromatogr. 1994, 716, 215-220. (14) Timperman, A. T.; Khatib, K.; Sweedler, J. V. Anal. Chem. 1995, 67, 139144. (15) Hirschfeld, T. Appl. Opt. 1976, 15, 2965-2966. (16) Dovichi, N. J.; Martin, J. C.; Jett, J. H.; Keller, R. A. Science 1983, 219, 845-847. (17) Dovichi, N. J.; Martin, J. C.; Jett, J. H.; Trkula, M.; Keller, R. A. Anal. Chem. 1984, 56, 348-354. (18) Nguyen, D. C.; Keller, R. A.; Jett, J. H.; Martin, J. C. Anal. Chem. 1987, 59, 2158-2160. (19) Cheng, Y. F.; Dovichi, N. J. Science 1988, 242, 562-564. (20) Peck, K.; Stryer, L.; Glazer, A. N.; Mathies, R. A. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 4087-4091. (21) Shera, E. B.; Seitzinger, N. K.; Davis, L. M.; Keller, R. A.; Soper, S. A. Chem. Phys. Lett. 1990, 174, 553-557. (22) 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. (23) Hahn, J. H.; Soper, S. A.; Nutter, H. L.; Martin, J. C.; Jett, J. H.; Keller, R. A. Appl. Spectrosc. 1991, 45, 743-745. (24) Whitten, W. B.; Ramsey, J. M.; Arnold, S.; Bronk, B. V. Anal. Chem. 1991, 63, 1027-1031. (25) Zhao, J. Y.; Chen, D. Y.; Dovichi, N. J. J. Chromatogr. 1992, 608, 117-120. (26) Ng, K. C.; Whitten, W. B.; Arnold, S.; Ramsey, J. M. Anal. Chem. 1992, 64, 2914-2929. (27) Barnes, M. D.; Ng, K. C.; Whitten, W. B.; Ramsey, J. M. Anal. Chem. 1993, 65, 2360-2365. (28) Goodwin, P. M.; Johnson, M. E.; Martin, J. C.; Ambrose, W. P.; Jett, J. H.; Keller, R. A. Nucleic Acids Res. 1993, 21, 803-810. (29) Castro, A.; Fairfield, F. R.; Shera, E. B. Anal. Chem. 1993, 65, 849-852. (30) Wilkerson, C. W.; Goodwin, P. M.; Ambrose, W. P.; Martin, J. C.; Keller, R. A. Appl. Phys. Lett. 1993, 62, 2030-2032. (31) Soper, S. A.; Mattingly, Q. L.; Vegunta, P. Anal. Chem. 1993, 65, 740-747. (32) Soper, S. A.; Davis, L. M.; Shera, E. B. J. Opt. Soc. Am. 1993, 65, 17611769. (33) Lee, Y. H.; Maus, R. G.; Smith, B. W.; Wineforner, J. D. Anal. Chem. 1994, 66, 4142-4149. 0003-2700/96/0368-0690$12.00/0
© 1996 American Chemical Society
THEORY A Monte Carlo simulation has been developed for a molecular shot noise dominated electrophoresis peak. This simulation is written in Matlab 4.2c and runs on a Macintosh PowerPC 8100/ 80 computer. The simulation consists of three main parts. The first part generates a Poisson-distributed random number with mean M; this routine is based on the program POIDEV.34 To speed execution, the calculation of random Poisson-distributed numbers with mean, M, greater than 100 is replaced with calculation of a Gaussian-distributed number with mean M and with standard deviation of M1/2. Matlab’s internal Gaussiandistributed random number generator, randn, is used in this case. The second part of the simulation generates a 101-point Gaussian peak defined as
{ [
a(1)(1/norm) exp -0.5
]}
i - a(2) a(3)
2
(1)
where i is a counter running from 50 to 150, a(1) is the number of molecules contained within the peak, a(2) is the peak center, a(3) is the standard deviation of the peak, and norm is a factor that normalizes the area of the Gaussian function. In the peak generation step, a(2) ) 100 and a(3) ) 10. The routine then constructs a simulated electrophoresis peak by generating a file of Poisson-distributed random numbers with mean given by the amplitude of the 101-point Gaussian peak. The third part of the simulation fits a Gaussian function to the simulated electrophoresis peak. The nonlinear regression analysis is performed with a set of routines modeled on the Fortran program CURFIT.35 An unweighted fit is used. The procedure is repeated 100 times for each set of conditions, and all analyses are performed in duplicate or triplicate. Simulations are performed for the injection of 10, 30, 100, 300, 1000, 3000, 104, and 105 molecules. The variance of the peak area increases linearly with the number of molecules injected, r ) 0.999, and with slope of 0.99 ( 0.01 and zero intercept. The variance of the peak center is inversely proportional to the number of injected molecules, r ) 0.999. The variance of the peak standard deviation is inversely proportional to the number of injected molecules, r ) 0.999. The relative precision of all three parameters is proportional to the inverse square root of the number of injected molecules. As a result, the precisions of peak area, center, width, and theoretical plate count are all dominated by molecular shot noise as the number of injected molecules becomes small. Because of molecular shot noise, at least 104 analyte molecules are required to define a peak with 1% accuracy. A second simulation has been performed to study the effect of background noise on estimates of peak area, center, standard deviation, and baseline offset. For this simulation, Gaussiandistributed noise is added to the Poisson-distributed Gaussian peak generated by eq 1. Background noise has little effect when the standard deviation of the noise is much less than 0.1M1/2, where M is the mean number of injected molecules. Higher background noise degrades the precision of each parameter. (34) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge, U.K., 1986; pp 206-208. (35) Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw Hill: Toronto, 1969; pp 237-239.
EXPERIMENTAL SECTION Electrokinetic Pumping System. A 10-µm-I.D., 142-µm-O.D., 28.9-cm-long fused silica capillary (Polymicro, Phoenix, AZ) introduced sample to the detector. To ensure uniform flow, 1 mm of the capillary was etched to ∼50 µm-diameter tip with hydrofluoric acid while nitrogen gas was pumped through the tip of the capillary at a pressure of 80 psi. Electrokinetic flow was driven by a Spellman CZE1000R high-voltage power supply. Sample Preparation. The pH 8.2 TBE buffer was prepared from 0.54 g of Tris, 0.275 g of boric acid, and 0.100 mmol of disodium EDTA, diluted to 50 mL with doubly distilled and deionized water (Barnstead Nanopure). The buffer was autoclaved before use. A stock B-phycoerythrin bulk solution was prepared from 2 mg of B-phycoerythrin in 60% saturated ammonium sulfate suspension (Molecular Probes) that was centrifuged at 5000 rpm for 10 min on a tabletop centrifuge. The pelleted B-phycoerythrin was dissolved in 50 mL of aqueous 0.1 M sodium phosphate containing 0.1 M NaCl in a volumetric flask. The concentration of the stock solution was calculated to be 1.67 × 10-7 M. The value was confirmed from the absorbance of the solution. All dilutions were prepared in a clean-air hood; the hood was operated for 20 min and then turned off immediately before sample handling. This procedure produced a clean work environment with minimal air flow to disperse aerosols. An aliquot of the bulk solution was diluted in TBE buffer to the desired concentration on the day of the experiment. A micropipet was used to dispense the sample; fresh tips were used at each dilution. Samples were dispensed into disposable plastic centrifuge tubes. All solutions were vortexed for at least 1 min after dilution. The sheath buffer, also TBE, was supplied by a siphon from a wash bottle seated on a lab jack. The surface of the buffer in the wash bottle was 2 cm higher than the surface of the waste fluid collection reservoir. To avoid a siphon in the analysis capillary, the height of the sample was matched to the height of the waste reservoir. Instrumentation. The instrument was similar to that described previously.12 A 2-mW green, λ ) 543.5 nm, HeNe laser (Melles Griot) was the excitation source. The beam was focused with a 6.3× microscope objective to a 30-µm spot in a sheath flow cuvette. The locally constructed cuvette had 1-mm thick quartz windows and a 150-µm square flow chamber. The cuvette was held in a grounded stainless steel holder. The capillary was inserted into the flow chamber. Fluorescence was excited ∼20 µm downstream from the tip of the capillary. The fluorescence from the sample was collected by two 60×, 0.7 NA microscope objectives (Universe Kogaku), one on each side of the cuvette. Each objective imaged fluorescence through separate interference filters (580DF30, Omega Optical) onto irises, adjusted in size to block scattered laser light. Fluorescence was detected with R1477 photomultiplier tubes (Hamamatsu), operated at 1 kV with a Hamamatsu high-voltage supply base. The tube outputs were summed and passed through a 1-kHz low-pass electronic filter (Ithaco). The filtered output was digitized at 1 kHz by a Macintosh Quadra 700 computer equipped with a high-resolution multifunction I/O board (National Instruments). The injection, electrophoresis, and data collection programs were programmed in LabVIEW (National Instruments). Single Molecule Data Analysis. Data analysis was performed with Matlab 4.2c on a Macintosh 8100/80 PowerPC Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
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Figure 1. Single-molecule detection. A solution of B-phycoerythrin was introduced into the fluorescence detector at a rate of 6 molecules/s. The data were convoluted with a 12-ms-wide Gaussian filter. Local maxima were located and are marked with a “+”. Those local maxima that exceed a threshold set at 0.1 V are denoted with a “O”. (A) Sample flowing. (B) Sample off.
computer. Data were first convoluted with a 101-point digital filter,
filter (count) )
1 exp{-0.5(count/12)2} 30.96
(2)
where count ranges from -50 to +50, and where the area under the filter has been normalized to unity. The data were then passed through a digital peak height analyzer; all local maxima in the filtered data were identified. Finally, those local maxima that exceed a threshold were identified as single-molecule events. Capillary Electrophoresis. Electrophoretic injections were performed by applying potential through a platinum wire to the solution. To avoid contamination of the running buffer, the capillary tip was rinsed in two different buffer-containing vials before the capillary was placed in the final running buffer. RESULTS AND DISCUSSION Single Molecule Detection. In our detection limit studies, analyte was continuously introduced into the detector for the first 30 s of an experiment. The polarity of the power supply was then reversed for 10 s, generating a background signal due to the sheath buffer alone. Figure 1A presents the first 5 s of data generated at a nominal molecular flow rate of 6 molecules/s. Local maxima are identified with a cross. Fifteen local maxima that exceed a threshold of 0.10 V are marked with a circle; these bursts were presumably generated by single analyte molecules passing through the detector. The burst at 2.25 s is anomalously large and may have been generated by two molecules passing simultaneously through the laser beam. There are two closely spaced local maxima at 4 s, which may be due to two molecules passing in quick succession. Figure 1B presents the 5 s of the background signal generated by flowing sheath buffer. Only one peak exceeds the threshold of 0.1 V. This photon burst resembles those observed in Figure 1A and may be due to a lone analyte molecule diffusing from the tip of the capillary. The average amplitude of a single-molecule photon burst was calculated for five experiments with nominal molecular flow rate from 3 to 18 molecules/s; higher flow rate data were not used to avoid bias due to multiple molecule events. The average photon burst was 0.028 V above the background. The noise in the 692 Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
Figure 2. Single-molecule detection calibration curve. The crosses represent the observed single-molecule events generated during a 30-s experiment. The smooth curve is the least-squares fit of the function a(1 - e-γf/a) to the data.
background was 0.004 V. The average photon burst was 6.8 times larger than the standard deviation in the background signal. Calibration Curve. Figure 2 presents a calibration curve generated by 10 experiments at molecular flow rates ranging from 3 to 120 molecules/s. The curve reaches a steady state of ∼375 counts at high flow rates. Saturation occurs in paralyzable detectors that cannot respond to simultaneous events.36-38 In our case, saturation arises when two or more molecules are present in the laser beam at the same time; their signals combine to generate a single, large-amplitude peak that is counted as a single molecule. The saturated calibration curve is described by Poisson statistics. The probability that one or more molecules is present in the illuminated volume is given by
P(N g 1) ) 1 - e-R
(3)
where R is the average number of molecules present in the laser (36) Johnson, F. A.; Jones, R.; McLean, T. P.; Pike, E. R. Phys. Rev. Lett. 1966, 16, 589-592. (37) Gilchrist, J. H.; Babey, S. K. IEEE Trans. Nucl. Sci. 1978, NS-25, 16551660. (38) Julanov, Y. V.; Lushnikov, A. A.; Nevskii, I. A. J. Aerosol Sci. 1984, 15, 69-79.
Figure 3. Injection of 30 000 B-phycoerythrin molecules. A 1.67 × 10-9 M solution of B-phycoerythrin was injected at 1 kV for 5 s. Separation proceeded at 29 000 V. Data were digitized at 1 kHz and convoluted with a Gaussian filter with σ ) 48 ms.
beam. The smooth curve in Figure 2 is the least-squares fit of the function a(1 - e-γf/a) to the data, where a is the saturated count rate, f is the molecular flow rate, and γ is the fraction of molecules that generate a detectable fluorescence burst; if all molecules are detected, then R ) f/a. In Figure 2, the saturated count rate is 375 molecules/s, and the fraction of detected molecules is 0.49; approximately half of the analyte molecules generate a photon burst that exceeds the 0.1 V threshold. The fraction of detected molecules can also be estimated from the linear portion of the calibration curve, corresponding to molecular flow rates less than ∼18 molecules/s. The average number of counts observed for five experiments was 45% of the expected number of counts. There are several possible causes for this loss in signal. Some of the analyte molecules may stick to the capillary walls and be lost from the system. However, our electrophoresis data suggest that adsorptive losses are not significant for this molecule in this buffer system. If the sample stream is much larger than the laser beam, then some of the molecules will not pass through the laser beam and will not generate detectable signals. A series of experiments was performed at higher sheath flow rates to force analyte into a narrower stream through the laser beam; there was no systematic variation in the number of counts with sheath flow rate. The laser beam was focused to a 30-µm spot, which is much larger than the sample stream radiussall analyte must flow through the laser beam.39 Some molecules will be lost if a portion of the photon burst distribution falls below our threshold. Finally, B-phycoerythrin is inhomogeneous, generating several electrophoretic bands; some components may be nonfluorescent. Capillary Electrophoresis. Figure 3 presents the capillary electrophoresis separation of a solution containing 3 × 104 molecules (50 zmol) of B-phycoerythrin. The peak consist of a relatively broad envelope, with evidence for at least five components; one component is roughly twice the height of the other four components, which are roughly equal in height. These peaks arise from the heterogeneous nature of the enzyme due to differences in posttranslational modifications.40 The data in this Figure were convoluted with a Gaussian-shaped filter matched to the resolution expected for this electrophoretic separation. Based (39) Zarrin, F.; Dovichi, N. J. Anal. Chem. 1987, 59, 846-850. (40) Wilbanks, S. M.; Glazer, A. N. J. Biol. Chem. 1989, 264, 17860-17867.
Figure 4. Injection of 30 analyte B-phycoerythrin molecules. A 1.67 × 10-12 M solution of B-phycoerythrin was injected at 1 kV for 5 s. Separation proceeded at 29 000 V. Data were recorded at 1 kHz and convoluted with a σ ) 12 ms Gaussian filter. Local maxima that exceeded 0.1 V are marked with a cross. (a) Two-minute portion of electropherogram. (b) Enlarged region corresponding to the electrophoretic peak.
on the full width at half-height of the sharp central feature, the electrophoresis system generates at least 100 000 theoretical plates for this compound. This high plate count demonstrates that adsorption of the compound on the capillary wall is minimal. Replicate injections of this amount of analyte lead to very similar electropherograms. Figure 4A presents the electropherogram of an injection of 30 analyte molecules. In this case, the data have been treated with a single-molecule detection filter. The analyte generates a much lower amplitude peak that appears as a noisy bump on the noisy baseline. Figure 4B presents a close-up of the electrophoresis peak, which dissolves into a set of discrete, sharp spikes, generated by individual B-phycoerythrin molecules. Those spikes that exceed the threshold for single-molecule detection are marked with a cross. The electrophoretic peak generated by a large number of molecules, Figure 3, is quite different from the peak generated by a small number of molecules, Figure 4. Molecular shot noise affects the peak shape in two ways. First, the total number of molecules injected is governed by Poisson statistics; the relative precision in peak area should be equal to the square root of the number of detected molecules. Second, for inhomogeneous analyte, the distribution of analyte among the different fractions is governed by sampling statistics from the inhomogeneous solution. That is, in this very dilute solution, the analyte molecules are not uniformly distributed within the solution. Injection is a sampling procedure wherein a number of molecules of each component is drawn from the parent population. As the number Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
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Figure 5. Replicate injection of varying amounts of B-phycoerythrin. Data were recorded at 1 kHz and convoluted with a σ ) 12 ms Gaussian filter. Local maxima that exceeded 0.1 V are marked with a cross. Only the portion of the data that contains migrating peaks is plotted. (A and B) A 1.67 × 10-13 M solution was injected at 5 kV for 5 s, injecting 15 molecules. (C and D) A 1.67 × 10-12 M solution was injected at 1 kV for 5 s, injecting 30 molecules. (E and F) A 1.67 × 10-11 M solution was injected at 1 kV for 5 s, injecting 300 molecules. (G and H) A 1.67 × 10-10 M solution was injected at 1 kV for 5 s, injecting 3000 molecules. (I and J) A 1.67 × 10-9 M solution was injected at 1 kV for 5 s, injecting 30 000 molecules.
of molecules injected becomes small, the sample generated by injection is not necessarily representative of the bulk solution. Figure 5 presents a set of electropherograms generated by injection of samples that contain, on average, 15 (top), 30, 3 × 102, 3 × 103, and 3 × 104 (bottom) analyte molecules; molecular shot noise introduces significant uncertainty into the injected amount for the smallest sample injection. Here, the data were treated to highlight single-molecule events. The data were generated in duplicate; peaks that exceed the threshold for singlemolecule detection are denoted with a cross, and only the region near the electrophoretic peak is plotted. The largest injections 694 Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
generate the familiar five-component peak for B-phycoerythrin; a smoothed version of the high-concentration data is shown as Figure 3. Injection of 3000 molecules generates a much less distinct peak. Injection of 300 molecules leads to collapse of the peak structure. Finally, injection of 30 or 15 analyte molecules leads to random spikes located within the B-phycoerythrin migration envelope. As in Figure 1, the number of detected molecules is equal to roughly half the number of molecules injected. The effects of molecular sampling are seen in the degradation of the peak shape. Let X be the area of each of the four minor
Figure 6. Sixteen replicate injections of 300 molecules. A 1.67 × 10-11 M solution of B-phycoerythrin was injected at 1 kV for 5 s. The data were recorded at 30 Hz after passing through a 20-Hz analog filter. The data were then convoluted with a Gaussian filter with σ ) 6 ms. Local maxima that exceeded 0.1 V are marked with a cross.
components and 2X the area of the major component. The total area of a peak is 6X; the analyte molecules are distributed within these components. The uncertainty of the amount of analyte contained within each component is given by the square root of the number of molecules contained within the component. Taking into account the 50% detection efficiency, the actual number of molecules detected in Figure 5 ranges from 7.5 to 15 000. The uncertainty due to molecular shot noise is listed in Table 1. When analyzing a large number of molecules, the uncertainty in the fraction of each component due to molecular shot noise is only a few percent. However, as the number of detected molecules decreases, the uncertainty becomes very large, approaching 100% for the smallest injections considered in Figure 6. To judge the effect of single-molecule sampling statistics on peak shape in electrophoresis, 16 replicate injections were
performed that contained 300 analyte molecules, Figure 6. Again, taking into account our 50% detection efficiency, these data correspond to 150 detected molecules. A least-squares routine was used to fit a Gaussian peak to the data. Two regression results gave poor fits and were rejected from further analysis. The relative standard deviation of the peak area was 13%, that of the peak standard deviation was 14%, and that of the theoretical plate count was 15%. These values are a bit larger than the 8% uncertainty expected for analysis of 150 B-phycoerythrin molecules, reflecting the contribution of background noise to the uncertainty in the estimated parameters and the precision associated with electrophoretic injection. Finally, replicate injections were performed of a sample volume that would nominally contain 15 analyte molecules, Figure 7. A different capillary was used in this experiment, which resulted in Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
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Table 1. Uncertainty in Quantitative Analysis due to Molecular Shot Noise total no. of molecules
no. of molecules in major band
no. of molecules in minor band
uncertainty in amount detected (%)
uncertainty in major band (%)
uncertainty in minor bands (%)
15 000 1500 150 15 7.5
5000 500 50 5 2.5
2500 250 25 2.5 1.25
0.8 2.5 8 25 35
1.4 4 14 45 45
2 6 20 63 90
Figure 7. Six replicate injections of 15 molecules. A 1.67 × 10-12 M solution of B-phycoerythrin was injected at 500 V for 5 s. Data were collected at 1 kHz and convoluted with a σ ) 12 ms Gaussian filter, and local maxima exceeding 0.1 V were marked with a cross.
a slightly longer migration time. The single-molecule events are denoted with a cross, and there are an average of 12.0 ( 3.3 molecules detected per injection. As expected from Poisson statistics, the standard deviation in the number of molecules equals the square root of the number of detected molecules. The molecules are distributed across the B-phycoerythrin peak profile and do not fall into discernible subfractions. There is an insufficient number of molecules to define the peak profile. CONCLUSIONS Sampling analyte solutions at the single-molecule level represents the ultimate level of quantitative accuracy. The uncertainty in both the number of detected molecules and the peak shape is determined by counting statistics due to molecular shot noise. Analysis at the yoctomole level must always suffer from poor precision, simply due to the small number of molecules contained within the sample. This uncertainty is a fundamental limit of (41) Craig, D. B.; Wong, J. C. Y.; Dovichi, N. Anal. Chem. 1996, 68, 697-700.
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analysis and can be improved only by increasing the number of molecules taken for analysis. In the particular case demonstrated in this paper, molecular shot noise is a result of the very small amount of sample taken for analysis. As demonstrated in the following paper in this issue, large amounts of very dilute solutions are also susceptible to molecular shot noise.41 ACKNOWLEDGMENT This work was supported by the Natural Sciences and Engineering Research Council. N.J.D. acknowledges a McCalla Professorship from the University of Alberta.
Received for review July 5, 1995. Accepted November 22, 1995.X AC950651R X
Abstract published in Advance ACS Abstracts, January 1, 1996.
