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Anal. Chem. 1986, 58, 2567-2568 Hemingway. R. W.; Laks, P. E.; McGraw, 0. W.; Kreibich, R. E. I n Proceedings IUFRO Conference , Forest Products Research International-Achievements and the Future ; Pretoria, South Africa, 1985;VOI. 17,pp 1-20. Howes. F. N. Vwefables Tanning Materlals; Butterworths: London, 1953;pp 152-156. Stafford, H. A,; Lester, H. H.; Porter, L. J. Phytochemktry 1985, 24,
333-338. Hemingway, R. W.; Laks, P. E. J. Chem. SOC., Chem. Commun. W85, 746-747. Foo, L. Y.; Hemingway, R. W. J. Chem. Soc., Chem. Commun. 1984, 85-86. Hemingway, R. W.; Karchesy, J. J.; McGraw, G. W.; Wielsek, R. A. Phyfochemistry 1883, 22, 275-281. Heminqway, R. W.; Foo, L. Y. J. Chem. Soc., Chem. Commun. 1983, -1 035-1 036. Nonaka, G.; Morimoto, S.; Nishioka, T. J. Chem. Soc., Perkin Trans. 1 l.-a- m 2139-214s. -, -. Porter, L. J.; Newman, R. H.; Foo, L. Y.; Wong, H.; Hemingway, R. W. J. Chem. Soc., Perkin Trans. 11982, 1217-1221. Tandem Mass Spectrometry; McLafferty, F. W., Ed. Wiley-Interscience: New York, 1983. Mass Specfromefry in the HeaM and Life Sciences : Burlingame, A. L.; Castagnoli, N.. Jr.. Eds.; Elsevier: New York, 1985; Chapters 9, IO,
12-14. McLafferty. F W.; Bockhoff, F. M. Anal. Chem. 1978, 50, 69-76. Haddon, W. F. I n High Performance Mass Spectrometry; Gross, M. L.. Ed.: American Chemical Society: Washington, DC, 1978;pp 97-1 19. Karchesy, J. J.; Hemingway, R. W. J . Agric. Food Chem., in press. Hemingway, R. W.; Foo, L. Y.; Porter, L. J. J. Chem. SOC.,Perkin Trans. 7 1982, 1209-1216. de Koster, C. G.; Heerma, W.; Dijkstra, G.; Niemann, G. J. Biomed. Mass Spectrom. 1985, 72, 596-601. Thompson, R. 9.; Jacques, D.; Haslam, E.; Tanner, R. J. N. J. Chem. SOC.,Perkin Trans. 7 1972, 1387-1399.
(28) Jacques, D.; Haslam, E.; Bedford, G. R.; Greatbanks, D. J. Chem. SOC., Perkin Trans. 11974, 2663-2671.
Joseph J. Karchesy* Department of Forest Products Oregon State University Corvallis, Oregon 97331 Richard W. Hemingway Southern Forest Experiment Station Pineville, Louisiana 71360 Yeap L. Foo Chemistry Division, DSIR Petone, New Zealand Elisabeth Barofsky Douglas F. Barofsky Department of Agricultural Chemistry Oregon State University Corvallis, Oregon 97331
RECEIVED for review December 26,1985. Resubmitted June 9, 1986. Accepted June 9, 1986.. This is paper 2040, Forest Research Laboratory, Oregon State University, Corvallis, OR 97331. Support from USDA Grant 85-CRSR-2-2555 is gratefully acknowledged (J.J.K.).
Statistical Sampling Errors as Intrinsic Limits on Detection in Dilute Solutions Sir: Sensitive detection schemes such as laser-induced fluorescence have reached levels of performance where single atoms can now be detected in the gas phase ( l ) ,and after complexation with fluorescein-labeled antibodies, single polymer chains have been detected on the surface of a microscope slide (2). Failure to detect single molecules in a liquid phase has been ascribed to large background signals (3). Impressive reductions in background signals from Raman scattering (4) and window fluorescence (5-7) have now been achieved. Dovichi et al. (3) have predicted that single molecules could be detected in solution with reasonable improvements to laser flow cytometer detectors (3,8, 9). It is the purpose of this work to show that statistical sampling theory predicts poor precision for sampling microvolumes of very dilute solutions and presents an ultimate lower limit on detection. It will be shown that solvent molecules play a key role in setting this statistical limit, in contrast to gas and solid phase detection where a solvent is not required. Sampling theory for chemical analysis (10) and in chemometrics (11, 12) has been well-reviewed in the literature. Sampling theory (13, 14) can be applied to a simplified model of a homogenous solution. Consider a binary system consisting only of analyte and solvent. A binomial distribution predicts an analyte relative standard deviation for any binary system as where nAis the number of analyte particles in the sample, uA is its standard deviation (heterogeneity between samples), nT is the total number of particles in the sample, including solvent particles, and p is defined as P =n ~ / n ~
(2)
0003-2700/86/0358-2567$01.50/0
and (1 - p ) is then the corresponding fraction for solvent particles. At concentrations typical of analysis, the solvent particles in liquid solutions far outnumber the analyte particles, and thus p is very small. A t the same time, nT is very large due to dense packing in liquids, even in the smallest possible volume. Equation 1 can now be simplified by the above conditions to give
(3) Readers will recognize this result as the Poisson limit of a binomial distribution, valid when n T is greater than 100 and p is smaller than 0.01 (15). Equation 3 has important ramifications. The analysis of microvolumes of very dilute solutions is ultimately limited, not by instrument sensitivity but by statistical errors in sampling. For example, if 1% relative standard deviation is desired for sampling a homogeneous solution that is M in analyte, a sample volume of no less than 2 pL is required. The impact of statistical sampling errors in detecting a very dilute analyte in solution is now examined. We choose fluorescence as an example technology because it has good potential for single molecule detection in solution and is well suited to miniaturization. Much of the discussion that follows could be modified for other detection schemes. As smaller volumes of solution are probed, diffusion times through the sample observation (probe) volume will be reduced. At some point, a plot of signal vs. time will have an additional noise component due to the sampling uncertainty. This occurs because diffusion replaces one “sample” with another. This effect will also be observed in flowing streams. Dovichi et al. made a distinction between the volume “sampled” by the detector and the probe volume ( 3 ) . This convention will be used here. Thus, the probe volume is the 0 1986 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 12, OCTOBER 1986
smallest possible volume observed by the detector and is conveniently used as a unit sample volume. The number, n, of these unit sample volumes swept through the detector during a signal integration time T can be related to the flow rate qp through the probed area and the probe volume Vp by n = qpT/Vp (41 When diffusion is the only means of transport and assuming a spherical probe volume, the Einstein-Stokes equation can be used to predict the number of probe volumes sampled as
n = 6DT/r2 (5) where r is the radius of the probe volume, T i s defined above, and D is the diffusion coefficient of the slowest diffusing sample component. The total number of analytical particles nA sampled and the number probed, nA,p,are related by nA = nnA,P (6) Equation 3 can now be written Since the right-hand term is related to the number of analyte particles actually detected, the total volume sampled during flow and the resulting nA are the parameters that determine the sampling error. Combining eq 4 and 6 yields
uA/nA = (vP/qPT)1/2nA,P-1/2
(8)
This expression implies that decreases in probe volumes will improve precision, which is merely an artifact of the concentration and volumetric flow terms, so eq 7 is best written in a new form as ‘ T A / n A = U A , c / C A = (qpT)-1i2cA-1i2 (9)
noise (16). At only single molecule levels, strongly fluorescent impurities in a flowing blank will lead to a sampling noise component in the total blank noise. A distinction can be made between a background signal due to fluorescing solvent or a few molecules of a very strongly fluorescent impurity. The solvent cannot contribute any significant sampling noise to the background. This is made readily apparent by remembering that the probability of drawing a solvent molecule in the sample is very large. While the sampling statistical effect is not instrumental and can be considered an intrinsic limit on detection, it should not be confused with the statistical fluctuations in signal discussed by Alkemade (17). His work addressed changes in signal due to statistical fluctuations in observed events, such as photoelectron emission or charge carrier collection. Since both gas-phase detection and solid-surface detection have been successfully carried out a t the single species level, it is instructive to predict sampling errors for these systems. In gas-phase detection, can obviously be small. At the same time, the experiments are carried out at low pressures so that p is essentially unity. Equation 1predicts negligible sampling error. In solid-surface detection of an immunocomplex ( 2 ) , removal of solvent and unbound solutes will reduce nT to unity, but here the sampling error vanishes altogether as p must also be unity. Statistical sampling errors place restrictions on the precise sampling of microvolumes of dilute solutions. Even in the total absence of instrumental noise, these sampling errors impose an ultimate lower limit on detection. LITERATURE CITED
where cA is the analyte concentration in species per unit volume, and uABc is the corresponding absolute standard deviation. Alternatively an expression for u ~can , be ~ written. We choose to emphasize the relative standard deviation because it accounts for the relative changes in precision and signal. It is easily shown by rearrangement of eq 4,6, and 8 that u ~ , ~ scales as the square root of nA,p, qp, and T and the inverse root of V,. However, the mean value nAscales as the first power of these variables so that the relative standard deviation improves with increasing flow rates and integration times. Combining eq 5 and 6 yields, for the case of sample diffusion through a spherical probe volume UA/nA
= r/(6DT)1/2nA,p-1/2
(10)
At this point it becomes obvious that improvements in sampling precision by increasing the integration time are not due to ordinary signal averaging, since the n1j2gain in precision obtained in signal averaging can be obtained by n repetitive observations of the same sample while the nl/*improvement in sampling precision requires some form of transport to generate n unique samples. Thus more analyte is required than is contained in the probe volume. The sampling error may also increase the blank or background noise level beyond that predicted for shot and flicker
Pan, C. L.; Prodan, J. V.; Fairbank, W. M., Jr.; She, C. Y. Opt. Left. 1980, 5 , 459-461. Hirschfeld, T. Appl. Opt. 1978, 15,2965-2966. Dovichi, N. J.; Martin, J. C.; Jett, J. H.; Trkuia, M.; Keller, R. A. Anal. Chem. 1984, 56, 348-354. Lytle, F. E. J. Chem. Educ. 1982, 59, 915-920. Diebold, G. J.; Zare, R. N. Science 1977, 196, 1439-1441. Lyons, J. W.; Faulkner, L. R. Anal. Chem. 1982, 54, 1960-1964. Folestad, S . ; Johnson, L.; Fosefsson, B.; Galle, B. Anal. Chem. 1982, 54, 925-929. Herschberger, L. W.; Cailis, J. B.; Christian, G. D. Anal. Chem. 1979, 51. 1444-1446. Kelly, T. A.; Christian, G. D. Anal. Chem. 1981, 5 3 , 2110-2114. Kratochvil, B.; Wallace, D.; Taylor, J. K. Anal. Chem. 1984, 56, 113R-129R. Kowalski, B. R. Anal. Chem. 1880, 50, 112R-122R. Frank, I.E.; Kowalski, B. R. Anal. Chem. 1982, 52, 232R-243R. Benedetti-Pichler, A. A. I n Pbysical Methods in Chemical Analysis; Berl, W. M., Ed.; Academic Press: New York. 1956; Vol 3, p 183. Baule, B.;Benedetti-Pichler, A. A. Z . Anal. Chem. 1928, 7 4 , 442. Rosner, 8. A. fundamentals of Biostatistics ; Duxbury Press: Boston, MA, 1982; p 91. Alkernade, C. Th. J.; Snelleman, W.; Boutiiier, G. D.; Pollard, B. D.; Winefordner, J. D.; Chester, T. L.; Ornenetto, N. Spectrochim. Acta, Part B 1978, 3 3 6 , 383. Aikernade, C . Th. J. Appl. Spectrosc. 1981, 3 5 , 1
James M. Hungerford Gars D. Christian* Department of Chemistry/BG-10 University of Washington Seattle, Washington 98195
RECEIVED for review March 27,1986. Accepted May 30,1986.
