1726
Anal. Chem. 1984, 56, 1726-1729
ment. Further, it is likely that immobilized FBP would have less affinity for folate than soluble FBP due to conformational restrictions on the protein. Unfortunately, no literature data exist for the binding of radiolabeled folate with immobilized FBP. A homogeneous type of assay (33) was also tried with the folate-enzyme conjugate prepared, but even high concentrations of soluble FBP were unable to inhibit the activity of the enzyme conjugates. The relatively low molecular weight of the bovine milk FBP (30000-40000) (34-38) compared to those of the antibodies that are normally used in homogeneous enzyme immunoassays (typically 160 000) may explain this observation (e.g., difference in bulkiness of the specific binder). In summary, preliminary data on a unique enzyme-labeled sequential binding assay for folate have been presented. The assay utilizes an immobilized binding protein rather than antibodies that are usually employed in enzyme-labeled assays. The assay can detect as little as 1ng of folate per assay tube or even less when the ascending portion of the dose reponse curve is used (e.g., 0.05 ng). Thus, the assay may be useful for the determination of folate in biological fluids and certain foods. Indeed, work in this direction including complete characterization of the selectivity of the assay is now in progress. Further, because of the apparent cooperative nature of the assay as viewed in the double reciprocal binding plot and the dose response curves, basic studies employing immobilized FBP and enzyme-folate conjugates may be useful in further understanding the complex nature of the folateFBP interaction. Registry No. Folic acid, 59-30-3;folate hydroxysuccinimide, 90605-84-8.
LITERATURE CITED Scharp6, S. L.; Cooreman, W. M.; Blomme, W. J.; Laekeman, G. M. Clin. Chem. (Winston-Salem, N . C . ) 1978, 22. 733-738. Masseyeff, R. I n “Nuclear Medicine and Biology Advances, Proceedings of the World Congress, 3rd”; Raynaud, C., Ed.; Pergamon: Oxford, 1983; pp 18-21. Van Weemen, B. K.; Schuurs, A. H. W. M. I n “Immunoenzymatlc Techniques”; Feldman, G., Druet, P., Blgnon. J., Avrameas, S., Eds.; Elsevier: New York, 1976; pp 125-133. Meyerhoff, M. E.; Rechnitz, G. A. Anal. Biochem. 1979, 95,483-493. Meyerhoff, M. E.; Rechnltz, G. A. Methods Enzymol. 1980, 70, 439-454. Scott, J. M.; Weir, D. G. Clin. Haematol. 1978, 5 , 547-568. Blakley, R. L. I n “Folic Acid, Proc. Workshop”; NAS: Washington, DC, 1975; pp 3-24. Stokstad, E. L. R. I n “Folic Acid, Proc. Workshop”, NAS Washington, DC, 1975; pp 122-135. Deacon, R.; Lumb, M. J.; Perry, J. Med. Lab. Sci. 1982, 3 9 , 171-178. Llndenbaum, J. Blood 1963, 61, 624-627. Branda, R. F. J. Nutr. 1981, 111, 618-623. Waxman, S.; Schrelber, C. Mefhods Enzymoi. 1980. 66, 468-483.
Anderson, G. W.; Zimmerman, J. E.; Caiiahan, F. M. J. Am. Chem. Soc.1964,86, 1839-1842. Cheng, K. L. I n “Spectrochemlcal Methods of Analysis; Vol. 9”; Wine fordner, J. D., Ed.; Wllley-Interscience: New York, 1971; pp 321-385. Olive, C.; Levy, H. R. J. Biol. Chem. 1971, 246, 2043-2046. “Affinity Chromatography, Principles and Methods”; Pharmacia Fine Chemicals, 1979. Waxman, S. Br. J. Haemafol. 1975, 2 9 , 23-29. Zettner, A.; Duly, P. E. Clin. Chem. (Wlnsfon-Salem,N . C . ) 1974, 20, 5-14. Salter, D. N.; Scott, K. J.; Siade, H.; Andrews, P. Biochem. J. 1981, 193, 469-476. Hansen, S. I.; Holm, J.; Lyngbye, J.; Pedersen, T. G.; Svendsen, I. Arch. Biochem. Biophys. 1983, 226, 636-642. Rodbard, D.; Bertino, R. E. A&. Exp. Med. Biol. 1973, 36, 327-341. Nlederer, W. J. Immunol. Mefhods 1974, 5 , 77-82. Rodbard, D.; Rodgers, R. C. I n “Hormones In Human Blood: Detection and Assay”; Antonlades, H. N.. Ed., Harvard Unlverslty Press: Cambridge, MA, 1976; pp 92-114. Matsukura, S.; West, C. D.; Ichikawa, Y.; Jubiz, W.; Harada, G.; Tyler, F. H. J. Lab. Ciin. Med. 1971. 77, 490-500. Arimura, A.; Sato, H.; Kumusaka, R. B.; Worobec, L.; Dunn, J.; Schaliy, A. V. Endocrinology 1973, 93, 1092-1103. Weintraub, B. D.; Rosen, S. W.; McCummon, J. A,; Perlman, R. L. Endocrlnology 1973, 92, 1250-1255. Imura, H.; Nakai, Y.; Matsukura. S.; Hirata, Y. Horm. Metab. Res., Suppi. Ser. 1974, 5 , 7-12. Ratanasthien, K.; Blair, J. A.; Leeming, R. J.; Cooke, W. T.; Meiiklan, V. J. Clin. Pafhol. 1974, 27, 875-879. Theobald, R. A,; Batcheider, M.; Sturgeon, M. F. Ciin. Chem. (Winston-Salem, N . C . ) 1981. 27, 553-555. Givas, J. K.; Gutcho, S. Clln. Chem. ( Winston-Salem, N.C .) 1975, 21, 427-428. Klotz, I.M. Science 1982, 217, 1247-1249. Hansen, S. I.; Holm, J.; Lyngbye, J. Clin. Chem. (Winston-Salem, N . C . ) 1962, 28, 117-118. Schneider, R. S. I n “Ligand Assay”; Langan, J., Ciapp, J. J., Eds., Masson Publishing USA, Inc.: New York, 1981; pp 151-181. Ford, J. E.; Salter, D. N.; Scott, K. J. J. Dairy Res. 1989, 36, 435-446. Salter, D. N.; Ford, J. E.; Scott, K. J.; Andrews, P. FEBS Left. 1972, 2 0 , 302-306. Waxman, S.; Schreiber, C. F€BS Left. 1975, 55, 128-130. Pedersen, T. G.; Svendsen, I.; Hansen, S. I.; Holm, J.; Lyngbye, J. Carlsberg Res. Common. 1980, 45, 161-166. Iwai, K.; Tani, M.; Fushikl, T. Agrlc. Bioi. Chem. 1983, 47, 1523-1530.
‘Present address:
Stanford Medical School, Palo Alto, CA
Leonidas G . Bachas P a u l F. Lewis’ Mark E. MeyerhofP Department of Chemistry University of Michigan Ann Arbor, Michigan 48109 RECEIVED for review February 2, 1984. Accepted April 17, 1984. We gratefully acknowledge the National Science Foundation for supporting this work (Grant No. CHE8118817). Presented at the 1984 Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy in Atlantic City, NJ, March 8, 1984.
AIDS FOR ANALYTICAL CHEMISTS Minimization of Errors in Measurement of Chromatographic Retention Tlme Leonid M. Blumberg Hewlett-Packard Co., Route 41 and Starr Road, Avondale, Pennsylvania 19311 Errors of measurement of RT (retention time) of chromatographic peaks are one of the contributors of instability of R T ( I ) . The errors could become the major source of instability of RT in tracing small peaks significantly corrupted by background noise. A study of different aspects of evaluation txi well as reduction of the errors might be found in literature 0003-2700/84/0356-1726$01.50/0
(1-4). The primary attention in this paper is given to consideration of theoretical limitations for reduction of RT measurement errors of chromatographic peaks corrupted by noise. An obvious way to measure RT of a positive chromatographic peak is to take the derivative of the peak and identify 0 1984 Amerlcan Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 9, AUGUST 1984
1727
\
T
Run time
RT
Block diagram of a basic RT measurement system: (D) differentiator, (C) ZC comparator, (T) time reading unit. Shown in circles are stages of a peak transformation. Figure 1.
downward direction. The mechanism is shown on the block diagram in Figure 1. The ZC (zero crossing) comparator there is a unit with two-state output: a positive input causes “high” output and a negative input causes “low” output. The time reading unit has two inputs: a control input, connected to comparator’s output, and time input, connected to the time base of the system. It reads and outputs RT from time input a t an instant when the state of control input changes from “high” to “low”. An RT measurement system as described above is far from being optimum precision-wise if a mixture of peak + noise applied to its input. The system’s performance might be improved (significantly in many cases) if an input filter (prefilter) is placed in front of differentiator, Figure 2a. It is convenient to approach to a cascade of prefilter and differentiator (dashed block on Figure 2a) as a single fiiter, named here a ZC filter. A ZC filter might be viewed as a generalization of a differentiator. A block diagram of the complete RT measurement system is shown on Figure 2b. In addition to the signal and time inputs, the RT measurement system has a third one (not shown on Figure l and Figure 2): a control input to activate and deactivate the system. The RT measurement system is activated (by peak detection system) only when a peak is rendered to its signal input. This prevents false RT outputs caused by base line noise etc. A study was undertaken to find a ZC filter which minimizes SD (standard deviation) of errors of measurement of RT for a given chromatographic peak and noise. The scope of the study was limited by the following assumptions: (1)noise is a Gaussian stationary random process (5)uncorrelated with the signal, (2) A ZC filter is a linear, time invariant one, (3) false RT outputs are prevented by an appropriate deactivation of R T measurement system. Displacements of true zero crossings by noise in a signal are the only source of RT measurement errors. No direct solution for the problem was found even within the limited scope. Therefore a detour was taken: a ZC criteria, p, was introduced and an optimum ZC filter which
RTierror t
t
RT a t an instant when the derivative crosses zero level in a
RT+error
Figure 2. Block diagrams of an improved RT measurement system: (P) prefilter, (F) ZC filter. Blocks P and D of (a) are replaced by F on (b). Shown in circles are stages of transformation of peak noise
+
mixture.
maximizes that criteria was found. A ZC criteria is a ratio of a slope of output of ZC filter at an instant of ZC to RMS of noise in that output. (A more accurate definition of ZC criteria is given in the theoretical section below.) To return from the detour to the original problem, it is shown that SD, u, of RT measurement errors might be approximated by a simple asymptoticallyaccurate formula u 1 / p . That means that, although the problem of minimization of RT measurement errors remains opened (in rigorous theoretical terms), we know that utilization of optimum ZC filter reduces those errors almost all the way down to the lowest possible level.
THEORY The following terminology is adopted throughout the paper. ZC image of a signal is output of a ZC filter when the signal is applied to its input. ZC noise is a noise component of a ZC image of any peak + noise mixture. Thus, on Figure 2, y ( t ) is ZC image of x ( t ) ;e ( t ) is ZC noise. Also, to simplify notations, we assumed that a chromatographic peak has RT = 0. Finally, we want to make it certain that spectral density (5) of noise is normalized such that
where u,, is the RMS of noise whose spectral density is W(f). RT Measurement Errors. To find statistical characteristics of RT measurement errors, let us reexamine a ZC image in time interval (-to, to), Figure 3. When a noiseless peeak, x , is applied to a ZC filter, a ZC image, y , of x (an output of the filter) crosses the zero level at an instant t = 0, Figure 3a. When noise, n, is added to a peak, x, many different ZC images y + e become possible for a given x , causing those images to lie within some stripe around y , Figure 3b. That, in turn, causes instants t of ZC to have a certain probability density,
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 9, AUGUST 1984
than using entire y ( t ) as in the case of eq 2 and 3. Let again x ( t ) + n(t)be a mixture of a chromatographic peak x ( t ) , and noise, n(t),applied to RT measurement system of Figure 2b and y ( t ) and e ( t ) be ZC images of x ( t ) and n(t) correspondingly. We define ZC criteria as c1
=
lu’(O)I/a,
(4)
where y’(t) = dy/dt, u, RMS of ZC noise e ( t ) . It seems to be intuitively obvious that ZC criteria must be inversely related to SD, u, of RT measurement errors. It has been shown in supplement 1 (See paragraph at end of paper regarding supplementary material ordering information.) that, indeed u
c
f
(5)
Inaccuracy of this equality falls below any practical importance when u is a small fraction (10% of less) of peak width. Optimum ZC Filter. A value, p , of ZC criteria depends on the ZC filter. Indeed, two filters might provide different noise level (specified by u, in eq 4) while giving the same slope, ly’(O)l, of ZC image for the same uncorrupted signal. It is shown in supplement 2 that given a peak and noise environment, there is a class of ZC filters which brings ZC criteria to its highest possible level. A filter from that class will be referred to as an optimum ZCfilter. A transfer function, Go@, of an optimum ZC filter, also found in Supplement 2, is
where real number G is a gain of the filter and X @ and IP@ are the peak’s spectrum and spectral density of input noise correspondingly; % * * means complex conjugation. In many cases of design of RT measurement systems it is desirable la deal with impulse response of ZC filter rather with its transfer function. A direct formula for calculation of impulse response of an optimum ZC filter could be easily obtained from eq 6. Nevetheless, we found it more convenient to present it indirectly via a well-known concept of a matched filter (6). It is shown in supplement 3 that impulse response, go(t) of an optimum ZC filter is go(t) = g’,(t)
Flgure 3. Uncertainty of a ZC: (a) noiseless environment, (b) noisy
environment, (c) linearized ZC image in noisy environment.
l/p
(7)
where
p,(t), around t = 0. Those t values are RT measurement errors. Let probability density of ZC noise be p,(e) then
Indeed, a ZC event, t, takes place when y ( t ) + e ( t ) = 0 , i.e., e ( t ) = -y(t). Substituting that into p,(e) and normalizing the result such that J+?p,(-y(t)) dt = 1one has eq 2. SD, u, and expectation, E, of RT measurement errors now could be found. They are
We look a t u of eq 3 as a criteria for minimization of R T measurement errors. For the criteria to be productive it must be simple in definition and calculation. This is not the case with the u of eq 3, since taking several integrals is involved on different levels of its calculation when input peak, input noise, and ZC filter are given. Zero-Crossing Criteria. Here we introduce more convenient criteria for minimization of RT measurement errors. The basic concept behind ZC criteria rests on dealing with the slope of ZC image, y ( t ) ,at an instant of ZC event rather
is an impulse response of a matched filter for that signal and noise environment. Standard Deviation of RT Measurement Errors. Based on eq 5, a SD, u, of RT measurement errors is found in supplement 4 for a given spectrum, X ( f ) ,of a peak, spectral density, IPO, of input noise, and transfer functions, G O , of a ZC filter. It is
(9)
Also found in supplement 4 is the minimum, uo,of u which is achieved when the ZC filter is an optimum one. In that case
ANALYTICAL CHEMISTRY, VOL. 56, NO. 9, AUGUST 1984
I
Flgure 4.
I
I N I T I A L SETPOINTS %OFFSET 10 0.500 RANGE T I M E SCALE 0.50 C 0.06 PEAK WIDTH 0.6 14 600 THRESHOLD
1729
1 DV
cdmin cdpcat)
mVi n a
Impulse response of a rectangular ZC filter.
EXPERIMENTAL SECTION Theoretical results of the previous section were tested in many ways. They were also used in practical design of integrators. One of the tests is described below. A train of 50 equally spaced Gaussian peaks corrupted by white noise was generated. AU peaks had the same heights and widths. The following are some parameters of the peak train: height of each peak, H = 500 rV; effective width of each peak, W = 5.12 s (0.08533 min); retention time of the first peak, T , = 0.34133 min; space between apexes of neighboring peaks, AT = 2T1 = 0.68166 min; spectral density . a parameter, effective of the noise, N = 28.3 ~ V / H Z ' / ~Here, width of a peak, W, is defined as W = A / H , where A is peak area and H is its height. Effective width of a Gaussian peak might be measured as its width at 45.6% of its height (exp(7/4) fraction of its height to be exact). To measure retention of peaks, a rectangular ZC filter, Figure 4,was used, The width of filter's shoulder, T , was the same as the peaks effective width, W. An annotated output of the RT measurement system is shown in Figure 5 (output channel had 0.8 Hz low pass filter). All measured RTs were automatically compared to their ideal positions, and the SD of the errors was min. estimated. The result was gest = 1.06 X The theoretical value for this SD is found in supplement 5 based min. A difference between those on (9). It is gtheor = 1.11X two values is well within an error of estimation based on 50 random samples. RESULTS AND DISCUSSION A RT (retention time) measurement is approached as filtering of a peak and capturing an instant when the filter's output crosses zero level. The filter was named a ZC (zero crossing) filter and a simply defined ZC criteria was introduced, eq 4. It was shown then that the ZC criteria is asymptotically inversely related to the SD (standard deviation) of R T measurement errors, eq 5. That relationship and a simple definition of ZC criteria make it a convenient criteria of minimization of R T measurement errors. A ZC filter which maximizes ZC criteria and, therefore, minimizes the SD of the RT measurement errors was defined as an optimum ZC filter. Its transfer function, and impulse response, as well as the minimum of R T measurement errors are found, eq 6, 7 , and 10. A simple formula (9) for calculation of the SD of the R T measurement errors for an arbitrary ZC filter is also found. Results of that formula were compared with experimental results and found to be well within a margin of precision of statistical estimation of the SD of experimental errors. It is important to emphasize a 2-fold usefulness of results of a concept of an optimum ZC filter. The first one is the very fact that the filter was found. The second one rests on the fact that since the theoretical minimum of R T measurement errors in known, it becomes possible to evaluate any ZC filter vs. an optimum one and make a judgement about a quality of a filter. As usually happens, an optimum ZC filter might be a complicated one and a replacement of it by a simple but slightly worse one might be a useful approach in a practice of R T measurement system design. Thus, if an optimum ZC filter were used to measure RTs in a peak train of the Experimental Section, SD of errors would have been (supplement 5) 60= 1.01 X min, which is only about 10% better than
END OF 06TA
Figure 5.
minutes.
An experimental peak train. Labels indicate peak RTs in
a theoretically expected performance of a simple rectangular filter actually evaluated in the experiment.
Supplementary Material Available: Derivation of eq 5-10 as well as calculation of theoretically expected values of errors in the Experimental Section (9 pages) will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper or microfiche (105 X 148 mm, 24X reduction, negatives) may be obtained from Microforms Office, American Chemical Society, 1155 16th Street, N.W., Washington, D.C. 20036. Orders must state whether for photocopy or microfiche and give complete title of article, names of authors, journal issue date, and page numbers, Prepayment, check or money order for $15.00 for photocopy ($17.00 foreign) or $6.00 for microfiche ($7.00 foreign), is required and prices are subject to change. LITERATURE CITED (1) Riiks, J. A. "Characterization of Hydrocarbons by Gas Chromatography: Means of Improving Accuracy": Drukkerij J. H. Pasmans: 's-Gravenhage, The Netheriands, 1973. (2) Goedert, M.;Guiochon, G. Anal. Chem. 1970, 42, 962. (3) Kaiser, R. Chromatographia 1970, 3, 127. (4) Peetre, I. B. Chromafograph/a 1973, 6 , 257. (5) Korn, G. A.; Korn, T. M. "Mathematical Handbook for Scientists and Englneers"; McGra'w-Hill: New York, 1968. (6) Schwartz, L. S. Principles of Coding, Filtering, and Information Theory"; Spartan Books: Baltimore, MD (Cleaver-hume Press: London), 1963.
RECEIVED for review December 20, 1982. Resubmitted February 10, 1984. Accepted April 2, 1984.
