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On the sampling variance of ultra-dilute solutions
Talanta 52 (2000) 711 – 715 www.elsevier.com/locate/talanta
On the sampling variance of ultra-dilute solutions Constantinos E. Efstathiou * Laboratory of Analytical Chemistry, Department of Chemistry, Uni6ersity of Athens, Panepistimiopolis, Athens 157 71, Greece Received 13 January 2000; received in revised form 6 April 2000; accepted 27 April 2000
Abstract Contemporary analytical methodology allows the determination of substances in ultra-dilute solutions, i.e. picomol per liter (pM) and lower, by employing a wide spectrum of techniques. The reduction of both sample volume and quantification limits made possible the determination of zeptomol amounts of analytes. Among the many problems associated with the analysis of this kind of sample is the relatively high intrinsic sampling variance which restricts the attainable overall analytical precision. This sampling variance is due to the quantized nature of matter. The expected sampling variance can be calculated considering that the limited number of analyte molecules per test-portion is subjected to Poisson distribution. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Sampling variance; Ultra-dilute solutions; Poisson distribution
1. Introduction The use of the submultiple unit prefixes zepto (z) (from the greek word ‘optd´’ for seven) to designate 10 − 21 ( =1000 − 7), and yocto (y) (from the greek word ‘oktw ´ ’ for eight) to designate 10 − 24 (= 1000 − 8), along with their multiple unit prefixes counterparts zetta (Z) for 1021 and yotta (Y) for 1024, was presented and accepted by the 19th General Conference on Weights and Measures in October 1990 and finally adopted by the International Union of Pure and Applied Chemistry (IUPAC) in 1991. The introduction of these submultiple prefixes in scientific literature can be considered as ‘a tribute to those scientists who * Tel.: +30-1-7274312; fax: + 30-1-7231608. E-mail address: [email protected] (C.E. Efstathiou).
worked hard to lower the achievable limits of detection by modern analytical methodology’ [1]. Lower submultiple prefixes to describe amounts of matter will not be necessary — at least in analytical chemistry — due to the limitations imposed by the Avogadro number, NA = 6.0221367(36)×1023 mol − 1, since the absolute minimum amount of any compound is one molecule corresponding to 1/(NA mol − 1)= 1.66× 10 − 24 mol or 1.66 ymol. Such a limit does not exist in the case of concentrations, e.g. the concentration of a 5.0 l solution containing one molecule of a solute is (1/NA mol − 1)/(5.0 l) = 3.32× 10 − 25 mol l − 1 or 0.332 yM. Ultra-small amounts of analytes can be detected and quantified as well, because of the reduction of sample volume and the development of highly sensitive analytical techniques. Commer-
0039-9140/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 9 1 4 0 ( 0 0 ) 0 0 4 2 9 - X
712
C.E. Efstathiou / Talanta 52 (2000) 711–715
cially available GC, HPLC and capillary electrophoresis systems can now routinely handle submicroliter sample volumes, whereas by means of sophisticated instrumentation even smaller samples can be handled. From historical perspective, Alkemade introduced the concept of single atom detection (SAD) as early as 1981 [2]. Winefordner et al., working in the field of laser induced fluorimetry (LIF) and ionisation spectrometry, proved that the detection of single atom (or molecule) [3,4] is an achievable goal and they have discussed thoroughly the associated issue of estimation of the detection limits in ultra-trace analysis [5–7]. Typical examples of ultrasensitive determinations include the determination of proteins in a single 86 fl erythrocyte [8] and of Rhodamine 6G in a 10 mm diameter (= 0.52 fl) levitated glycerol microdroplet where a single molecule of the dye can be detected [9]. Although measurements of zmol amounts of analytes became feasible, the direct measurement of analytes in ultra-dilute solutions (i.e. pM or less) is still a challenge, since the lower limits of quantification of conventional instrumental techniques are only seldom lower than the concentration range 10 − 8 – 10 − 10 M. Well known are also the problems associated with the stability, storage and handling of extremely dilute samples. At the present time, measurements of ultra-dilute samples of practical importance concern mainly samples of biological fluids. The determination of sub-pM concentrations of a variety of proteins, enzymes and of DNA and RNA fragments, as well as of more complex and organized entities such as antibodies and viruses, is of utmost clinical and diagnostic significance. These analytes can be indirectly determined by employing enzyme amplification techniques, where each analyte molecule is labelled with an enzyme which, in the presence of excess of substrate, releases millions of product molecules at measurable concentrations within a few minutes. Direct analyte amplification can only be utilized in the unique case of DNA or RNA fragments by taking advantage of the polymerase chain reaction (PCR). PCR primarily intended as a sample preparation technique has been proposed recently for quantitative determinations of extraordinary
analytical sensitivity allowing the determination in samples containing a few thousand of the target biomolecules [10].
2. Sample homogeneity and sampling variance Chemistry textbooks describe solutions as homogeneous phases, i.e. any volume segment of a solution of any substance Y has the same concentration. However, this description becomes less realistic as the volume segment becomes smaller, and/or the concentration of Y decreases. This inevitable consequence of the quantized nature of matter introduces an intrinsic uncertainty in analytical determinations utilizing extremely small volumes and/or ultra-dilute solutions as samples. High sample homogeneity is a prerequisite for a meaningful quantitative analytical result. For an analytical procedure free of systematic errors the overall variance of the analytical result, s 2o, taking into account the rule of additivity of variance, is given by Eq. (1): s 2o = s 2a + s 2s 2 a
(1)
where s is the analytical variance, due to random errors occurring in all steps involved during the analysis (e.g. sample weighting and treatment, instrument electrical noise, calibrating procedure), and s 2s is the sampling variance, a significant source of uncertainty when one works with poorly homogenized samples [11]. Analytical and sampling variances are totally unrelated and they are only combined through Eq. (1) to yield the overall analytical variance for a given size of a particular sample. Sampling variance is an intrinsic property of the final sample and a measure of its homogeneity. Perfectly homogeneous samples (s 2s =0) are expected to yield an overall precision reflecting exclusively the analytical variance, whereas repeated determinations on samples of equal size using an ideally precise analytical procedure (i.e. s 2a = 0) would result in distributions reflecting exclusively the sample homogeneity. If either s 2a or s 2s is sufficiently larger than the other, there is little point trying to reduce the smaller one [11]. When one works with samples of low homogeneity the following rule of thumb
C.E. Efstathiou / Talanta 52 (2000) 711–715
applies: ‘if the analytical variance is equal to or less than about one-third of the sampling variance, any attempt to reduce analytical variance by using more expensive and/or sophisticated analytical techniques is meaningless’ [12,13]. In this case the only practical way to improve the overall precision of the analytical result is by employing a fast, handy and inexpensive analytical technique of limited precision and perform as many replicate measurements as possible [13]. Sample homogeneity depends on the size of the sample test-portion (or aliquot), normally defined as the amount of sample obtained from a bulk sample and completely consumed for a single analytical determination which may include steps like sample dissolution, obtaining of further aliquots etc. In the present case as sample test-portion is defined as the amount of solution finally introduced into a device generating an analytical signal by any integrating type mechanism, i.e. the analytical signal is a function of the number of analyte entitities present in the test-portion. In many cases, the analytical method itself does not allow the use of large test-portions despite the availability of large amounts of bulk samples. Typical examples include the sample volume introduced in the graphite tube during a determination with flameless atomic absorption spectroscopy, the sample injected into a GC or HPLC chromatograph, and the sample introduced in the wells of a luminometer plate for performing a highly sensitive immunochemical assay. Generally, for partially homogeneous samples the following equation, known as Ingamells’ equation, applies: W[sr(%)]2 =Ks
(2)
where W is the weight of the test-portion, sr(%) is the sampling relative (%) standard deviation (sr(%)= 100 ss/y¯, where y is the average analytical content) and Ks is the sampling constant. Ks depends on the nature of the sampled material and numerically corresponds to the weight (in g) of a test-portion with a sampling relative (%) standard deviation equal to 1% [14 – 16].
713
3. Ultra-dilute solutions as samples Let X be a discrete random variable which can take values 0, 1, 2, …, such that the probability function of X is given by Eq. (3): f(x)=P(X=x)=
m xe − m x!
(x= 0, 1, 2, …) (3)
where m is a given positive constant. The distribution described by Eq. (3) is known as Poisson distribution. Characteristic property of a variable X following Poisson distribution is that its mean value and its variance are both equal to m [17]. This distribution describes the probability of random occurrence of discrete events (x) in given segment of time or space at certain average rate (m). An ultra-dilute solution containing 3200 molecules of analyte Y per liter has a bulk concentration of Y (3200 l − 1)/(NA mol − 1)= 5.31× 10 − 21 mol l − 1 or 5.31 zM. On average, a 1.00-ml test-portion of this solution contains 3.20 molecules, but individual test-portions ultimately contain an integer number of molecules. Here, the molecules of Y are the discrete events and the volume of the test-portion is the segment of space. The probabilities of obtaining a 1.00-ml test-portion containing 0, 1, 2, … molecules of Y have been calculated using Eq. (3) and the corresponding discrete distribution diagram is shown in Fig. 1. An individual 1.00-ml test-portion of this solution would never have the expected bulk concentration of 5.31 zM, but concentration multiples of 1.66 zM ( = 1 molecule ml − 1). The closer multiple to the actual concentration of the bulk sample is 4.98 zM (three molecules per 1-ml test-portion) and this has only a 22.3% chance to occur, whereas there is also a 4.1% chance to obtain a test-portion containing no analyte molecules. This example describes a rather extreme case, as this distribution would eventually become relatively more narrow and symmetric approaching a Gaussian shape as the average number of molecules per test-portion increases. Since the mean and variance of populations exhibiting a Poisson distribution are arithmeti-
714
C.E. Efstathiou / Talanta 52 (2000) 711–715
cally the same, the sampling variance of a given size (i.e. volume) test-portion containing on average m molecules of analyte Y is: (4) s 2s =m= Vs CY NA where Vs is the volume of sample (in l), CY is the concentration of the analyte Y (in mol l − 1) and NA is the Avogadro number. From Eq. (4) the relative (%) standard deviation, sr(%), exclusively attributed to the non-homogeneity of a solution can also be calculated:
Fig. 1. Poisson distribution diagram showing the probabilities of occurrence of 0, 1, 2, … molecules in a 1-ml test-portion of a solution containing 3200 molecules l − 1 (= 5.31 zM).
Fig. 2. Sampling relative (%) standard deviation as a function of bulk-analyte concentration for various test-portion volumes.
sr(%)= (ss/m)×
100
Vs CY NA
(5)
For example, the sampling uncertainty of a 1.0-ml test-portion of an 1.0 fM solution of analyte is sr(%)= 100× [(1.0× 10 − 6 l)(1.0× 10 − 15 mol l − 1)(6.0221× 1023 mol − 1)] − 1/2 = 4.1%. Therefore, regardless of what efforts have been made to improve the overall analytical precision, measurements at 1 fM level with any technique using a test-portion of 1.0 ml, or generally a test-portion containing (1× 10 − 6 l)(1× 10 − 15 mol l − 1)= 1× 10 − 21 mol or 1 zmol amount of analyte, would never yield results with a relative (%) standard deviation less than 4%. This lower limit of uncertainty can only be reduced by increasing the size of the test-portion and/or by using the average of more replicates. By using a 1.00 ml test-portion, at the same concentration level, the expected sampling uncertainty would have been only 0.13%, which is generally considered as negligible especially for an analytical technique providing results at so low concentration levels. Unfortunately, increasing the size of the test-portion is not always feasible, particularly when one works with enzyme amplification and PCR techniques, because of the nature of these techniques, the available instrumentation and sometimes because of sample availability problems. It should be stressed that the assumption of Poisson distribution is valid as far as analyte entities behave as freely and randomly moving particles without any interaction between each other and the walls of the container as well. Clustering effects are not considered here. Obviously, any type of aggregation phenomena between the analyte entities would cause worse spreading of the analytical results. The expected relative (%) standard deviation for various sizes of test-portions (0.1 ml–10.0 ml) of ultra-dilute solutions (CY 5 10 − 12 M) can be quickly estimated from the log–log diagram shown in Fig. 2. The sampling constants, as defined by Ingamells’ equation (Eq. (2)) and now expressed in terms of volume, i.e. the volume of test-portion of an ultra-dilute solution, bearing a sampling uncer-
C.E. Efstathiou / Talanta 52 (2000) 711–715
tainty of 1% can be calculated from Eq. (5). Thus, the sampling constants for 1 fM, 1 aM and 1 zM sample solutions are 16.6 ml, 16.6 ml and 16.6 l, respectively. Actually, these are the volumes containing 10 000 analyte entities, since: sr(%)= (sY/y¯ ) × 100 =( m/m) × 100 = ( 10 000/10 000) × 100 = 1.00
4. Conclusions A high sampling variance affecting the overall analytical precision is expected when one has to work with very small volume test-portions of ultra-dilute solutions, or generally with test-portions containing a small number of analyte molecules. Running more replicates would be the only way to obtain more precise results. The traditional concept of homogeneity of solutions cannot be applied on small volumes of ultra-dilute solutions and any effort to push the limits of analytical determinations towards lower concentrations and smaller sample volumes is an intrinsically self-restricted goal.
.
715
References [1] M.A. Cousino, T.B. Jarbawi, H.B. Halsall, W.R. Heineman, Anal. Chem. 69 (1997) 544A. [2] C.T.J. Alkemade, Appl. Spectrosc. 35 (1) (1981) 1. [3] B.W. Smith, J.B. Womack, N. Omenetto, J.D. Winefordner, Appl. Spectrosc. 43 (5) (1989) 873. [4] J.D. Winefordner, B.W. Smith, N. Omenetto, Spectrochim. Acta Part B 44B (12) (1989) 1397. [5] C.L. Stevenson, J.D. Winefordner, Appl. Spectrosc. 45 (8) (1991) 1217. [6] C.L. Stevenson, J.D. Winefordner, Appl. Spectrosc. 46 (3) (1992) 407. [7] C.L. Stevenson, J.D. Winefordner, Appl. Spectrosc. 46 (5) (1992) 715. [8] T.T. Lee, E. Yeung, Anal. Chem. 64 (1992) 3045. [9] M.D. Barnes, K.C. Ng, W.B. Whitten, J.M. Ramsey, Anal. Chem. 65 (1993) 2360. [10] S. Bortolin, T.K. Christopoulos, M. Voerhagen, Anal. Chem. 68 (1996) 834. [11] D. Harris, Quantitative Chemical Analysis, fourth ed., W.H. Freeman, New York, 1996, p. 758. [12] W.J. Youden, J. Assoc. Off. Anal. Chem. 50 (1967) 1007. [13] B. Kratochvil, J.K. Taylor, Anal. Chem. 53 (1981) 924A. [14] C.O. Ingamells, P. Switzer, Talanta 20 (1973) 547. [15] C.O. Ingamells, Talanta 21 (1974) 141. [16] C.O. Ingamells, Talanta 23 (1976) 263. [17] J.R. Green, D. Margerison, Statistical Treatment of Experimental Data, Elsevier, Amsterdam, 1978, p. 35.
On the sampling variance of ultra-dilute solutions Constantinos E. Efstathiou * Laboratory of Analytical Chemistry, Department of Chemistry, Uni6ersity of Athens, Panepistimiopolis, Athens 157 71, Greece Received 13 January 2000; received in revised form 6 April 2000; accepted 27 April 2000
Abstract Contemporary analytical methodology allows the determination of substances in ultra-dilute solutions, i.e. picomol per liter (pM) and lower, by employing a wide spectrum of techniques. The reduction of both sample volume and quantification limits made possible the determination of zeptomol amounts of analytes. Among the many problems associated with the analysis of this kind of sample is the relatively high intrinsic sampling variance which restricts the attainable overall analytical precision. This sampling variance is due to the quantized nature of matter. The expected sampling variance can be calculated considering that the limited number of analyte molecules per test-portion is subjected to Poisson distribution. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Sampling variance; Ultra-dilute solutions; Poisson distribution
1. Introduction The use of the submultiple unit prefixes zepto (z) (from the greek word ‘optd´’ for seven) to designate 10 − 21 ( =1000 − 7), and yocto (y) (from the greek word ‘oktw ´ ’ for eight) to designate 10 − 24 (= 1000 − 8), along with their multiple unit prefixes counterparts zetta (Z) for 1021 and yotta (Y) for 1024, was presented and accepted by the 19th General Conference on Weights and Measures in October 1990 and finally adopted by the International Union of Pure and Applied Chemistry (IUPAC) in 1991. The introduction of these submultiple prefixes in scientific literature can be considered as ‘a tribute to those scientists who * Tel.: +30-1-7274312; fax: + 30-1-7231608. E-mail address: [email protected] (C.E. Efstathiou).
worked hard to lower the achievable limits of detection by modern analytical methodology’ [1]. Lower submultiple prefixes to describe amounts of matter will not be necessary — at least in analytical chemistry — due to the limitations imposed by the Avogadro number, NA = 6.0221367(36)×1023 mol − 1, since the absolute minimum amount of any compound is one molecule corresponding to 1/(NA mol − 1)= 1.66× 10 − 24 mol or 1.66 ymol. Such a limit does not exist in the case of concentrations, e.g. the concentration of a 5.0 l solution containing one molecule of a solute is (1/NA mol − 1)/(5.0 l) = 3.32× 10 − 25 mol l − 1 or 0.332 yM. Ultra-small amounts of analytes can be detected and quantified as well, because of the reduction of sample volume and the development of highly sensitive analytical techniques. Commer-
0039-9140/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 9 1 4 0 ( 0 0 ) 0 0 4 2 9 - X
712
C.E. Efstathiou / Talanta 52 (2000) 711–715
cially available GC, HPLC and capillary electrophoresis systems can now routinely handle submicroliter sample volumes, whereas by means of sophisticated instrumentation even smaller samples can be handled. From historical perspective, Alkemade introduced the concept of single atom detection (SAD) as early as 1981 [2]. Winefordner et al., working in the field of laser induced fluorimetry (LIF) and ionisation spectrometry, proved that the detection of single atom (or molecule) [3,4] is an achievable goal and they have discussed thoroughly the associated issue of estimation of the detection limits in ultra-trace analysis [5–7]. Typical examples of ultrasensitive determinations include the determination of proteins in a single 86 fl erythrocyte [8] and of Rhodamine 6G in a 10 mm diameter (= 0.52 fl) levitated glycerol microdroplet where a single molecule of the dye can be detected [9]. Although measurements of zmol amounts of analytes became feasible, the direct measurement of analytes in ultra-dilute solutions (i.e. pM or less) is still a challenge, since the lower limits of quantification of conventional instrumental techniques are only seldom lower than the concentration range 10 − 8 – 10 − 10 M. Well known are also the problems associated with the stability, storage and handling of extremely dilute samples. At the present time, measurements of ultra-dilute samples of practical importance concern mainly samples of biological fluids. The determination of sub-pM concentrations of a variety of proteins, enzymes and of DNA and RNA fragments, as well as of more complex and organized entities such as antibodies and viruses, is of utmost clinical and diagnostic significance. These analytes can be indirectly determined by employing enzyme amplification techniques, where each analyte molecule is labelled with an enzyme which, in the presence of excess of substrate, releases millions of product molecules at measurable concentrations within a few minutes. Direct analyte amplification can only be utilized in the unique case of DNA or RNA fragments by taking advantage of the polymerase chain reaction (PCR). PCR primarily intended as a sample preparation technique has been proposed recently for quantitative determinations of extraordinary
analytical sensitivity allowing the determination in samples containing a few thousand of the target biomolecules [10].
2. Sample homogeneity and sampling variance Chemistry textbooks describe solutions as homogeneous phases, i.e. any volume segment of a solution of any substance Y has the same concentration. However, this description becomes less realistic as the volume segment becomes smaller, and/or the concentration of Y decreases. This inevitable consequence of the quantized nature of matter introduces an intrinsic uncertainty in analytical determinations utilizing extremely small volumes and/or ultra-dilute solutions as samples. High sample homogeneity is a prerequisite for a meaningful quantitative analytical result. For an analytical procedure free of systematic errors the overall variance of the analytical result, s 2o, taking into account the rule of additivity of variance, is given by Eq. (1): s 2o = s 2a + s 2s 2 a
(1)
where s is the analytical variance, due to random errors occurring in all steps involved during the analysis (e.g. sample weighting and treatment, instrument electrical noise, calibrating procedure), and s 2s is the sampling variance, a significant source of uncertainty when one works with poorly homogenized samples [11]. Analytical and sampling variances are totally unrelated and they are only combined through Eq. (1) to yield the overall analytical variance for a given size of a particular sample. Sampling variance is an intrinsic property of the final sample and a measure of its homogeneity. Perfectly homogeneous samples (s 2s =0) are expected to yield an overall precision reflecting exclusively the analytical variance, whereas repeated determinations on samples of equal size using an ideally precise analytical procedure (i.e. s 2a = 0) would result in distributions reflecting exclusively the sample homogeneity. If either s 2a or s 2s is sufficiently larger than the other, there is little point trying to reduce the smaller one [11]. When one works with samples of low homogeneity the following rule of thumb
C.E. Efstathiou / Talanta 52 (2000) 711–715
applies: ‘if the analytical variance is equal to or less than about one-third of the sampling variance, any attempt to reduce analytical variance by using more expensive and/or sophisticated analytical techniques is meaningless’ [12,13]. In this case the only practical way to improve the overall precision of the analytical result is by employing a fast, handy and inexpensive analytical technique of limited precision and perform as many replicate measurements as possible [13]. Sample homogeneity depends on the size of the sample test-portion (or aliquot), normally defined as the amount of sample obtained from a bulk sample and completely consumed for a single analytical determination which may include steps like sample dissolution, obtaining of further aliquots etc. In the present case as sample test-portion is defined as the amount of solution finally introduced into a device generating an analytical signal by any integrating type mechanism, i.e. the analytical signal is a function of the number of analyte entitities present in the test-portion. In many cases, the analytical method itself does not allow the use of large test-portions despite the availability of large amounts of bulk samples. Typical examples include the sample volume introduced in the graphite tube during a determination with flameless atomic absorption spectroscopy, the sample injected into a GC or HPLC chromatograph, and the sample introduced in the wells of a luminometer plate for performing a highly sensitive immunochemical assay. Generally, for partially homogeneous samples the following equation, known as Ingamells’ equation, applies: W[sr(%)]2 =Ks
(2)
where W is the weight of the test-portion, sr(%) is the sampling relative (%) standard deviation (sr(%)= 100 ss/y¯, where y is the average analytical content) and Ks is the sampling constant. Ks depends on the nature of the sampled material and numerically corresponds to the weight (in g) of a test-portion with a sampling relative (%) standard deviation equal to 1% [14 – 16].
713
3. Ultra-dilute solutions as samples Let X be a discrete random variable which can take values 0, 1, 2, …, such that the probability function of X is given by Eq. (3): f(x)=P(X=x)=
m xe − m x!
(x= 0, 1, 2, …) (3)
where m is a given positive constant. The distribution described by Eq. (3) is known as Poisson distribution. Characteristic property of a variable X following Poisson distribution is that its mean value and its variance are both equal to m [17]. This distribution describes the probability of random occurrence of discrete events (x) in given segment of time or space at certain average rate (m). An ultra-dilute solution containing 3200 molecules of analyte Y per liter has a bulk concentration of Y (3200 l − 1)/(NA mol − 1)= 5.31× 10 − 21 mol l − 1 or 5.31 zM. On average, a 1.00-ml test-portion of this solution contains 3.20 molecules, but individual test-portions ultimately contain an integer number of molecules. Here, the molecules of Y are the discrete events and the volume of the test-portion is the segment of space. The probabilities of obtaining a 1.00-ml test-portion containing 0, 1, 2, … molecules of Y have been calculated using Eq. (3) and the corresponding discrete distribution diagram is shown in Fig. 1. An individual 1.00-ml test-portion of this solution would never have the expected bulk concentration of 5.31 zM, but concentration multiples of 1.66 zM ( = 1 molecule ml − 1). The closer multiple to the actual concentration of the bulk sample is 4.98 zM (three molecules per 1-ml test-portion) and this has only a 22.3% chance to occur, whereas there is also a 4.1% chance to obtain a test-portion containing no analyte molecules. This example describes a rather extreme case, as this distribution would eventually become relatively more narrow and symmetric approaching a Gaussian shape as the average number of molecules per test-portion increases. Since the mean and variance of populations exhibiting a Poisson distribution are arithmeti-
714
C.E. Efstathiou / Talanta 52 (2000) 711–715
cally the same, the sampling variance of a given size (i.e. volume) test-portion containing on average m molecules of analyte Y is: (4) s 2s =m= Vs CY NA where Vs is the volume of sample (in l), CY is the concentration of the analyte Y (in mol l − 1) and NA is the Avogadro number. From Eq. (4) the relative (%) standard deviation, sr(%), exclusively attributed to the non-homogeneity of a solution can also be calculated:
Fig. 1. Poisson distribution diagram showing the probabilities of occurrence of 0, 1, 2, … molecules in a 1-ml test-portion of a solution containing 3200 molecules l − 1 (= 5.31 zM).
Fig. 2. Sampling relative (%) standard deviation as a function of bulk-analyte concentration for various test-portion volumes.
sr(%)= (ss/m)×
100
Vs CY NA
(5)
For example, the sampling uncertainty of a 1.0-ml test-portion of an 1.0 fM solution of analyte is sr(%)= 100× [(1.0× 10 − 6 l)(1.0× 10 − 15 mol l − 1)(6.0221× 1023 mol − 1)] − 1/2 = 4.1%. Therefore, regardless of what efforts have been made to improve the overall analytical precision, measurements at 1 fM level with any technique using a test-portion of 1.0 ml, or generally a test-portion containing (1× 10 − 6 l)(1× 10 − 15 mol l − 1)= 1× 10 − 21 mol or 1 zmol amount of analyte, would never yield results with a relative (%) standard deviation less than 4%. This lower limit of uncertainty can only be reduced by increasing the size of the test-portion and/or by using the average of more replicates. By using a 1.00 ml test-portion, at the same concentration level, the expected sampling uncertainty would have been only 0.13%, which is generally considered as negligible especially for an analytical technique providing results at so low concentration levels. Unfortunately, increasing the size of the test-portion is not always feasible, particularly when one works with enzyme amplification and PCR techniques, because of the nature of these techniques, the available instrumentation and sometimes because of sample availability problems. It should be stressed that the assumption of Poisson distribution is valid as far as analyte entities behave as freely and randomly moving particles without any interaction between each other and the walls of the container as well. Clustering effects are not considered here. Obviously, any type of aggregation phenomena between the analyte entities would cause worse spreading of the analytical results. The expected relative (%) standard deviation for various sizes of test-portions (0.1 ml–10.0 ml) of ultra-dilute solutions (CY 5 10 − 12 M) can be quickly estimated from the log–log diagram shown in Fig. 2. The sampling constants, as defined by Ingamells’ equation (Eq. (2)) and now expressed in terms of volume, i.e. the volume of test-portion of an ultra-dilute solution, bearing a sampling uncer-
C.E. Efstathiou / Talanta 52 (2000) 711–715
tainty of 1% can be calculated from Eq. (5). Thus, the sampling constants for 1 fM, 1 aM and 1 zM sample solutions are 16.6 ml, 16.6 ml and 16.6 l, respectively. Actually, these are the volumes containing 10 000 analyte entities, since: sr(%)= (sY/y¯ ) × 100 =( m/m) × 100 = ( 10 000/10 000) × 100 = 1.00
4. Conclusions A high sampling variance affecting the overall analytical precision is expected when one has to work with very small volume test-portions of ultra-dilute solutions, or generally with test-portions containing a small number of analyte molecules. Running more replicates would be the only way to obtain more precise results. The traditional concept of homogeneity of solutions cannot be applied on small volumes of ultra-dilute solutions and any effort to push the limits of analytical determinations towards lower concentrations and smaller sample volumes is an intrinsically self-restricted goal.
.
715
References [1] M.A. Cousino, T.B. Jarbawi, H.B. Halsall, W.R. Heineman, Anal. Chem. 69 (1997) 544A. [2] C.T.J. Alkemade, Appl. Spectrosc. 35 (1) (1981) 1. [3] B.W. Smith, J.B. Womack, N. Omenetto, J.D. Winefordner, Appl. Spectrosc. 43 (5) (1989) 873. [4] J.D. Winefordner, B.W. Smith, N. Omenetto, Spectrochim. Acta Part B 44B (12) (1989) 1397. [5] C.L. Stevenson, J.D. Winefordner, Appl. Spectrosc. 45 (8) (1991) 1217. [6] C.L. Stevenson, J.D. Winefordner, Appl. Spectrosc. 46 (3) (1992) 407. [7] C.L. Stevenson, J.D. Winefordner, Appl. Spectrosc. 46 (5) (1992) 715. [8] T.T. Lee, E. Yeung, Anal. Chem. 64 (1992) 3045. [9] M.D. Barnes, K.C. Ng, W.B. Whitten, J.M. Ramsey, Anal. Chem. 65 (1993) 2360. [10] S. Bortolin, T.K. Christopoulos, M. Voerhagen, Anal. Chem. 68 (1996) 834. [11] D. Harris, Quantitative Chemical Analysis, fourth ed., W.H. Freeman, New York, 1996, p. 758. [12] W.J. Youden, J. Assoc. Off. Anal. Chem. 50 (1967) 1007. [13] B. Kratochvil, J.K. Taylor, Anal. Chem. 53 (1981) 924A. [14] C.O. Ingamells, P. Switzer, Talanta 20 (1973) 547. [15] C.O. Ingamells, Talanta 21 (1974) 141. [16] C.O. Ingamells, Talanta 23 (1976) 263. [17] J.R. Green, D. Margerison, Statistical Treatment of Experimental Data, Elsevier, Amsterdam, 1978, p. 35.