ANALYTICAL
BIOCHEMISTRY
79,635-638
(1977)
The Use of Linear Regression Analysis in the Isotope Dilution Assay of Biotin The use of linear regression analysis for calculating the results of isotope dilution assays is described, using an assay of biotin as an example. The technique affords an inbuilt check on the accuracy of the assay and corrects for quenching in liquid scintillation counting systems.
This investigation describes a new method for calculating the results of assays employing the principle of isotope dilution and is illustrated using a recently reported procedure (1) for the radiochemical assay of biotin in biological materials. Two methods, using the stoichiometric binding of biotin to avidin, for the radiochemical assay of biotin have been documented (1,2). Although the principle is the same in both procedures, different mathematical approaches are used in the calculation of the results. Linear regression analysis, besides allowing the calculation of the dilution of an isotope by the nonradioactive compound under assay, also facilitates the calculation of a correction factor which accounts for quenching, a problem commonly encountered in liquid scintillation counting systems containing aqueous biological material. The reliability of the assay is monitored by comparing the correlation coefficient for each set of data with that required for a certain level of confidence. MATERIALS
AND METHODS
Materials. D-Carbonyl-[14C]biotin (45 mCi/mmol) was obtained from the Radiochemical Centre Ltd. D-Biotin and ,avidin (0.54 unit/mg) were obtained from Sigma Chemical Co. and Nutritional Biochemicals Corporation, respectively. [14C]Biotin and avidin preparation. The [14C]biotin solution (9.5 ng of biotin.60~1) was prepared and calibrated as described previously (1). A solution of avidin in 0.9% NaCl (solution 5) was prepared such that the avidin in it bound approximately 80% of the [14C]biotin, when 50 ~1 of the [14C]biotin standard was added to 100 ~1 of solution 5. Avidin solutions 0 to 4 were prepared by quantitative dilution of avidin solution 5 to yield concentrations which were relative to that in solution 5, e.g., the concentration of avidin in solution 4 was twice that of solution 2. Assay procedure. The reagents which were added to a series of blank and assay tubes are listed in Table 1. The avidin solutions were added only after the two biotin solutions were thoroughly mixed. In this investigation, 50~1 of a nonradioactive biotin standard were added to tubes A, to A,. However, 635 Copyright 0 1977 by Academic Press, Inc. All rights of reproduction in any form reserved,
ISSN WO3-2697
SHORT COMMUNICATIONS
636
TABLE REAGENTS
1
FOR A TYPICAL
BIOTIN
ASSAY
Blank Centrifuge tube
Assay Avidin solutiod
[14C]Biotin (/a
0.9% NaCl (/a
(No.1
Q.4
50 50 50 50 50 50
50 50 50 SO 50 50
0 1 2 3 4 5
100 100 100 100 100 100
Bb Bl B2 B3 B4 B5
a The procedure for the preparation and Methods.
Cenfuge tube A0 Al A2 : A&
FROM AN ISOTOPE
0 1 2 3 4 5
Blank 2.939 2.509 2.061 1.487 1.049 0.595 Mathematical
Correlation coefficient Slope Intercept, x = 0 Significance [Y]Biotin added (ng) Z uncorrectedc Z quench correctedd Biotin (ng)
-0.999 -0.476 2.964 0.0001 9.5
(No.)
@I)
50 50 50 50 50 50
0 1 2 3 4 5
100 100 100 100 100 100
SO 50 50 50 50 50
2
DILUTION
Radioactivity Concentration of avidin=
Avidin solutioP
Biotin unknown b.4
of the avidin solutions is described under Materials TABLE
RESULTS
[‘*C]Biotin 6.4
Standard biotin 2.990 2.752 2.575 2.339 2.157 1.951
ASSAY
FOR BIOTIN
in supematant (cpm x 10e3) Standard biotin quenched sampleb 2.634 2.425 2.269 2.061 1.900 1.718
Standard biotin: 20% error in one determination 2.990 2.752 2.060 2.339 2.157 1.951
parameters from above results -0.999 -0.206 2.976 0.0001 9.5 0.433 0.431 12.5
-0.999 -0.182 2.622 0.0001 9.5 0.382 0.432 12.5
-0.870 -0.191 2.854 0.05 9.5 0.401 0.416 13.3
(1The units are arbitrary, and each solution was prepared by dilution from a common solution. b Color quenched with 500 ~1 of liver digest. c Z uncorrected = AJB,. d 2 quench-corrected is calculated from Eq. [I].
SHORT COMMUNICATIONS
637
in biological samples with a low biotin content, this volume can be increased, providing an equal volume of O.% NaCl is added to tubes B, to B5. The final volume in each series of tubes is not important, although it should be kept to a minimum for convenience. The incubation and estimation of [14C]biotin in the supernatant were carried out as previously described (1). Quenching was demonstrated in the liquid scintillation counting system by the addition of 500 ~1 of a solution which resulted from the digestion of 2 g of liver in 2 N H&SO, (1). The biotin was removed from the digest by treatment with avidin. The quantity of biotin in an unknown sample was calculated, using Eq. [l] and [2], from the slope and intercept of the straight line determined by linear regression analysis on the relative avidin concentration (X) and the counts per minute in the supernatant (Y) (Table 2).
Z = I(&) &MO%) WI Amount (ng) of unknown
biotin = C x (1 - 2)/Z
El1 PI
B, and A, are the slopes of the regression lines for the blank series (BOto B,) and assay series (A, to A& of tubes, respectively; BI and A, are the intercepts calculated for the respective regression lines when the relative avidin concentration is zero; C is the amount (ng) of [14C]biotin added to each tube. RESULTS AND DISCUSSION In the biotin assay, the amount of [14C]biotin in the supernatant, after precipitation of the avidin-biotin complex, is dependent on the dilution of the [14C]biotin by the biotin unknown. This dilution and the linear binding relationship between avidin and biotin are shown in Fig. 1. The proportion of [14C]biotin in the supernatant (Z) in a series of assay tubes is given by A$B,, whereas the factor B,/A, is included in Eq. [l] to correct for quenching which may occur during liquid scintillation counting.
r, C 2 z
g ,s 5 d
A 0 H
Blank Standard Quenched
standard
I
1 Avidin
2 3 Concentration
4
I
5
FIG. 1. The effect of quenching on an isotope dilution assay for biotin.
638
SHORT COMMUNICATIONS
The effects of quenching on the regression analysis are shown in Fig. 1 and Table 2. Quenching diminishes the slope of the regression line and causes an error in the calculation of Z (Table 2), unless corrected for by taking into account the value of the intercept when the avidin concentration is zero. This correction (Br/AJ adjusts the counting efficiency to that obtained from the blank series of assay tubes. That is, the results for tubes B,,-B5 become directly comparable to those for tubes A,,-AS. If this correction had not been made, the results of the quenched biotin standard in Table 2, a result of 15.4 ng of biotin rather than 12.5 ng, would have been obtained. Variation in the avidin concentration within a series of six determinations provides an inbuilt assessment of the reliability of the values used in the regression analysis. By using six avidin concentrations, correlation coefficients better than -0.99 are routinely obtained. If the correlation coefficient is not better than a value routinely obtained, the regression analysis is repeated after eliminating a value which is considered incorrect or which has been calculated by a statistical procedure, to fall outside desired confidence limits. If, after elimination of the value, the correlation coefficient is not better than a value routinely obtained, the assay requires repeating. An example of the results expected after a 20% error was simulated in one determination is shown in Table 2. The use of linear regression analysis provides a reliable and convenient method for the calculation of results from isotope dilution assays. REFERENCES 1. Hood, R. L. (1975) J. Sci. Food Agr. 26, 1847-1852. 2. Dakshinamurti, K., Landman, A. D., Ramamurti, L., and Constable, R. J. (1974) Anal. Biochem. 61,225-231.
Ross L. HOOD CSIRO Division of Food Research P.O. Box 52 North Ryde. New South Wales 2113, Australia Received November I, 1976; accepted January 18, 1977
BIOCHEMISTRY
79,635-638
(1977)
The Use of Linear Regression Analysis in the Isotope Dilution Assay of Biotin The use of linear regression analysis for calculating the results of isotope dilution assays is described, using an assay of biotin as an example. The technique affords an inbuilt check on the accuracy of the assay and corrects for quenching in liquid scintillation counting systems.
This investigation describes a new method for calculating the results of assays employing the principle of isotope dilution and is illustrated using a recently reported procedure (1) for the radiochemical assay of biotin in biological materials. Two methods, using the stoichiometric binding of biotin to avidin, for the radiochemical assay of biotin have been documented (1,2). Although the principle is the same in both procedures, different mathematical approaches are used in the calculation of the results. Linear regression analysis, besides allowing the calculation of the dilution of an isotope by the nonradioactive compound under assay, also facilitates the calculation of a correction factor which accounts for quenching, a problem commonly encountered in liquid scintillation counting systems containing aqueous biological material. The reliability of the assay is monitored by comparing the correlation coefficient for each set of data with that required for a certain level of confidence. MATERIALS
AND METHODS
Materials. D-Carbonyl-[14C]biotin (45 mCi/mmol) was obtained from the Radiochemical Centre Ltd. D-Biotin and ,avidin (0.54 unit/mg) were obtained from Sigma Chemical Co. and Nutritional Biochemicals Corporation, respectively. [14C]Biotin and avidin preparation. The [14C]biotin solution (9.5 ng of biotin.60~1) was prepared and calibrated as described previously (1). A solution of avidin in 0.9% NaCl (solution 5) was prepared such that the avidin in it bound approximately 80% of the [14C]biotin, when 50 ~1 of the [14C]biotin standard was added to 100 ~1 of solution 5. Avidin solutions 0 to 4 were prepared by quantitative dilution of avidin solution 5 to yield concentrations which were relative to that in solution 5, e.g., the concentration of avidin in solution 4 was twice that of solution 2. Assay procedure. The reagents which were added to a series of blank and assay tubes are listed in Table 1. The avidin solutions were added only after the two biotin solutions were thoroughly mixed. In this investigation, 50~1 of a nonradioactive biotin standard were added to tubes A, to A,. However, 635 Copyright 0 1977 by Academic Press, Inc. All rights of reproduction in any form reserved,
ISSN WO3-2697
SHORT COMMUNICATIONS
636
TABLE REAGENTS
1
FOR A TYPICAL
BIOTIN
ASSAY
Blank Centrifuge tube
Assay Avidin solutiod
[14C]Biotin (/a
0.9% NaCl (/a
(No.1
Q.4
50 50 50 50 50 50
50 50 50 SO 50 50
0 1 2 3 4 5
100 100 100 100 100 100
Bb Bl B2 B3 B4 B5
a The procedure for the preparation and Methods.
Cenfuge tube A0 Al A2 : A&
FROM AN ISOTOPE
0 1 2 3 4 5
Blank 2.939 2.509 2.061 1.487 1.049 0.595 Mathematical
Correlation coefficient Slope Intercept, x = 0 Significance [Y]Biotin added (ng) Z uncorrectedc Z quench correctedd Biotin (ng)
-0.999 -0.476 2.964 0.0001 9.5
(No.)
@I)
50 50 50 50 50 50
0 1 2 3 4 5
100 100 100 100 100 100
SO 50 50 50 50 50
2
DILUTION
Radioactivity Concentration of avidin=
Avidin solutioP
Biotin unknown b.4
of the avidin solutions is described under Materials TABLE
RESULTS
[‘*C]Biotin 6.4
Standard biotin 2.990 2.752 2.575 2.339 2.157 1.951
ASSAY
FOR BIOTIN
in supematant (cpm x 10e3) Standard biotin quenched sampleb 2.634 2.425 2.269 2.061 1.900 1.718
Standard biotin: 20% error in one determination 2.990 2.752 2.060 2.339 2.157 1.951
parameters from above results -0.999 -0.206 2.976 0.0001 9.5 0.433 0.431 12.5
-0.999 -0.182 2.622 0.0001 9.5 0.382 0.432 12.5
-0.870 -0.191 2.854 0.05 9.5 0.401 0.416 13.3
(1The units are arbitrary, and each solution was prepared by dilution from a common solution. b Color quenched with 500 ~1 of liver digest. c Z uncorrected = AJB,. d 2 quench-corrected is calculated from Eq. [I].
SHORT COMMUNICATIONS
637
in biological samples with a low biotin content, this volume can be increased, providing an equal volume of O.% NaCl is added to tubes B, to B5. The final volume in each series of tubes is not important, although it should be kept to a minimum for convenience. The incubation and estimation of [14C]biotin in the supernatant were carried out as previously described (1). Quenching was demonstrated in the liquid scintillation counting system by the addition of 500 ~1 of a solution which resulted from the digestion of 2 g of liver in 2 N H&SO, (1). The biotin was removed from the digest by treatment with avidin. The quantity of biotin in an unknown sample was calculated, using Eq. [l] and [2], from the slope and intercept of the straight line determined by linear regression analysis on the relative avidin concentration (X) and the counts per minute in the supernatant (Y) (Table 2).
Z = I(&) &MO%) WI Amount (ng) of unknown
biotin = C x (1 - 2)/Z
El1 PI
B, and A, are the slopes of the regression lines for the blank series (BOto B,) and assay series (A, to A& of tubes, respectively; BI and A, are the intercepts calculated for the respective regression lines when the relative avidin concentration is zero; C is the amount (ng) of [14C]biotin added to each tube. RESULTS AND DISCUSSION In the biotin assay, the amount of [14C]biotin in the supernatant, after precipitation of the avidin-biotin complex, is dependent on the dilution of the [14C]biotin by the biotin unknown. This dilution and the linear binding relationship between avidin and biotin are shown in Fig. 1. The proportion of [14C]biotin in the supernatant (Z) in a series of assay tubes is given by A$B,, whereas the factor B,/A, is included in Eq. [l] to correct for quenching which may occur during liquid scintillation counting.
r, C 2 z
g ,s 5 d
A 0 H
Blank Standard Quenched
standard
I
1 Avidin
2 3 Concentration
4
I
5
FIG. 1. The effect of quenching on an isotope dilution assay for biotin.
638
SHORT COMMUNICATIONS
The effects of quenching on the regression analysis are shown in Fig. 1 and Table 2. Quenching diminishes the slope of the regression line and causes an error in the calculation of Z (Table 2), unless corrected for by taking into account the value of the intercept when the avidin concentration is zero. This correction (Br/AJ adjusts the counting efficiency to that obtained from the blank series of assay tubes. That is, the results for tubes B,,-B5 become directly comparable to those for tubes A,,-AS. If this correction had not been made, the results of the quenched biotin standard in Table 2, a result of 15.4 ng of biotin rather than 12.5 ng, would have been obtained. Variation in the avidin concentration within a series of six determinations provides an inbuilt assessment of the reliability of the values used in the regression analysis. By using six avidin concentrations, correlation coefficients better than -0.99 are routinely obtained. If the correlation coefficient is not better than a value routinely obtained, the regression analysis is repeated after eliminating a value which is considered incorrect or which has been calculated by a statistical procedure, to fall outside desired confidence limits. If, after elimination of the value, the correlation coefficient is not better than a value routinely obtained, the assay requires repeating. An example of the results expected after a 20% error was simulated in one determination is shown in Table 2. The use of linear regression analysis provides a reliable and convenient method for the calculation of results from isotope dilution assays. REFERENCES 1. Hood, R. L. (1975) J. Sci. Food Agr. 26, 1847-1852. 2. Dakshinamurti, K., Landman, A. D., Ramamurti, L., and Constable, R. J. (1974) Anal. Biochem. 61,225-231.
Ross L. HOOD CSIRO Division of Food Research P.O. Box 52 North Ryde. New South Wales 2113, Australia Received November I, 1976; accepted January 18, 1977
