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Anal. Chem. IQQl, 63, 2581-2586
Mathematical Theory of Complex Ligand-Binding Systems Applied to Free Triiodothyronine Immunoassays Kaj R.Blombe.rg*J Department of Physical Chemistry, Abo Akademi, Porthansgatan 3, SF-20500 Abo, Finland
Sten 0. Engblom Department of Analytical Chemistry, Abo Akademi, Biskopsgatan 8, SF-20500 Abo, Finland
A theoretical bask for direct hnnunoassap of free hormoneg In serum Is presented. The muHlple-llgand/mu~lpl~~e blndlng theory employed makes H possible to predlct the dletrlbutlon of hormones among exogenous and endogenous blndkrg proteins In the assay mixture. The model allows sknu l a t h of assay systems InvOMng any "her of Ugands and blndlng sites. The slmulatlon of an assay of free trllodothyronlne Illustrates the way In wMch assay parameters such as antlbody concentratlon, antlbody affinity, serum dllutlon, and labeled her", Interactlono with serum blndlng protelns affect the valldlty of free hormone assays. The simultaneous equatlons descrlblng these complex blndlng systems at equiilbrium were solved on a personal computer, with the use of commercial mathematics sofiware. This general method for oolvlng and modellng free hormone assay systems provldes a tool for predlctlng the behavlor of free-hormone assays.
INTRODUCTION In serum, structurally related hormones of low molecular weight (thyroid hormones, steroids) compete for the same binding sites on endogenous binding proteins. The concentration of free (unbound) hormone is measured in direct immunoassays by incubating the serum sample with an antibody specific for the hormone to be measured. In this complex ligand-binding system the amount of hormone bound to the antibody is under certain conditions a function of the free hormone concentration in the serum. Determination of antibody sites occupied gives the free-hormone concentration. The mathematical theory of free hormone immunoassay has been developed by several researchers. Ekins has given dose-response curves for the situation in which one univalent antibody binds two antigens (labeled and unlabeled ligand) with the same avidity and has also considered the equilibrium system in which one antigen reacts with many binding sites (1). The same author has also extensively discussed the pros and cons of different free-hormone methodologies and assay designs (2). Geiseler and Ritter have studied the system in which labeled and unlabeled analyte reads with any number of binding sites, employing a model assuming one strong binding site and a simplified expression for the rest of the binding sites (3). Midgley and Wilkins have used equations based on the mathematical model of complex ligand binding by Feldman ( 4 ) for computer simulation of a free thyroxine assay based on the use of a thyroxine-analogue label (two ligands, many binding sites) (5, 6). In this paper the mathematical theory of complex ligandbinding systems a t equilibrium, introduced and used by Feldman et al. (7) for modeling of immunoassays for deterCurrent address: Wallac Biochemical Laboratory, P.O.Box 10, SF-20101 Turku. Finland. 0003-2700/9 1/0363-2581$02.50/0
mination of total analyte concentrations, has been applied to immunoassays of free hormones. The mathematical model formulated is applicable to free-hormone assay systems involving any number of hormones and binding sites. This technique made it possible to simulate an assay system for free triiodothyronine in which four ligands react with up to 12 binding sites. The mass law equations determining the composition of the assay mixture at equilibrium were solved using a personal computer equipped with commercial software. The general method for simulation of the complex equilibrium state, together with the standard computer equipment used, provides a convenient tool for predicting sufficient assay conditions.
THEORY The mathematical model is based on the following assumptions: each reaction is reversible and proceeds to equilibrium obeying the law of mass-action, binding sites compete independently for univalent ligands, and no reactions other than
Hi + Bj i = l , 2 ,..., n
HiBj
(1)
j = l , 2 ,..., m
take place. The number of ligands involved in the reaction is n, and they are denoted by H1 to Hn. Similarly, the number of binding sites involved is m and the denotations used are B1to B,. The equilibriumcomposition of the solution is given by (2) K H ~ ,=B[HiBjl/ ~ [ m i l [fBjl
i = l , 2,..., n
j = l , 2 ,..., m
[tHi] = [ m i ] + g [ H i B j ] j=l
(3) (4)
where KHi,B.denotes the affinity constant for each reaction, [mi] is the hee concentration of the ith hormone, [fsi] is the concentration of the j t h unreacted binding site, [tHi] is the total concentration of the ith hormone, [tB,] is the total concentration of the j t h binding site, [HiBi] is the concentration of the complex formed between the tth hormone and jth binding site. Combination of eqs 2-4 yields a set of n equations:
i = 1, 2, ..., n When [tHi], [tBj], and KHi,g are known, this system of n nonlinear equations can be solved numerically to find the free concentrations of the hormones involved. 0 1991 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 63,NO. 22, NOVEMBER 15, 1991
Using the relation [tHJ - [Mi] = C m [HiBj] (from eq 3), is given by the concentration of bound hormone
latter is unreactive with serum binding proteins. Assuming again a divalent antibody with identical binding sites, we obtain the concentration of antibody-bound analogue from eq 6 as [H,*Ab] = [H,*Abl] + [H,*Ab2] =
i = 1, 2, ..., n Assume now that an antibody Ab with two binding sites, Abl and Ab2, is present in the solution. This situation is reflected in the simulations described in a later section of this paper. It follows from eq 6 that the concentration of antibody-bound hormone can be expressed as [HiAb] = [HiAbl]
+ [HiAb,]
=
\
i = 1, 2, ..., n where [tAb] = [tAbl] = [tAb2] is the total concentration of the antibody and [HiAb] is the concentration of hormone Hi bound to the antibody. If the antibody is absolutely specific for the hormone Ha we wish to measure and if the two affiiity constants are equal, then eq 7 reduces to [&Ab] -[tAb1
2KH,,Ab[mel
+ KH,,Ab[mal
Since total concentrations of antibody and analogue are identical in each tube in an away run and since only analogue and analyte compete for binding sites on the antibody, it is the free-hormone concentration in the solution that determines the amount of analogue sequestered onto the antibody. Measurement of [H,*Ab] in eq 10 gives the free-hormone concentration in the sample, provided A is close to 1. The amount of antibody-bound analogue is related to the freehormone concentration by constructing a dceeresponse curve from standards with known free-hormone concentrations. Useful Equations and Numerical Methods. When eq 5 is solved for the free-hormone concentrations [mi],the concentrations of free binding sites [fBj] can easily be calculated from [tBjl
j = 1, 2,
(8)
The fraction of antibody binding sites occupied is determined solely by the free-hormone concentration in the solution. Thus, the free-hormone concentration can be quantified by measuring the fraction of binding sites occupied. A commonly employed technique for determination of occupied binding sites is the titration of unoccupied binding sites with labeled hormone after the serum binding proteins have been removed (49). Constructing a dose-response curve from standards with known free-hormone concentrations relates the antibody occupancy to the free-hormone concentration. The sequestering antibody need not be absolutely specific for the analyte of interest. If [H,Ab] >> [HiAb], i # a, holds, then binding of hormones other than the one to be measured to the antibody, is negligible. This is the case when the antibody has a sufficiently low affinity for cross-reacting hormones present in high concentrations or when the antibody binds cross-reacting substances, present in sufficiently low concentrations, with a high affinity. Dilution of the sample and addition of antibody reduces the free-hormone concentration initially present in the sample. Nevertheless, the free-hormone concentration in the sample can be quantified with a neglible systematic error if the alteration of the free-hormone concentration is insignificant. The extent of the alteration can be expressed as A = [fKl/[fH’al (9) where [M’,] is the free-hormone concentration in the sample, [W,] is the free-hormone concentration of the assay mixture, and A is their ratio. The unavoidable alteration of the freehormone concentration affects the accuracy of the assay. In a well-optimized assay with maximized accuracy, the ratio A should be close to 1. Another technique which is also widely used for determination of the free-hormone concentration is the one-step, analogue assay (6),where sample, antibody, and a hormoneanalogue label are incubated simultaneously. A criterion for an analogue assay is that the antibody reacts only with the hormone of interest and the labeled hormone, and that the
(11)
n
..., m
With known values of the variables [mi] and [fBj] and the affinity constants, the concentrations of complexes formed are obtained from eq 2. The numerical solutions of the equations used for simulations were obtained on a personal computer equipped commercially available mathematics software. Information on the computer program and the algorithm it used to solve simultaneous nonlinear equations can be found in the program manual (IO).
EXPERIMENTAL SECTION We used the reagents of the DELFIA Free T3 (Wallac OY, Turku, Finland) solid phase, fluoroimmunoassay,for the quantitative measurement of free T3 in serum. The buffer used for the incubations was a 0.05 M tris(hydroxymethy1)aminomethane buffer, pH 7.4, containing 0.9% sodium chloride and 0.05% sodium azide. This buffer was also used for the washings after addition of 0.01% Tween. Specimens. Serum samples from various patient groups with known concentrations of total thyroxine (T4), thyroxine-binding globulin (TBG), and prealbumin (PA) were obtained from Dr. Med.Sc. Tom Petterson, Danderyd Hospital, Danderyd, Sweden. The methods used for the determination of these parameters have been reported (11). Total triiodothyronine (T3) was determined with the DELFIA T3 kit (Wallac OY). Apparatus. The fluorescence was measured with a time-resolved fluorometer (1232 DELFIA, Wallac OY). Principle of the Assay. Standards and samples are first reacted with anti-T3 monoclonal antibody (derived from mouse) which binds to anti-mouse IgG immobilized onto the wells of microtitration strips. After incubation, buffer and serum are washed away and in a second incubation step the remainingempty sites on the anti-T3 antibody are back-titrated with europiumlabeled T3 (Eu-T3). Then the bound europium is dissociated into DELFIA enhancement solution where it forms highly fluorescent chelates with componentsof the enhancement solution. The fluorescence is measured in the time-resolved fluorometer. Assay Procedure. In the assay, 50 p L of standard, serum sample, and diluted serum sample (diluted with buffer solution) were pipetted in duplicate into the coated wells and 200 p L of antibody solution was added to each well. In order to demonstrate the effect of antibody concentration on the free-T3 concentration the samples were assayed using two concentrations of antibody,
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 22, NOVEMBER 15, 1991
Table I. Affinities of T4, T3, and rT3 for Serum Binding Proteins affinity constants (M-l) at 37 O C , pH 7.4, phosphate buffer, 0.1 M NaCl ref T3 ref rT3
protein
binding site
T4
TBG' PA
1 1 2 1 2-6
1.0 x 1010 2.2 x loe 3.5 x 106 7.0 x 105 4.8 x 104
Alb
126 14d 14d 15 15
4.6 X lo8 1.8 x 107 8.8 X lo5 1.2 x 105 8.0 x 103
126 14d 14d 16, 178 16, 178
3.1 X l@ 1.1 x 10' 3.0 x 105 7.0 x 105 4.8 x 104
ref 18c 19.
f
16" 16"
"TBG, thyroxine binding protein; PA, prealbumin; Alb, human serum albumin. bAffinityconstant in phosphate buffer at 25 O C , adjusted to 37 "C by Robbins and Rall (23). cValuein phosphate buffer. dLiterature value at 25 "C multiplied by 0.7 to adjust to 37 O C (13). eValue at 37 "C, pH 8, in 0.1 M Tris-NaC1. 'Estimated. #For each binding site, the affinity of T3 for albumin is assumed to be 16.7% of that of T4. The affinity of rT3 for albumin is assumed to be equal to that of T4. Table 11. Serum Concentration of Binding Proteins, T4, T3, and rT3 protein TBG PA Alb
hormone
normal serum" low TBG serumo 340 nM 5.0 pM 640 pM 100 nM 2 nM
3.4 nM 5.0 pM 640 pM 30 nM 1.2 nM
T4 T3 rT3 500 pM 400 pM "Values are within the range of mean values, for various reference populations, quoted by Pettersson ( 1 0 , Ramsden et al. (20), and authors listed bv Robbins and Rall (13). . . 4.4 X lo-" and 4.4 X M. The wells were incubated for 90 min at 37 OC and then washed two times with wash solution. After dispensing 200 p L of Eu-T3 buffer solution into each well, the wells were incubated for 30 min at 4 "C and then washed four times. Before the fluorescence was measured, 200 p L of enhancement solution was added to each well. Mathematical Calculation. The calculation of the free-T3 concentration in each sample was made from the equilibrium model using the affinity constants in Table I and the measured total T3, T4, TBG, and PA concentrations. The concentration of albumin was not measured for each individual sample but given a value of 640 pM. Since the concentration of reverse T3 has no effect on the calculated T3 concentration, a value of 0.5 nM was used in the calculations. The affinity of the antibody was 1Olo M-l. Calculated and experimentally obtained free-T3 values for the serum samples from the various groups at different dilutions and two antibody concentrations are presented.
RESULTS AND DISCUSSION Free concentrations of thyroid hormones triiodothyronine (T3) and thyroxine (T4) are of diagnostic value in assessing thyroid function. In serum they compete with different affinities for the same binding sites on binding proteins. Affinities of T3, T4, and the thyroid hormone metabolite 3,3',5'-triiodothyronine (reverse T3, rT3) for the major serum binding proteins are presented in Table I. The serum concentrations of these compounds and the major serum binding proteins in normal serum and in serum from subjects with congenitally low thyroxine-binding globulin concentrations are listed in Table 11. Minor binding proteins and thyroid hormone metabolites have been excluded, since very little data on their binding parameters is available and their low concentration and affinity are not expected to significantly affect the distribution of T3 among the binding proteins. This seems to be a justifiable assumption since excluding rT3 from the calculations has an insignificant effect on the computed results presented here. The calculated free-T3 concentration in the normal serum is 6.28 pM, and in the serum with low TBG, 5.96 pM. The corresponding free-T4 concentrations are 22.5 and 17.8 pM, which is in agreement with measured values (21-23). The binding parameters, if not separately measured, listed in Tables I and I1 have been used in the computations throughout this paper.
1s 14
-
13
-
12
-
B
11 -
10
-
987
-13
-12
-11
-10
-9
-8
-1
i
Figwe 1. Effect of dilution, antibody amity, and antibcdy concentration on the free T3 concentration in serum was simulated using the parameters listed in Tables I and 11. The decrease in the free 13 concentration is less than 3 % in a free T3 assay if antibody affinity and concentration are in the area delimited by the x axis (antibody concentration), the y axis (antibody affinity), and the curves Showing antibody affinities and concentrations causing a decrease of 3 % in the free T3 concentration. The curves refer to different serum dilutions. Figure 1A is for normal serum: (solid line) undiluted serum, (dashed iine) serum diluted by a factor of 5 in the assay; (dotted iine) serum diluted by a factor of 10 in the assay. Figure 18 Is for serum with low TBG (solid line) undiluted serum; (dashed line) serum sample diluted by a factor of 5 in the assay.
Introduction of antibody and dilution of a serum sample causes a decrease in the free-hormone concentration. The fall in the free-T3 concentration is less than 3% if antibody affinity, antibody concentration, and the dilution factor are selected as depicted in Figure 1. A 10-fold dilution of the TBG-deficient serum decreases the free-T3 concentration by 4.570, and addition of antibody, even more. The computation
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ANALYTICAL CHEMISTRY, VOL. 03, NO. 22, NOVEMBER 15, 1991
Table 111. Effect of Dilution and Antibody Addition on the Free T3 Concentration, pM (Percent Decrease) in Normal Seruma antibody affinity, M-'
antibody concentration dilution factor
lo+' M
10"
10'2
M
lW1* M
10l9 M
6.28 (-) 6.20 (1.3) 6.11 (2.8) 4.76 (24) 6.28 (-) 6.19 (1.4) 6.09 (3.1) 4.63 (26) 6.28 (-) 6.18 (1.7) 6.06 (3.6) 4.39 (30)
6.28 (-1 6.21 (1.2) 6.11 (2.8) 4.78 (24) 6.28 (-) 6.20 (1.3) 6.11 (2.8) 4.17 (24) 6.28 (-) 6.20 (1.3) 6.10 (2.8) 4.74 (29.5)
6.28b 6.21 (1.2) 6.11 (2.8) 4.79 (24)
1 5 10 100 1 5 10 100 1 5 10 100 1 5 10 100
10'0
lo-"
M
5.93 (5.6) 4.79 (24) 3.85 (39) 0.83 (87) 4.37 (30) 1.70 (73) 0.93 (85) 0.10 (98) 2.06 (67) 0.24 (96) 0.11 (98) 0.01 (100)
6.25 (0.6) 6.03 (4.1) 5.78 (8.1) 3.27 (48) 6.05 (3.8) 5.15 (18) 4.28 (32) 0.89 (86) 5.75 (8.5) 3.75 (40) 2.02 (68) 0.11 (98)
6.28 (-) 6.19 (1.5) 6.07 (3.3) 4.58 (27) 6.26 (0.4) 6.09 (3.1) 5.88 (6.4) 3.54 (44) 6.23 (0.9) 5.94 (5.5) 5.59 (11) 1.74 (72)
" Binding parameters from Tables I and I1 have been used in these computations. Concentration of free T3 in undiluted normal serum. Table IV. Measured and Calculated Effect of Antibody Concentration and Sample Dilution on Free T3 Values (pM) in Various Groups" antibody concn, M normal n = 10
hypothyroidism n=2
hyperthyroidism n=6
low TBG n = 10
4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4
x 10-11 x 10-10 x 10-11 x 10-10 x 10-11 x 10-10 x 10-11 x 10-10
measured free T3 dilution factorb
calculated free T 3 dilution factof
1
2
4
6
10
20
1
10
20
30
50
100
5.01d 5.05 2.60 2.60 19.0 15.8 4.77 4.39
4.81 4.13 2.53 2.15 18.8 10.5 4.69 3.14
4.59 3.05 2.63 1.80 16.9 6.14 4.20 2.06
4.25 2.43 2.49 1.42 15.0 4.50 3.75 1.50
3.67 1.92 2.24 1.18 12.5 3.06 3.11 0.98
2.73 1.52 1.98 1.40 8.09 2.32 2.08 0.86
5.57 5.51 5.27 2.55 2.50 15.2 14.4 3.68 3.56
5.41 5.24 4.24 2.43 2.00 14.4 11.4 3.43 2.62
5.24 4.95 3.43 2.31 1.63 13.5 8.96 3.19 2.01
5.09 4.70 2.87 2.20 1.38 12.7 7.43 2.98 1.63
4.79 4.25 2.17 2.00 1.05 11.4 5.45 2.63 1.19
4.20 3.46 1.35 1.66 0.66 9.00 3.26 2.04 0.67
"The mean concentrations of total T3, T4,TBG, and PA in the normal, hypothyroid, hyperthyroid, and low TBG groups were 1.74 nM, 105 nM, 360 nM, and 4.37 pM; 0.87 nM, 28 nM, 361 nM, and 4.49 pM; 3.94 nM, 216 nM, 346 nM, and 3.55 p M and 0.84 nM, 61 nM, 124 nM, and 4.39 pM, respectively. *The samples were diluted by these factors before measurement. 'Sample dilution factor in the assay mixture. Mean values are shown. suggests that a dilution factor of about 5 should be used if one wants the fall in the free-T3 concentration to be less than 3% also in samples with very low TBG concentrations. Increasing the dilution factor and antibody concentration may cause a considerable decrease in the free-T3 concentration (Table 111). As long as the reduction (even a large one) in the free-T3 concentration in standards and unknowns is similar or negligible, a reliable estimate of the free-T3 concentration is obtained. However, if too much antibody is used, a substantial variation in the extent of the reduction of the free-T3 concentration between samples occurs due to the presence of serum samples with altered protein binding. Hence, too much antibody results in a biased estimate of the free-T3 concentration in samples with abnormal binding protein affinities and in samples with concentrations of binding proteins differing from the average level. Therefore it is essential in a free-T3 assay (and in all free-hormone assays) that the antibody concentration and dilution factor are selected in such a way that the alteration of the freehormone concentration in standards and samples is negligible. A simple way to test the effect of the antibody concentration on a free-T3 assay is to perform the dilution test (24). In an assay where the antibody concentration causes a negligible fall in the free-T3 concentration, normal serum diluted with buffer free of binding substances follows at moderate dilutions the theoretically predicted dilution curve for normal serum. If antibody is used at a concentration resulting in a substantial reduction in the free-hormoneconcentration, a deviation from
the theoretically predicted dilution curve is seen also for low sample dilution factors. However, even with high concentrations of antibody, the estimated free-T3 concentration in normal serum will be close to that initially present in the undiluted serum provided the dilution factor and the binding parameters in standards and normal serum are identical. The results in Table IV show that the measured free-T3 concentration in normal samples and samples from hypothyroids is unchanged after a 10-fold increase in the antibody concentration (standards and samples are diluted 5-fold in the assay, and the standards are made in normal human serum). An increased antibody concentration causes a decrease in the estimated free-T3concentrationin the low TBG group because binding parameters in standards and samples are different. There is, overall, a fairly good agreement between calculated and measured free-T3 concentrations for the groups investigated, although the computed values for the hypothyroid and low TBG groups are somewhat lower than the experimentally obtained result. The magnitude of the theoretically predicted decrease in the FT3 concentration in diluted serum samples is similar to that obtained experimentally. Increasing the antibody concentration results in a further reduction in the free-T3 concentration in diluted samples, wich is in accordance with the calculated results. Figure 2A shows theoretical dose-response curves for analogue assays of free T3 where antibody and labeled analyte concentrations and affinities have been varied. Under these assay conditions and the assumption of a sample dilution
ANALYTICAL CHEMISTRY, VOL. 63, NO. 22, NOVEMBER 15, 1991
1
If
B
-
tml IPM Flgure 2. Theoretical dose-response curves for free T3 assays. A divalent antibody is assumed. 6 = T3-analogue label bound to antibody, Bo = T3-analogue label bound to antibody at zero T3 dose, T = total concentration of T3-analogue label. Figure 2A is for the analogue assay: (solid line)the antibody binds T3 and T3-analogue label wlth the same a m , K = 8.0 X 10'' M', [tAb] = 1.25 X lo-'' M, T = 2.5 X lo-'' M. B o l l = 0.5; (dashed line) the antibody binds T3 and the 13-analogue label with the same affinity, K = 8.0 X 10" M', [tAb] = 1.87 X lo-" M, T = 5.0 X lo-'' M, 6dT= 0.7; (dotted line) the antibody affinity for the T3-analogue label is 1.0 X 1O'O M-' and for T3, 1.0 X 10" M', [tAb] 1.0 X lo-'' M, T = 5.0 X M, 6oIT= 0.12; (dashdotted he) the antibody Mnds the T3-analogue and T3 wlth the same affinity, K = 8.0 X 10" M-', [tAb] = 1.87 X lo-'* M, T = 5.0 X lo-'* M, B o l l = 0.5. Figure 28 is for the back-titration assays: (sdM line) the affinity of 13 for the antibody is 5.0 X 10" M-'; (dashed line) the affinity constant for T3-antibody binding is 1.6 X 10'' M', which Is the reciprocal of the calculated free T3 cmsntratbn (8.28 pM) h a " a i sennn; (dotted line)the affinity constant for the T8antibody binding is 1.0 X lo'* M-'.
factor of 5, the free-T3 concentration in normal serum is reduced by 1.8-3.2% and 0.5-2% of the total amount of T3 available is sequestered onto the antibody. Theoretical dose-response curves for back-titration assays of free T3 are depicted in Figure 2B. These over-simplified curves are based on the assumption that all unoccupied binding sites on the antibody can be titrated with labeled T3 without any dissociation of antibody-bound "3. For back-titration assays, Ekins has suggested an affinity constant of the antibody that is roughly the reciprocal of the free-hormone concentration (2). Among the 'best" curves is the one where the antibody poss e - an ~ affinity ~ that is the reciprocal of the calculated free-T3 concentration (6.28 pM) in normal serum. In order to obtain an acceptable measurement range and sufficient slope of the standard curve, the computed results suggest that in a free-T3 assay the antibody should possess an affinity that is in the
2585
range 1011-1012M-' and the antibody concentration should be less than lo-" M when the sample is diluted by a factor of 5. Working assays can be constructed using antibodies of lower affinity but then the clinically important dose region is at the very beginning of the standard curve. In the computations in this paper a decrease of 3% or less in the free-T3 concentration is considered arbitrarily to be insignificant and acceptable. If a greater deviation is considered acceptable, higher antibody concentrationsthan those recommended can be used and still the estimated free-T3 concentration in samples with normal binding parameters will be close to that in undiluted serum. In samples with abnormal binding parameters the free-T3 estimate will be biased, depending on the antibody concentration, dilution factor, and binding parameters, to a degree that may or may not be clinically important. Under assay conditions as in Figure 2A (dotted line) and with the assumption that the affinity of the T3 specific antibody for T4 is 1.0 X lo8 M-l, the calculated concentrations of antibody-boundT3 and antibody-boundT4 are 5:9 X 10-l2 and 2.2 X 10-14M, respectively. The cross-reactivity of the antibody is insignificant since [T3Ab] >> [T4Ab]. The antibody cross-reacts insignificantly with T4 in all assays in Figure 2 when the antibody affinity for T4 is lo00 times lower than for T3, since in comparison to the amount of T3 bound to antibody, about 270 times less T4 is bound. In an analogue assay where the analogue reacts with serum binding proteins, the amount of analogue bound to antibody is not only determined by the free-T3 concentration but is also affected by variations in binding protein concentrations and affinities. This fact can be illustrated in the following way. Assay conditions are as in Figure 2A (dotted line), except for the tracer concentration,which is doubled. When normal serum samples are analyzed, 59% of the T3-analogue label is bound to the serum albumin, because of the reaction of the analogue with albumin. Two samples are assayed, both with 6.28 pM of free T3. One of the samples is a low-albumin sample (2.0 X lo4 M), the other one contains normal concentrations of binding proteins. The calculated values for antibody-bound T3-analogue label are 5.9 X M for the normal sample and 1.4 X M for the low-albumin sample. In a valid free-T3 assay with negligible cross-reactivity of the analogue with serum albumin, these two concentrations are identical. The approach presented here makes is possible to quantitatively describe and solve the equations of the complex binding system free-hormone assays constitutes. The effect of important assay parameters, such as antibody concentration and affinity, dilution factors, altered protein binding, reactivity of hormone-analogue label with serum binding proteins, drug interference, and cross-reactivity of the antibody can be predicted. In combination with experiments this theoretical method provides a powerful tool for optimization and evaluation of free-hormone assays. Registry No. T3, 6893-02-3.
LITERATURE CITED Eklns, R. P. In Free lfof"s In Bkod; Albert(ni, A., Eklns, R. P., Ed.; Elsevier Biomedical Press: Amsterdam. 1982; pp 73-90. Eklns, R. P. NwComp8ct 1985, 16, 305-313. Geiseler, D.; RMter. M. Anal. Chem. 1982. 54. 2062-2067. Feldman, A. H. Anal. Blochem. 1972, 48. 317-338. Wilkins. 1.A.; Midgley, J. E. M. in R a M m m s s a y end R a t e d Rocedves in MedIchs; M E A Vlenna. 1982 pp 221-240. Wikins, 1. A.; Mldgley, J. E. M.; Barron. N. Clh. 0".1985. 31. 1644-1653. Feldman, H.; Rodbard, D.; Levine, D. Ana/. B(0Chem. 1972. 45, 530-556. BUt7tbrg, J. In Free HamoneJ h Bkod; Aibertinl, A,. Ekins. R. P., Eds.; Eleevkr Biomedlcel Press: Amsterdam, 1982; pp 139-149. Ekins. R. P. In Immunoassays for Chical &emisby. 2nd ed.; Hunter, W. M.,Corrle, J. E. 1..Eds.; Chwchlli. Llvlngstone: Edinburgh, 1983; pp 319-339.
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w:
(10) U w ' s Mathcad; MathSoR Inc.: Cambridge. MA, 1989. (11) P e ~T. M. ~ Ph.0. , Thesis, Kardinska Institutet. Stockholm, S w e den, 1989. (12) Kwcek, L.; Tabachnlck. M. J. W . Chem. 1976, 257, 3558-3562. (13) Robbins. J.; Rail. J. E. I n M " s In Bbcd I , 3rd ed.; Gray, C. H., James, V. H. T., Eds.; Academlc Press: London, 1979; pp 575-688. (14) Cheng, S. Y.; Pages, R. A.; Saroff, H. A.; Edelhoch, H.; Robbins, J. Bbchemistry 1977* 76, 3707-3713. (15) Tebachnick, M. J. Bid. Chem. 1967, 242, 1646-1650. (16) Tabachnlck, M.; Wrgio, N. A. Arch. Bkxhim. Bbphys. 1984, 705, 563-569. (17) Steiner, R. F.; Roth, J.; Robbins, J. J. Biol. Chem. 1966, 247, 560-565. (18) Tabachnick, M.; Korcek. L. Biochem. Bbphys. Acta 1978, 537, 169- 175.
(19) Andrea, A. T.; Cavaiki, R. R.; Goldfine, I. D.; Jorgenaen. C. J. Biochemstry 1980, 19, 55-63. (20) Ramsdsn, D. B.; Sheppard, M.C.; Hoffenberg. R. In Frw Mwmones in Blood; Abertini. A., Eklns, R. P., Eds.; Elsevler Biomedical Press: A m sterdam, 1982; pp 187-193. (21) Liewendahl K.; Tikanoja, S.; Helenius, T.; Valimiki.
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RECEIVED for review December 10,1990. Revised manuscript received July 9, 1991. Accepted August 13, 1991.
A High-Performance Liquid Chromatography System with an Immobilized Enzyme Reactor for Detection of Hydrophilic Organic Peroxides Hans-Hagen Kurth,* Siegmar Gab, Walter V.T u r n e r , a n d Antonius Kettrup
GSF-Znatitut fur Okologische Chemie, Schulstrasse 10, 8050 Freising-Attaching, Federal Republic of Germany
A short reactor column containing horseradlsh peroxidase bnmob"d on contrdled-pore glass can replace the continuous flow of a solution of the enzyme In the detection of hydroperoxldes In an HPLC system, thereby allowing the dMnatkn of one ofthe three punps prevbudy required. The Immobllized enzyme catalyzes the oxidation by hydroperoxides of (phydroxypheny1)acetlc acid to a fluorescent blphenyl ckrlvaihe, whkh k the speck actually detected. The rimpiifled HPLC system is optlmlzed for the analysis of H202 and a number of alkyl and 1-hydroxyalkyl hydroperoxldes. The detection limn of the H,02 analyds is 5 X lo-' M (34 pg In a 20-pL sample), and the response Is llnear down to at least lo-' M.
3 to 2 the number of pumps necessary for the HPLC analysis of hydroperoxides, to considerable economic advantage. In making an enzyme reactor, the enzyme-immobilization strategy is crucial. The application of peroxidase as a marker in immunoassays (4) and in FIA systems has led to the development of a variety of ways to attach the enzyme to both synthetic and biological supports (5). We selected a method developed by Nakane and Kawaoi (6) for coupling peroxidase to biological materials and extended by Hayashi et al. (7) to coupling to controlled-pore glass (CPG). In this .method carbohydrate side chains of the glycoprotein peroxidase are oxidized with periodate to aldehyde groups, which react with aminopropyl groups of the CPG. Since we followed the scheme of Hayashi et al. with little change, we are concentrating in this report on the optimization of the HPLC system with the enzyme reactor.
INTRODUCTION
EXPERIMENTAL SECTION Apparatus (Figure 1). Two Gilson Model 302 pumps were utilized for the HPLC system, the eluent pump being connected with a Gilson Model 802 C manometer and a Rheodyne Model 7125 injection valve with a 20-fiL sample loop. The fluorescence detector and integrator were Hewlett Packard Models HP 1046 A (Aex = 285 nm, A,, = 410 nm) and HP 3390, respectively. The column, 250 mm X 4 mm i.d., was packed with 5-pm Shandon ODS Hypersil and surrounded by a circulating mixture of water/methanol cooled to 1 "C by a Lauda Model K2R cryostat. All connections and the 2-m reaction coil were of stainless-steel capillaries, 1/16 in. o.d., 0.12mm i.d. An HPLC column 17 mm x 4 mm i.d. was used for the enzyme reactor. The eluent flow rate was optimized at 0.5 mL/min, and that of the reagent at 0.4 mL/min (see text). Reagents. Sodium borohydride and aminopropyl-CPGbeads (mean pore size 1400 A, 120-200 mesh) were purchased from Riedel-de Haen (Seelze, Germany) and Serva (Heidelberg, Germany), respectively. Horseradish peroxidase (EC 1.11.1.7)was from Merck (Darmstadt, Germany); all other chemicals were of reagent grade and were also from Merck. Water was deionized, distilled from KMnO,, redistilled and stored in glass bottles protected from light. For use as the mobile phase it wae brought to pH 3.5 with H3P01. The reagent solution was optimized (see
Our research group recently described an HPLC system for the quantitative analysis of HzOz and hydrophilic organic peroxides ( I ) . The peroxides are separated on a cooled RP18 column with dilute H3P04(pH 3.5) as eluent and then allowed to react with peroxidase and (p-hydroxypheny1)acetic acid (PHOPA). This postcolumn reaction is specific for hydroperoxides, and the product is detected by its fluorescence (2). The great advantages of the system are the selectivity afforded by the enzyme reaction and the sensitivity of fluorescence detection. The detection limit of around 5 x lo* M peroxide means that the method is excellently suited for determining H202 and hydrophilic organic peroxides from air, in precipitation and in laboratory simulations of natural peroxideforming processes without a preconcentration step. Enzyme reactors in flow injection analysis (FIA) systems are routine nowadays, but their application to HPLC is an active area of research (3). The present work was undertaken to ascertain the characteristics of the HPLC system when a reactor containing immobilized peroxidase is substituted for the continuous addition of peroxidase to the eluate. The use of such a peroxidase reactor would allow us to reduce from 0003-2700/91/0363-2586$02.50/0
0 1991 American Chemical Society
