Journal of Immunological Methods, 131 (1990) 91-98 Elsevier

91

JIM 05625

A simple theoretical treatment of a competitive enzyme-linked immunosorbent assay (ELISA) and its application to the detection of human blood group antigens V. C h a p m a n , S.M. Fletcher * and M.N. Jones Department of Biochemistry and Molecular Biology, Biomolecular Organisation and Membrane Technology Group, Department of Biochemistry and Molecular Biology, School of Biological Sciences, University of Manchester, Manchester M13 9PT, U.K. (Received 20 November 1989, revised received 23 February 1990, accepted 3 April 1990)

A theory has been developed to explain the behaviour of a competitive enzyme-linked immunosorbent assay (ELISA) in which an immobilized antigen competes with a liquid-phase antigen for a limiting amount of antibody. The binding of antibody to antigen on a solid surface (microtitre well) is described in terms of Langmuirian adsorption with a binding constant k. Two equations are presented to describe the behaviour of the ELISA signal as a function of competing antigen concentration; an exact equation and an approximate equation which can be used when the surface coverage of the immobilized antigen is not known. It is shown how curves of ELISA signal vs. competing antigen concentration depend on K/k[antibody]. The theory has been tested using several immobilized blood group A antigens competing with ovarian cyst fluid A substance and found to adequately describe these competitive ELISAs which have a detection limit of approximately 1 ng of blood group antigen. Key words: ELISA; Competitive assay; Theoretical model; Ovarian cyst glycoprotein blood group activity; Human blood group antigen

Introduction

Enzyme-linked immunosorbent assays (ELISA) have been very broadly divided into the competitive and the non-competitive (Wood, 1982) and a variety of ELISA procedures for carrying out both types of assay have been described (Schuurs and Van Weemen, 1977; Voller et al., 1978; Clark and

Correspondence to: M.N. Jones, Department of Biochemistry and Molecular Biology, Biomolecular Organisation and Membrane Technology Group, School of Biological Sciences, University of Manchester, Manchester M13 9PT, U.K. * Present address: Central Research Establishment, Home Office Forensic Science Service, Aldermaston, Reading, Berksline RG7 4PN, U.K.

Engvall, 1980; Voller, 1980, Wehmeger et al., 1985). One of the simplest competitive ELISA methods for antigen assay involves the adsorption of an antigen on a microtitre well and after appropriate washing and blocking for non-specific binding the 'unknown' free antigen is allowed to compete for a limited amount of antibody with the adsorbed (immobilized) antigen. At equilibrium the residual free antibody-antigen complex is removed and the immobihzed antigen-antibody complex is assayed with a suitable enzymeantibody conjugate. The procedure is illustrated in Fig. 1. The inhibition of the signal by increasing concentrations of competing 'unknown' antigen usually follows a sigmoidal curve; the relative location of the curve with respect to competing 'unknown' antigen concentration will reflect the

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92

• A d s o r b e d Antigen Primary A n t i b o d y '~ Competing A n t i g e n ,~Anti body £onjugate

Fig. 1. A competitiveELISAsystem. magnitudes of the binding constants between the free 'unknown' antigen and antibody and the immobilized antigen-antibody. The limits of detection of the antigen to be assayed will thus be dependent on the relative binding constants as well as the extent of adsorption of the antigen on the surface of the microtitre wells. In order to investigate the factors which determine the characteristics and sensitivity of such a competitive ELISA a simple theoretical model has been formulated based on the equilibria between the binding of the antibody to immobilized antigen and the binding to the free 'unknown' antigen. The theory has been applied to the results of a number of competitive ELISA systems for the detection of human blood group antigens.

Theory The adsorption of antibody (Ab) to antigen (Ag) immobilized on a solid surface is assumed to obey the Langmuir isotherm (Langmuir, 1916). If the moles of antigen binding sites per unit area of the surface (the microtitre well) is n and the binding constant between antibody and immobilized antigen is k, then the surface concentration of adsorbed antibody (~, mol per unit area) is given by nk[Abl P= 1 + k[Ab]

(1)

where [Ab] is the free (solution) concentration of antibody at equilibrium. The use of equation (1) assumes that the antibodies are binding at a single binding site as depicted in Fig. 1. If the antibodies used two sites on the immobilized antigen then n in equation (1) would become n/2 but such a change would not

affect the form of the equation or the manipulations described below. The form of the equation would also be unchanged if antibody bound by one binding site and bound competing antigen at the other site. In practice, as discussed below the solid surface is very far from being saturated with antigen ( - 3 % of saturation) and very far from saturation with antibody so that binding by one antibody binding site (as in Fig. 1) will be predominant. In the presence of a liquid phase competing antigen (Ag c) the above adsorption equilibrium will be disturbed by the formation of equilibria between antibody and Agc. In the most general case with a divalent antibody and an oligovalent antigen the problem is one of multiple equilibria. The mathematics of such antibody-antigen equilibria have been discussed by Steward and Steensgaard (1983). Interactions between oligovalent antigens and antibodies in which all the antigenic determinants are alike are defined as type O / a and lead to the formation of a distribution of antigen-antibody complexes of general form AgiAbj where i and j represent the numbers of antigen and antibody molecules in the complexes respectively. If it is assumed that the association constants for all the equilibria are identical it is possible to estimate the relative concentrations of all possible species provided the valency of the antigen is defined. Such calculations show that for a system in which the antibody and antigen concentrations are comparable (as required in a competition assay) the predominant species are free antigen and antibody and the 1:1 complex. For example using the Goldberg theory Steward and Steensgaard (1983) calculate for a pentavalent antigen with equal numbers of antibody and antigen molecules present the free concentrations are 28% and 35% respectively and the 1 : 1 complex incorporates 22% of the total antibody and antigen. Thus 50% of the antibody molecules and 57% of the antigen molecules are represented by the equilibrium between free molecules and the 1 : 1 complex. The remainder of the molecules are distributed over a large number of other associations of variable complexity each of which on average incorporate less than 2% of the total antibody and antigen. It is clearly not possible to obtain a precise distribution of the species present in such

93 complex systems even if the antigen valency is precisely known and it is assumed that the association constants are all identical. However given the species distributions as estimated by the methods described by Steward and Steensgaard (1983) it seems reasonable to assume that to a first approximation the antibody-antigen interactions can be represented by the predominant equilibrium for the formation of the 1 : 1 complex. Furthermore it has been argued that on a solid surface interactions of antibody with immobilized antigen can be simplified to a biomolecular reaction scheme since the overall forward and reverse rates are limited by one particular rate constant (Stenberg and Nygren, 1988). Thus considering the formation of the 1 : 1 complex according to the reaction. Ab + Agc ~ Ab - Agc

v

A

[Ab]-r k(n_v)(l+K[AgC])+v--~

(5)

Manipulation of equation (5) gives a quadratic equation for v, p 2 __

Br + C = 0

(6)

where 1 v

v

B = ~- -~-(1 + K [Age]) + n + ~- [Ab]T v c = n ~- [Ab]T

(2)

the equilibrium constant may be written K = [Ab-Ag¢] [Abl[Ag¢l

pressed in grams per unit volume and r is in grams per unit area (see below). Both [Ab] and [Ab - Ag c] can be eliminated in equation (4) by use of equation (1) and (3) to give

(3)

A necessary physical condition for the solution of equation (6) is that v < n. Numerical solution of equation (6) with appropriate parameters showed that the root 1

It is assumed that at the low concentrations involved the solutions are ideally dilute (i.e., activity coefficients are unity) and that the competing antigen does not adsorb to the surface. In practice the blocking of adsorption sites on the surface after antigen immobilization should ensure that no competing antigen is immobilized. The ELISA 'signal' will be directly proportional to r since once the balance between antib o d y adsorption according to equation (1) and equilibrium (2) has been reached, the solution phase is removed leaving only adsorbed antibody (v) to bind the antibody-enzyme conjugate. At equilibrium, since the total amount of antibody in the system is constant, material balance requires that [Ab]TV = [Ab]V + [Ab - Agc] V + vA

B - ( B2-4C) ~ v

2

(7)

is required. In practice the E L I S A 'signal' is compared with that obtained in the absence of competing antigen. Defining P0 as the value of v in the absence of competing antigen the relative signal at a given competing antigen concentration is given by !

v

B - ( B 2- 4 C ) ~

Vo Bo_( B2_4C)½

(8)

where B0 = ~-(~1 + [AbT]) + n

(4)

where [Ab]T is the total antibody concentration (immobilized and in solution) in a microtitre well, volume V, and A is the area of the well exposed to solution when the well contains a volume V. The units of the terms in equation (4) are moles or more conveniently grams if concentrations are ex-

The plot of relative signal v / v o as a function of log [Ag ¢] will be sigmoidal and the slope at any point can be obtained from the differential o f equation (7) with respect to B to give dv d-B =

1 2

B 2(B 2 _4C) -~2

(9)

94

and the differential of B with respect to [Ag c] from equation (6) to give dB d[Ag c ]

KV kA

value of log [Ag c] when v/v o = 0.5 is dependent on K/k. These properties of equation (12) are illustrated in Fig. 2. Fig. 2a shows some theoretical plots of v/v o vs. log [Ag c] for a range of values of the ratio of equilibrium constant K to adsorption constant k from 10 to 104. In these calculations the value of VIA was taken as 9.0818 x 10 -4 m (see experimental section), the total antibody concentration, [Ab]T as 10 g. m -3 and n as 0.002 g. m -2. The latter figure was based on literature values for protein adsorption on ELISA wells (Cantarero et al., 1980; Sorensen and Brodbeck, 1986). The values of K/k[Ab]T range from 1 to 10 3 and the larger the value of this parameter the lower the detection limit for competing antigen. The dependence of the log [AgC]at the points of inflection of the curves is a linear function of log K/k. It should be noted that although the numerical value of d(v/vo)/d log lAg c] at v/v o = 0.5 is independent of the value of K / k , the maximum (negative) value of d(v/Vo)/d[Ag c] is markedly dependent on K / k (Fig. 2b) because this derivative depends on [Age]at the point of inflection. The application of equation (8) can be simplified by binomial expansion of the square root terms provided 2C/B 2 > n A / V . _._v= 1 + k[Ab]T 1,0 1 + k [ A b ] T + K [ A g ~]

(15)

This approximate equation has the desired limits i.e. when [Ag c] = 0, v / v o = 1 and when [Ag ~] ~ o0, p / v o ~ O.

Experimental Materials Purified A substance from ovarian cyst fluid (OCF-A) was kindly supplied by Professor W.M. Watkins, M.R.C. Clinical Research Centre, Harrow, U.K. The properties of this material have been previously described (Kabat, 1956; Watkins, 1972; Pigman, 1977). A-Hepta-BSA (blood group A-BSA conjugate) product no. 60/27 was from BioCarb Chemicals (Sweden). Mouse anti-human A antibody was from Biscot (Scotland) and BioClone anti-human A antibody was from Ortho Diagnostic Systems (U.S.A.). Alkaline phosphatase antibody-enzyme conjugate was goat anti-mouse IgM product no. A-7784 and phosphatase substrate tablets product no. 104-105 were from Sigma Chemical Company (London). Bovine serum albumin (fraction V) and 1-0-n-octyl-fl-Oglucopyranoside (OBG) were from Boehringer Mannheim (U.K.). All other reagents were of analytical grade and solutions were made up in double distilled water. Packed blood cells were kindly supplied by the Manchester Regional Blood Transfusion Centre (U.K.). Competition E L I S A Antigen solutions (ovarian cyst fluid A substance (OCF-A), A-BSA or blood extract) were

added to microtitre plate wells and incubated at 37°C for 2 h. The amounts of antigen per well were 20 ng (OCF-A), 200 ng (A-BSA) or 1/10 dilution of blood extract (see below) in a total volume of 200/sl of phosphate-buffered saline pH 7.4 (PBS). After antigen adsorption the wells were washed three times with PBS and non-specific binding blocked by addition of 1% (w/v) BSA for 30 min at room temperature followed by one wash with PBS. The antibody (50 btl), antibody-enzyme conjugate (50/xl) and competing antigen solutions (100/~1) over an appropriate concentration range, were added to the wells and incubated at 37 °C for 2 h after which the wells were washed five times with PBS. Substrate solution (200 #1 of a solution of 5 mg phosphatase in 10 ml of diethanolamine buffer) was added and the microtitre plate was incubated for 90 min at 37 ° C. The absorbance was measured at 410 nm using a Dynatech (MR 610) plate reader and processed with an Apple IIe microcomputer. A similar procedure was used for the direct (non-competitive) ELISA except that antigens were added in a total volume of 100/~1 and no competing antigen was present. Preparation of blood stain extracts Blood stains were prepared by spotting 10/~1 of whole blood on clean boiled cotton cloth. The stains were allowed to dry at room temperature for 3 days and stored in polythene bags in the dark. The stains were cut out and put into 400/tl of 0.5% (w/v) OBG in 0.2 M Tris (pH 10.6) in Eppendorf tubes and vortexed for 30 rain at room temperature. The resulting extracts were frozen overnight a t - 2 1 ° C ; after thawing they were diluted with PBS as required for the competition ELISA. The surface area of microtitre wells The geometric surface area of microtitre wells exposed to solution when the wells contained 200 /L1 aliquots was determined by the addition of a coloured dye solution (Coomassie blue) and measuring the width and height of the solution in a number of wells using a cathetometer. The geometric surface area was found to be 2.2022 ( + 0 . 0 1 0 9 ) X 1 0 - 4 m 2 which gives a volume to surface area of 9.0818 × 1 0 - 4 m when the wells contain 200 gl of solution.

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Protein assay and antigen adsorption

TABLE I

The protein content of antibodies and blood stain extracts was determined by the method of Bradford (1976). Estimates of the adsorption of antigen on the microtitre plate wells were made by first preparing a calibration curve of the ELISA signal in a direct ELISA as a function of the solution concentration of antigen in the well. Antigen was then allowed to adsorb to microtitre wells from a solution of antigen at the same concentration as used in the competitive ELISA under the same conditions (2 h at 37°C). After adsorption the supernatant was removed and aliquots used in a further direct ELISA. From the observed signal and the calibration graph, the solution concentration before and after adsorption and the surface area of the microtitre wells (as determined above) the surface coverage of antigen on the microtitre wells could be estimated. In four independent experiments this method gave 17 + 7% adsorption from 200/~1 of solution containing 20 ng OCF-A which corresponds to a surface coverage of approximately 2 × 10 -5 g . m-=. Under these conditions adsorption is well below saturation coverage which occurs at much higher solution concentration of OCF-A ( - 1 0 # g / 2 0 0 /xl).

P A R A M E T E R S D E T E R M I N E D BY N O N - L I N E A R REG R E S S I O N A N A L Y S I S F R O M C O M P E T I T I V E ELISA O F I M M O B I L I Z E D O V A R I A N CYST F L U I D A - S U B S T A N C E W I T H BIOSCOT A N T I - H U M A N A A N T I B O D Y

Results and discussion

In order to assess the validity of equation (8) a series of competitive ELISAs were carried out using immobilized OCF-A and competing OCF-A with an anti-human A antibody (Bioscot). Equation (8) contains three unknown parameters k, K and [Ab]x and two variables (V/Vo) and [Age]. Because of the mathematical form of the equation attempts to fit data with three unknowns were not successful as it is only the ratio of parameters K/k[Ab]T which determines the fit and not the absolute values of the individual parameters. This arises because equation (8) is a ratio of similar terms. Data can, however, be statistically fitted by assigning an arbitrary value to one of the parameters e.g. k = 100 m 3- g-1 and determining K and [Ab]T. This was done using the non-linear regression algorithm 'Multifit' of Walmsley and Lowe (1985). This procedure gives the best statistical fit

V/A =

Fitted with k = 1 0 0 ma.g -1, 1.6)