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\centerline{METHOD FOR DETERMINING THE AFFINITY OF MONOCLONAL ANTIBODY}
\centerline{USING NON-COMPETITIVE ELISA : A COMPUTER PROGRAM}
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\parindent 0em
\centerline{G. P. S. Raghava\footnote{*}{\rm {\bf Correspondence to:} G. P. S. Raghava,
Scientist, Computer Center, IMTECH, Post Box No.~1304,
Sector 39A, Chandigarh 160 014, India. Phone/Fax (+91 172) 44252,
Email: raghava@imtech.ernet.in } and Javed N. Agrewala}
\vskip 0.1in
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\centerline{Institute of Microbial Technology,  Post Box No. 1304}
\centerline{Sector 39A, Chandigarh 160 014, India.}
\parindent 2.5em
\bigskip\centerline{\bf ABSTRACT}

A simple and reliable method based upon law of mass action for calculating
affinity of a monoclonal antibody using non-competitive ELISA, is described.
In this method, the binding of an antibody (Ab) with an antigen (Ag) is measured
by ELISA using serial dilutions of both antigen (coated on the plate) as well
as antibody. When the OD measured after the antigen antibody interaction was
plotted against the concentration of Ab, added to the wells, a hyperbolic curve
was obtained. The OD, at any point of the curve, was considered as a direct reflection of
the amount of antibody bound to the antigen. The OD-100 denotes the occupancy of
maximum no. of epitopes available on the antigen molecules, accessible to the
antigen. The concentration of antibody (Ab, Ab$'$) at corresponding levels of
antigen concentration (Ag, Ag$'$), presents the value obtained at OD-50.
The [Ag] and [Ag$'$] are not the true antigen concentrations but are the
measurement of antigen density on the plate. The affinity constant $K_{aff}$ was calculated by
using the  formula $K_{aff}$ = (n - 1)/2(n[Ab$'$] - [Ab]), derived from
law of mass action,  where n = [Ag]/[Ag$'$]. A computer program to calculate
the affinity of antibody to the antigen using method  described in this manuscript has been
developed and discussed.

\vskip0.1in
\noindent{\bf Key words:} Affinity constant; Monoclonal antibody;
Non-competitive; ELISA; Computer program

\baselineskip=2.00\normalbaselineskip

\bigskip\centerline{\bf INTRODUCTION}

Affinity constant is one of the most important parameter for determining the
strength of an antibody-antigen interaction. Several competition methods
are available in the literature for calculating affinity of the antibodies and
prominent amongst them are radioimmunoassay (RIA) [1,2] and  enzyme-linked
immunosorbent assay (ELISA) [3,4]. ELISA, is preferred for measuring
antibody concentrations as it is relatively easier and does not employ the use of
radioactive isotopes. Numerous reports in the past have shown that affinity
plays a significant role in the quantification of antibodies [5,6]. However,  the main
problem in determining the affinity constant of antibody by competitive ELISA,
is fluid phase equilibrium disruption by solid phase [7].


A method for measuring affinity of a monoclonal  antibody by using
\-non-comp\-etitive ELISA has been previously described by Beatty, {\it et al.}
[8]. In this method a new equation  was derived based upon the law of mass action for
calculating affinity of antibody from ELISA data. Beatty {\it et al.} [8,9] estimated
the maximum OD (OD-100), and antibody concentration at OD-50 (half of OD-100)
by fitting ELISA data in sigmoid curve (logit-log), for calculating the
affinity. Subsequently, it was shown that the ELISA data can be fitted better by
hyperbolic curve as compared to the sigmoid curve [10,11]. Recently, we have demonstrated that accuracy of antibody concentration
measured from ELISA data may be further  increased by using a graphical method [12].
In the graphical method ELISA data is fitted by linear regression in semilogarithmic
linear range (Sl-range), and to fit the data beyond Sl-range the
hyperbolic curve fitting method is used. Thus, in the light of observations made
in ref. 10, 11, 12, their is need to modify the method described by Beatty
{\it et al.} [8].

In the present paper, we describe a method for calculating the affinity of
antibody by non-competitive ELISA. Our method is similar to that of Beatty
{\it et al.} [8], except that we have calculated OD-100 and derived antibody
concentrations at OD-50 for various concentrations of the antigen, by using
graphical method[12, 13], which is more accurate for determining the
affinity. A computer program has also been developed which calculates 
the affinity of antibody to antigen by using non-competitive ELISA, described in
this contribution.

\vfill\eject

\bigskip\centerline{\bf MATERIALS AND METHODS }

\bigskip\noindent{ELISA Reagents}

The MAb GORK was kindly provided by Erik Wiersma, University of Uppsala
Biomedical Center. The ovalbumin (OVA), bovine serum albumin (BSA) and
substrate orthophenylene diamine (OPD) were obtained from Sigma, St.~Louis, USA.
Trinitrophenyl (TNP) and peroxidase conjugated antibody were obtained from
Pierce and Sera Lab (Sussex, England), respectively.

\bigskip\noindent{ELISA Method}

2.4 ml of TNP-OVA was prepared  in \-carbonate-bi\-carbonate buffer
(0.05M,pH 9.6) at a concentration of 625 ng/ml. A series of 4 doubling
dilutions were made so that the final concentration was 1/8 of the starting one.
In a 96-well microtiter plate (costar), rows A, B, C and D were coated with
100 $\mu$l antigen of concentrations 625 ng/ml, 312.5 ng/ml, 156.75 ng/ml and 78.37 ng/ml
respectively. 100 $\mu$l of OVA (625 ng/ml) was added in row E, this row
worked as a control. After overnight incubation at 4$^\circ$C, the plates
were washed three times with PBS(0.01M,pH 7.2)-Tween-20 buffer.
The remaining protein binding sites of the wells were saturated with
150 $\mu$l/well of 2\% (w/v) bovine serum albumin (Sigma) for 2 h at
37 $^\circ$C. The plates were again washed with PBS-Tween-20 buffer.


MAb GORK against 2,4,6-trinitrophenyl was prepared at an initial
concentration of 70 $\mu$g/ml and added in each well of column 1. It was
then serially double diluted and transferred into the remaining wells of
column 2, 3, 4  etc., respectively. The plates were incubated at
37 $^\circ$C for 2 h and then washed as described earlier.
100 $\mu$l/well of peroxidase labeled antibody conjugate (1:1000) diluted in 0.5\%
BSA was added to the plate and incubated for 2 h at 37 $^\circ$C. Assay
wells were then washed 5 times and 100 $\mu$l of freshly prepared substrate
orthophenylene diamine was added. The color was allowed to develop for 20
minutes and the reaction was stopped using 100 $\mu$l 7\% H$_2$SO$_4$.

The optical density (OD) of each well was read at a wavelength of 492 nm using
a MPR-A4 Eurogenetics EIA reader. This reader can be interfaced to the
serial communication port of any computer via RS-232 interface.

\bigskip\centerline{\bf MATHEMATICAL DERIVATIONS}

The assumptions and detailed derivation based on law of mass action is
described earlier by Beatty {\it et al.} [8]. The Law of Mass
Action for two identical antibody binding sites that have no cooperativity
can be expressed as:

$K_{aff} = K_1 = K_2$, Thus

\eqn1{{K} ={{\left[ Ag Ab \right]}\over 2{\left [Ag\right]\left[Ab\right]}}
={2{\left[ Ag_2 Ab \right]}\over {\left [Ag\right]\left[Ag Ab\right]}}}


\noindent{where K is the Ag-Ab affinity constant and [Ag], [Ab], [Ag Ab], [$Ag_2$ Ab]
are the concentrations of free antigen, free antibody having 2[Ab] antigen
binding sites, antibody bound to one antigen and antibody bound to two antigens
respectively. $[Ab]_t, [Ag]_t$ are the total antibody and antigen concentration
in a well and can be defined by the equation:}

\eqn2{{\left[Ab\right]_t} = \left[Ab \right] + \left[Ag Ab \right] + \left[ Ag_2 Ab \right]}

\eqn3{{\left[Ag\right]_t} = \left[Ag \right] + \left[Ag Ab \right] + 2 \left[ Ag_2 Ab \right]}

\noindent{Equ.~1 can be expressed as}

\eqn4{{K\left[Ag\right]} ={2{\left[Ag_2 Ab\right]}\over {\left[Ag Ab\right]}}}

As the quantity of [Ab] in the well increases, the OD approaches a maximum value
OD-100, and free antigen conc. [Ag] approaches to near zero. Thus from equ.~(4), [$Ag_2$ Ab] also
approaches to zero at OD-100 . From equ.~(3) we thus derive that
[Ag]$_t$ = [Ag Ab] at OD-100. The OD directly reflects the amount of antibody
bound to the antigen in the well ([Ag Ab] + [$Ag_2$ Ab]). At 50\% of OD-100,
OD-50, the amount of antibody bound to the antigen in the well is one half
the amount of antibody bound ([Ag Ab] or [Ag]$_t$) at OD-100. Thus, at OD-50:


\eqn5{{\left[Ag\right]_t \over 2} = \left[ Ag Ab \right] + \left[ Ag_2 Ab \right]}

\noindent{By solving equ.~(3) and equ.~(5) }

\eqn6{{\left[ Ag\right]} = \left[ Ag Ab \right] }

\noindent{By solving equ.~(1) and equ.~(6)}

\eqn7{{\left[K_{aff}\right]} = {1\over{2\left[ Ab \right]}}}

\noindent{By solving equ.~(2), equ.~ (3), equ.~(5) and equ.~(7)}

\eqn8{{\left[K_{aff}\right]} = {1\over{\left( 2\left[ Ab \right]_t -
\left[Ag\right]_t\right)}}}

By using two different concentrations of Ag in coating solutions, one [Ag$'$]
being half the other [Ag], we find at OD-50 that:

\eqn9{{\left[ Ag\right]_t} = 4\left(\left[ Ab\right]_t - \left[
Ab'\right]_t\right)}

\noindent{Thus, from equ.~(8) and equ.~(9)}

\eqn{10}{{\left[K_{aff}\right]} = {1\over{2\left( 2\left[ Ab' \right]_t -
\left[Ab\right]_t\right)}}}

\noindent{The general formula is thus}

\eqn{11}{{\left[K_{aff}\right]} = {\left(n - 1\right)\over{2\left( n\left[ Ab'\right]_t -
\left[Ab\right]_t\right)}}}

Equ.~(11) represents the general formula for calculating affinity of monoclonal
antibody to antigen, where n = $[Ag]\over{[Ag']}$ and K$_{aff}$ affinity constant.

\bigskip\noindent{\bf READING AND CALCULATING ELISA DATA}

The OD data were collected using a 96-well microplate reader
(Eurogenetics) and fed to the microcomputer from the keyboard. The computer
can also be directly linked to microplate reader  using RS-232 interface, for
inputting the data.
The program uses the OD data information to complete $8\times 12$ table of
the data points which is then printed on the output sheet.
The OD data was fitted by graphical method for given concentrations of
Ag and maximum OD (OD-100), and concentrations at OD-50 for different Ag
concentrations is described as below.

In graphical method ELISA data is fitted by linear regression
in semilogarithimic linear range (Sl-range) and by hyperbolic curve fitting
method beyond the Sl-range. The detail derivation of graphical method is
described previously by Raghava {\it et al.} [12].

a) The semilogarithmic curve of OD  ${\it vs}$ logarithm of
concentration for given Ag concentration was used for
calculating linear interpolation formula. In this curve, the Sl-range
was depicted, denoting the linear portion of the curve with maximum slope
and data in this range fitted by linear regression [12,13].
Sl-range of the curve was fitted by using following linear equation

\eqn{12}{\hbox{OD} = A_0 + A_1\times \log_{10}\left(C\right)}


Where C, $A_0$ and $A_1$ represent the concentration, constant and slope of
the curve respectively. The value of $A_0$ and $A_1$ was then calculated by
fitting the data in the range using least square curve fitting method.

\eqn{13}{C = 10^{ \left(M \times \hbox{OD} + K \right)}}

Where $K = - {A_0 \over A_1}$ and $M = {1\over A_1}$ are constants.


Equ.~(13) represents the  linear interpolation formula obtained
for given concentration of [Ag] coated on the plate.

b) The Hyperbolic interpolation formula was calculated as below. The detailed
derivation has been described previously [10,11,12]. The hyperbolic equation
can be written as

\eqn{14}{\left( C - X_0\right )\left(\hbox{OD} - Y_0\right) = C_0}


\eqn{15}{C ={C_0\over{\left(\hbox{OD} - Y_0\right)}} + X_0}


\noindent{ where variables C and OD represent antibody concentration and
optical density respectively.
The regression constants which define the curve are $X_0, Y_0$ and $C_0$
represent capacity, (signal+background) and flatness. Constants could be
obtained for N set of standard data points as described below.}

\eqn{16}{A = N\sum{X_i}Y_i - \sum{X_i}\sum{Y_i}}

\eqn{17}{B = N\sum{X_i}X_i - \sum{X_i}\sum{X_i}}

\eqn{18}{C = N\sum{Y_i}Y_i - \sum{Y_i}\sum{Y_i}}

\eqn{19}{D = N\sum{X_i}X_iY_i - \sum{X_i}Y_i\sum{X_i}}

\eqn{20}{E = N\sum{X_i}Y_iY_i - \sum{X_i}Y_i\sum{Y_i}}


These three equations (for $X_0, Y_0$ and $C_0$), along with the definitions
of five constants ( $A$ -- $E$ ) above are sufficient to perform hyperbolic
regression on any experimental data set


\eqn{21}{{X_0} ={{DA - EB}\over {A^2 - CB}}}

\eqn{22}{{Y_0} = {{AE - CD}\over {A^2 - CB}}}

\eqn{23}{{C_0} = {\sum X_iY_i - X_0\sum Y_i - Y_0\sum X_i\over N} + X_0Y_0}

Equ.~(13) represents the hyperbolic interpolation formula, where C is the
concentration of the Ab the values of constants $X_0, Y_0$ and $C_0$ were
calculated by using equ.~(16-- 20). The signal value $Y_0$ (OD-100) was calculated
from equ.~23.



The total Ab concentration at OD-50 was calculated by using equ.~(13),
when OD-50 is in Sl-range, otherwise by using
equ.~(15). Similarly, the total Ab concentration at OD-50, for different Ag
concentrations coated on the plate, were also calculated. Finally, the Ab
affinity  was
calculated by putting the value of Ab concentration at OD-50 in equ.~(11).

\bigskip\centerline{\bf COMPUTER PROGRAM}

The menu-driven computer program for calculating the affinity of monoclonal
antibody from non-competitive ELISA data, has been written in GW-BASIC.
OD data obtained from the Microplate reader can be fed to computer either
directly by interfacing or by using Keyboard and data could be stored for
future use. Before starting the calculation, a `dilution template'
should be defined and the information about the serial dilutions of antibody
added to the wells and antigen coated on the plate  must also be supplied.
The dilution template consists of arrays of dilution factor (DF),
corresponding to the given [Ag] coated on the assay plate. It calculates the
average OD of samples, which are in duplicate, triplicate, etc.

The computer program, 1) Fits the OD data for given Ag concentration by
hyperbolic curve-fitting method and calculates OD-100, OD-50 and also hyperbolic
interpolation formula; 2) It depicts the Sl-range and calculates the  linear
interpolation formula; and 3.) Checks whether OD-50 is in Sl-range and then
calculates accordingly the concentration of [Ab] for a given [Ag]
concentration coated on the plate using equ.~(13) and equ.~(15). In a similar
way the program calculates the [Ab] concentration at OD-50 for other
concentrations of [Ag] coated on the plate. Lastly, it calculates the
antibody affinity by using equ.~(11). It also calculates the capacity and
flatness of hyperbolic curve, which plays a vital role in ELISA optimization.
The program also allows the display and printing of the results. Fig.~1 shows
an example printout from one of such ELISA plate.

\bigskip\centerline{\bf RESULTS}

The affinity constant of MAb GORK was calculated by using equ.~(11) for
different ratios of antigen coated on the plate (n = $[Ag]\over{[Ag']}$).
Based on the curve in Fig.~2

(a) 6.82$\times 10^8$, 4.46$\times 10^8$ and 3.045$\times 10^8$ for antigen
coated ratio n = 2.

(b) 5.04$\times 10^8$ and 3.41$\times 10^8$ for antigen coated ratio n = 3.

(c) 3.67$\times 10^8$ for antigen coated ratio n = 4.

Similarly, a number of experiments were performed for calculating the affinity
constant of MAb GORK for different values of `n'. The affinity constant
calculated by this method was comparable with soluble-phase affinity constants
measured by inhibition RIA (competition RIA) for this antibody [2]. The solid
phase affinity constants calculated by our method  had an excellent correlation coefficient
of 0.98. We also calculated the affinity constant of MAb GORK, by Beatty {\it et al.} [8]
method. The overall results, calculated from our method, a previously described
non-competitive method,  and RIA are summarized in table~I.

The affinity calculated by our method had higher correlation coefficient
0.98 as compared to 0.96 obtained with competition RIA, where
affinity is calculated by Beatty {\it et al.} [8] method. This shows the affinity
calculated by our non-competition ELISA method is more accurate then the
previous non-competitive method. The standard error in affinity calculated by
our method is very less in comparison to the earlier method, (Table~I), showing
thereby that our method is more reliable. The OD-100 calculated in our case is
the representation of full ELISA data where in Beatty {\it et al.} [8]
it was average of upper values only.




\bigskip\centerline{\bf DISCUSSION}

The objective of this study was to describe a simple, reliable
and rapid method for determining affinity of monoclonal antibodies
by using non-competitive ELISA and to develop a computer program using
this strategy. A variety of
competitive methods using RIA [1,2,14] and ELISA [3,4,15,16] for determining
the affinity of the antibody have been described in literature. These
approaches rely on the measurement of bound $vs$ free antigen ratio,
requiring an additional separation step of
free and bound entities. This may intern falsify the analytical results,
especially when equilibrium disturbing procedures are included. Another
problem in determining the affinity constant of antibody by competitive ELISA
method is fluid phase equilibrium disruption by solid phase [7].

 To address some of these issues, a method, which does not require
any purification of reactants or separation steps at equilibrium has been
described earlier [17] for determing the affinity constants of monoclonal
antibodies to enzymes. However, this approach has limitations as it can be used
for calculating the affinity of only those antibodies which are inhibitory in nature.
To understand  the impact  of  affinity
upon  solid-phase and for determining the affinity by non-competitive ELISA
method, Beatty {\it et al.} [8] have derived a new equation from the principle of
law of mass action.
This method compares the OD-50 of
two sigmoid curves of antibody serial dilutions on plate coated  with two
different concentrations of the antigen. In this approach the exact value
of the antigen adsorbed onto the plate are unknown and the relationship of
the antigen adsorptions is therefore assumed. This method [8] for
determining the affinity constant of the antibody, by using non-competitive
ELISA, is an alternative method of competitive methods, described earlier.


In this manuscript, we have utilized the similar approach as described
earlier [8], which calculates the affinity of antibody using
ELISA. However, in our approach contrary to logit-log method used by Beatty
{\it et al.} [8], the ELISA data is fitted by graphical method
for calculating the signal (OD-100) and the total Ab
concentration at OD-50. It has been shown
that the most reliable representation of ELISA data is obtained by graphical method
rather than logit-log method[10,11,12]. In graphical method ELISA data, falling in
Sl-range is fitted by
linear regression while that is falling beyond Sl-range, is fitted by
hyperbolic curve fitting method. The main drawback in the approach of the Beatty
et al. [8] was that it calculates the maximum OD (OD-100) by taking
the average of upper values of OD. It was also assumed in the derivation that
the OD-100 is the maximum value when free concentration of the antigen approaches to
zero. Accordingly, the value calculated by taking average would always be less
than OD-100. The OD-100 calculated is directly affected by upper values
(saturated portion of the curve). It may be pointed here that normally the errors occur
in saturated portion of the curve. Therefore, calculation based on saturated
portion of the curve may not necessarily always reflect the actual data of
the curve. In the present report, the above problems of calculating OD-100
have been eliminated by
fitting the ELISA data by graphical method and the signal (OD-100) of the
curve is calculated, which is the asymptote of hyperbola.  As OD-100
calculated in our method represents the actual ELISA data and the small
errors occurring in the upper portion of the curve will not affect accuracy
significantly in calculating
the  affinity. The accuracy
has further been increased by calculating the antibody concentration at
OD-50 by using graphical method [12].


In conclusion, it may be stated that we have improved the non-competition ELISA method described
earlier by Beatty et al. [8] for calculating the affinity. This method
is ideal for screening and testing
hybridoma products, determining the affinity and avidity.
Further, the data analysis was simplified by a
computer program. The principles employed in this method should have
application in a broad range of immunoassays.


     The computer program was developed to run on an IBM compatible
microcomputer. It is written in the GW-BASIC. The program is simple to
use, has been designed to be run by users
with little knowledge of mathematics or computers. Program provides
the data storage facility on Disk. The Ab$_-$affi program is freely avaliabe from
authors on request. The Authors will prefer to distribute the source code 
via e-mail (contact Raghava@imtech.ernet.in).


\bigskip\centerline{\bf ACKNOWLEDGMENTS}

This is a communication no. 002/92 from Institute of Microbial Technology,
Chandigarh supported by grants from Council of Scientific and Industrial
Research \& Department of Biotechnology, Government of India. The authors
are thankful to Dr.~G. C. Mishra \& Dr. G. C. Varshney for critically
evaluating the manuscript.

\baselineskip=1.15\normalbaselineskip

\bigskip\centerline{\bf REFERENCES}
\def\ref{\hangindent\parindent\noindent}


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\vfill\eject

\bye

