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Allowance for antibody bivalence in the determination of association rate constants by kinetic exclusion assay
Analytical Biochemistry 441 (2013) 214–217
Contents lists available at ScienceDirect
Analytical Biochemistry journal homepage: www.elsevier.com/locate/yabio
Allowance for antibody bivalence in the determination of association rate constants by kinetic exclusion assay Donald J. Winzor ⇑ School of Chemistry and Molecular Biosciences, University of Queensland, Brisbane, Queensland 4072, Australia
a r t i c l e
i n f o
Article history: Received 7 May 2013 Received in revised form 22 June 2013 Accepted 24 June 2013 Available online 12 July 2013 Keywords: Antigen–antibody interaction Antibody bivalence Association rate constant Kinetic exclusion assay Reacted site probability
a b s t r a c t This investigation completes the amendment of theoretical expressions for the characterization of antigen–antibody interactions by kinetic exclusion assay—an endeavor that has been marred by inadequate allowance for the consequences of antibody bivalence in its uptake by the affinity matrix (immobilized antigen) that is used to ascertain the fraction of free antibody sites in a solution with defined total concentrations of antigen and antibody. A simple illustration of reacted site probability considerations in action confirms that the square root of the fluorescence response ratio, RAg/Ro, needs to be taken in order to determine the fraction of unoccupied antibody sites, which is the parameter employed to describe the kinetics of antigen uptake in the mixture of antigen and antibody with defined initial composition. The approximately 2-fold underestimation of the association rate constant (ka) that emanates from the usual practice of omitting the square root factor gives rise to a corresponding overestimate of the equilibrium dissociation constant (Kd)—a situation that is also encountered in the thermodynamic characterization of antigen–antibody interactions by kinetic exclusion assay. Ó 2013 Elsevier Inc. All rights reserved.
Of the chromatographic methods available for the thermodynamic characterization of immunochemical reactions, the kinetic exclusion assay (KinExA)1 affords a direct and hence rapid approach for quantifying the composition of a solution comprising an equilibrium mixture of a univalent antigen, bivalent antibody, and complexes thereof [1–7]. Basically, the procedure entails assessment of the extent of complex formation in such mixtures from the decrease in antibody uptake by an affinity matrix bearing a high concentration of immobilized antigen. However, whereas the difference between the consequent fluorescence response (RAg) and that for the same total concentration of antibody in the absence of antigen (Ro) has been taken to reflect the concentration of antibody sites occupied by antigen [1–7], it actually reflects the concentration of antibody sites present as the fully saturated complex AbAg2 [8]. This leads to a situation where the KinExA results for equilibrium mixtures need to be analyzed according to an expression in which [1–(RAg/Ro)]1/2 describes the rectangular hyperbolic dependence of response on free antigen concentration [8,9]. A similar problem pervades use of the KinExA approach to determine the association rate constant for an antigen–antibody interaction from time dependence of the decrease in RAg/Ro [7,10]. This investigation explores the changes required to convert the current invalid kinetic expression based on antibody univa-
lence into one that takes into account the bivalence of antibody (immunoglobulin G, IgG) in its interaction with immobilized antigen on the affinity matrix. Theoretical considerations In the determination of an association rate constant by KinExA, there are two time courses to consider. First, there is the time course of antibody site occupancy by antigen in the reaction mixture with defined total antigen and antibody concentrations ([Ag]tot and [Ab]tot, respectively). Second, there is the time course of KinExA response, which monitors the decrease in antibody uptake from the mixture by immobilized antigen (Ag⁄) on the affinity matrix. Previous determinations of rate constants by KinExA have entailed the assumption that the ratio of fluorometric responses reflecting matrix-bound antibody for the same total antibody concentration in the presence and absence of antigen reflects the fraction of unoccupied antibody sites in the reaction mixture [7,10]. However, this assumption is untenable because the two 1:1 Ab–Ag complexes still retain affinity for immobilized antigen; only the Ag–Ab–Ag complex exhibits no affinity for the affinity matrix [8]. The two events are considered in turn. Time course of complex formation in reaction mixture
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E-mail address: [email protected] Abbreviation used: KinExA, kinetic exclusion assay.
0003-2697/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ab.2013.06.020
The time course of antigen uptake by antibody in the reacting solution can be described by a 1:1 interaction between antigen
Rate constant determination by KinExA / D.J. Winzor / Anal. Biochem. 441 (2013) 214–217
(presumed univalent) and antibody sites, with the relevant differential rate equation being 0
0
0
0
d½Ab Ag=dt ¼ ka ½Agt ð½Ab tot ½Ab Agt Þ kd ½Ab Agt ;
ð1Þ
0
where [Ab ]tot denotes the total antibody site concentration (= 2[Ab]tot), [Ab0 Ag]t denotes the concentration of complexed antibody sites ([–Ab–Ag] plus [Ag–Ab–]) at time t, and ka and kd are the respective rate constants for complex formation and dissociation. For illustrative purposes, it suffices to consider a situation in which the concentration of antigen in the mixture greatly exceeds that of antibody sites ([Ag]tot >> [Ab0 ]tot)—a simplification that allows the replacement of [Ag]t in Eq. (1) by [Ag]tot for all t. Eq. (1) then becomes a pseudo-first-order kinetic expression, 0
0
0
0
d½Ab Ag=dt ¼ ka ½Agtot ð½Ab tot ½Ab Agt Þ kd ½Ab Agt ;
ð2Þ
which is readily integrated. In the current context, it is convenient to use the following integrated form of the expression for fractional saturation of antibody sites, Fs = [Ab’Ag]t/[Ab0 ]tot,
F s ¼ ðka ½Agtot =kobs Þ½1 expðkobs Þt
ð3aÞ
kobs ¼ ka ½Agtot þ kd ;
ð3bÞ
which is the counterpart of that introduced by O’Shannessy and coworkers [11] for analysis of the adsorption stage of Biacore sensorgrams. The corresponding expression for the fraction of unoccupied antibody sites, therefore, becomes
ð1 F s Þ ¼ 1 ðka ½Agtot =kobs Þ½1 expðkobs tÞ
ð4Þ
Time course of corresponding KinExA response ratio As noted above, the current assumption [7,10] that (RAg)t/Ro can be substituted for (1 Fs) in Eq. (4) is clearly incorrect because the experimental parameter being monitored is the time dependence of [Ag–Ab–Ag]t instead of [Ab0 Ag]t. From reacted site probability theory [12–14], the concentration of AbAg2 is P 2Ab [Ab]tot, where PAb is the probability that an antibody site is occupied by an antigen molecule. On the basis that the fluorometric response ratio is thus monitoring ð1 F 2s Þ, the KinExA equivalent of Eq. (4) becomes 2
½ðRAg Þt =Ro ¼ 1 fðka ½Agtot =kobs Þ½1 expðkobs tÞg
ð5Þ
Although Singer [15] introduced the concept of reacted site probability theory into immunochemistry nearly 50 years ago, immunologists still seem reluctant to accept its validity. Confirmation of the inference that [1 (RAg)t/Ro]1/2 is the experimental counterpart of the fraction of unoccupied antibody sites, [Ab0 ]t/ [Ab0 ]tot, in the reaction mixture has thus been provided by the following illustrative example of its application. Relative contribution of AbAg2 to fractional antibody saturation To illustrate the reacted site probability approach, the proportion of antigen bound as AbAg2 is first determined for each successive antigen attachment in the pathway to saturation of all antibody molecules in the system. For simplicity, it is assumed that ka[Ag]tot kd to allow the disregard of complex dissociation (Eq. (3b)). Consider initially a system with only five bivalent acceptor (antibody) molecules A. For the first attachment of a univalent ligand (antigen) molecule S, there are clearly 10 possible ways of forming a 1:1 complex AS, all of which are equally probable (first line of Table 1). The first opportunity for AS2 formation occurs during attachment of the second ligand molecule. However, such formation of S–A–S is restricted to the attachment of ligand to only 1 of the 9 unoccupied acceptor sites; the alternative distribution
215
with S attached to sites on 2 different acceptor molecules is thus 8 times more likely (Table 1). For 20% acceptor site saturation (2 sites occupied), the probability of AS2 formation is thus 1/9 (i.e., 0.111); this probability, denoted as P(2,2) in Table 1, follows a general terminology P(i,j), where i refers to the total number of ligand molecules bound and j refers to the number of molecules bound as AS2. During the third ligand attachment, there are two more opportunities for AS2 formation by its location on either of the 2 AS molecules in the lower distribution for second ligand attachment. Values of P(3,j) for the three possible ligand distributions are listed in the penultimate column of Table 1. This availability of P(i,j) values for each possible ligand distribution then allows the determination of the fraction of bound antigen present as AS2, Fs(AS2), from the expression
F s ðAS2 Þ ¼
X ðj=iÞPði; jÞ
ð6Þ
which has been used to calculate the value of this parameter listed in the final column of Table 1. The results of similar considerations for the subsequent steps to saturation of the 5 acceptor molecules (10 sites) are summarized in the remaining lines of Table 1. On the grounds that those values of Fs(AS2) refer specifically to the system with 5 acceptor molecules, the whole exercise has been repeated for systems with 10, 15, 20, and 40 molecules of A in order to ascertain the likely variation of this parameter with number of acceptor sites. That variation in Fs(AS2) for 20, 40, 60, and 80% site saturation is shown in Fig.1, where the abscissa is expressed as the reciprocal of the number of acceptor sites to facilitate extrapolation to the ordinate intercept; the value in the limit of an infinite number of acceptor molecules should be more applicable to the experimental situation in KinExA studies, where the antibody concentration (typically 20– 60 pM) requires the consideration of systems with 1013 to 1014 as the order of magnitude for the number of acceptor molecules. Although the extent of that extrapolation is clearly underemphasized in Fig. 1, the results conform with the conclusion that the fraction of ligand (antigen) bound as AS2 is numerically equal to Fs, the fractional saturation of acceptor (antibody) sites. As foreshadowed above, the inference from Fig. 1 that Fs(AS2) is a measure of F 2s is merely an illustrative manifestation of reacted site probability considerations [12–15] that has been included to provide reassurance that the ratio of fluorometric responses in a kinetic experiment is, indeed, monitoring the square of the fractional saturation of antibody sites—the condition incorporated into Eq. (6). Experimental ramifications The consequences of the above theoretical considerations are illustrated by employing Eq. (4) to calculate the time dependence of fractional site saturation at 10-min intervals for a mixture with [Ab]tot = 10 pM, [Ag]tot = 500 pM, ka = 3 106 M–1 s–1, and kd = 1.5 10–4 s–1 (an equilibrium dissociation constant Kd of 50 pM). In that regard, the restriction of [Ab]tot (which does not appear in the calculation) to 10 pM (20-pM sites) ensures a decrease of less than 4% in [Ag]t and thus provides reasonable support for the approximation inherent in the substitution of [Ag]tot for [Ag]t that renders the system amenable to pseudo-first-order kinetic analysis. Delineation of the time course for the decrease in the fraction of unoccupied antibody sites, (1 Fs) = [Ab’]t/[Ab’]tot (shown as open symbols in Fig. 2), allows the calculation of F 2s and hence the corresponding decrease in (RAg)t/Ro (closed symbols in Fig. 2). The current practice [7,10] of regarding (RAg)t/Ro as (1 Fs) rather than (1 F 2s ) clearly leads to underestimation of kobs, or of ka if ka[Ag]tot kd, as well as overestimation of Kd from the time-indepen-
216
Rate constant determination by KinExA / D.J. Winzor / Anal. Biochem. 441 (2013) 214–217
Table 1 Deduction of probability of AS2 formation after each ligand (S) addition to 5 molecules of a bivalent acceptor A.
a b
Possible distributions of species
Probability of distributiona
Value
Fs(AS2)b
1st addition S–A– –A– –A– –A– –A–
P(1,0) = 10/10
2nd addition S–A–S –A– –A– –A– –A– S–A– S–A– –A– –A– –A–
P(2,2) = 1/9 P(2,0) = 8/9
0.1111 0.8889
0.1111
3rd addition S–A–S S–A– –A– –A– –A– S–A– S–A– S–A– –A– –A–
P(3,2) = (8/8)P(2,2) + (2/8)P(2,0) P(3,0) = (6/8)P(2,0)
0.3333 0.6667
0.2222
4th addition S–A–S S–A–S –A– –A– –A– S–A–S S–A– S–A– –A– –A– S–A– S–A– S–A– S–A– –A–
P(4,4) = (1/7)P(3,2) P(4,2) = (6/7)P(3,2) + (3/7)P(3,0) P(4,0) = (4/7)P(3,0)
0.0476 0.5714 0.3810
0.3333
5th addition S–A–S S–A–S S–A– –A– –A– S–A–S S–A– S–A– S–A– –A– S–A– S–A– S–A– S–A– S–A–
P(5,4) = (6/6)P(4,4) + (2/6)P(4,2) P(5,2) = (4/6)P(4,2) + (4/6)P(4,0) P(5,0) = (2/6)P(4,0) 0.1270 0.4444
0.2381 0.6349
6th addition S–A–S S–A–S S–A–S –A– –A– S–A–S S–A–S S–A– S–A– –A– S–A–S S–A– S–A– S–A– S–A–
P(6,6) = (1/5)P(5,4) 0.0476 P(6,4) = (4/5)P(5,4) + (3/5)P(5,2) P(6,2) = (2/5)P(5,2) + (5/5)P(5,0)
0.5714 0.3810 0.5556
7th addition S–A–S S–A–S S–A–S S–A– –A– S–A–S S–A–S S–A–S–A– S–A–
P(7,6) = (4/4)P(6,6) + (2/4)P(6,4) P(7,4) = (2/4)P(6,4) + (4/4)P(6,2)
0.3333 0.6667
0.6667
8th addition S–A–S S–A–S S–A–S S–A–S –A– S–A–S S–A–S S–A–S S–A– S–A–
P(8,8) = (1/3)P(7,6) P(8,6) = (2/3)P(7,6) + (3/3)P(7,4)
0.1111 0.8889
0.7778
9th addition S–A–S S–A–S S–A–S S–A–S S–A–
P(9,8) = (2/2)P(8,8) + (2/2)P(8,6)
1.0000
0.8889
10th addition S–A–S S–A–S S–A–S S–A–S S–A–S
P(10,10) = (1/1)P(9,8)
1.0000
1.0000
P(i,j) denotes the probability that j of the i bound ligand molecules are present as AS2. Calculated from Eq. (6).
Fig.1. Dependence of the fraction of ligand (antigen) bound as AS2, Fs(AS2), on the reciprocal of the number of acceptor (antibody) sites used to calculate the most probable distribution of species (A, AS, AS2) at each successive ligand addition. Lines denote the linear extrapolation of Fs(AS2) values for the indicated extents of acceptor site saturation (Fs) to demonstrate the identity of Fs and Fs(AS2) for mixtures with an experimentally realistic number of acceptor molecules.
dent asymptote [8]. For example, nonlinear least squares analysis of the time dependence of (RAg)t/Ro ( , Fig. 2) in terms of Eq. (4) instead of Eq. (5) yields values (± standard deviations [SD]) of 1.77 (± 0.08) 106 M–1 s–1 for ka and 1.06 (± 0.05) 10–3 s–1 for kobs, which on substitution into Eq. (3b) gives rise to a kd of 1.77 (± 0.09) 104 s1. The consequent apparent value of 100 (±10) pM for Kd, which encompasses that of 102 pM obtained from the time-independent asymptote in Fig. 2, thus overestimates the input equilibrium constant by a factor of approximately 2—the con-
Fig.2. Time dependence of the fraction of unoccupied antibody sites (e), (1 Fs), calculated by means of Eq. (4) for an antigen–antibody mixture with [Ag]tot = 500 pM and [Ab]tot = 10 pM for which complex formation is governed by respective association and dissociation rate constants of 3 106 M–1 s–1 and 1.5 10–4 s–1 (an equilibrium dissociation constant of 50 pM). Closed symbols ( ) denote the corresponding dependence of fluorometric response ratio, (RAg)t/Ro, calculated from Eq. (5), with additional points having been used to ascertain the form of the dependence during the first 10 min.
clusion reached previously from considerations of equilibrium KinExA data [8]. In an experimental context, the time dependence of (RAg)t/Ro is the starting point for rate constant determination; but the expressions incorporated into the KinExA software are in terms of [Ab0 ]t/ [Ab]tot because of the above-mentioned misinterpretation of the fluorometric response ratio. Rectification of this mistake merely entails the substitution of [1 (RAg)t/Ro]1/2 for [1 (RAg)t/Ro] in
Rate constant determination by KinExA / D.J. Winzor / Anal. Biochem. 441 (2013) 214–217
the KinExA program, which can accommodate not only the pseudo-first-order kinetic situation considered above ([Ag]t [Ag]tot) but also the more general second-order kinetic situation where allowance must be made for the progressive decrease in free antigen concentration from [Ag]tot to [Ag]e, its equilibrium value associated with the limiting value of (RAg)t/Ro as t ? 1. However, such allowance for variation in [Ag]t is at the expense of that for reversibility of complex formation; it necessitates the assumption that kd ka[Ag]t, which renders the reaction essentially irreversible over the time scale of the experiment. The magnitude of kd can then be determined as the product of ka and the equilibrium constant Kd [7,10]. Discussion This investigation completes the amendment of theoretical expressions for the characterization of antigen–antibody interactions by kinetic exclusion assay—an endeavor that has been marred previously by inadequate allowance for the consequences of antibody bivalence in its uptake by the affinity matrix (immobilized antigen Ag⁄) that is used to ascertain the fraction of free antibody sites in a solution with defined total concentrations of antigen and antibody [1–7,10]. The fact that the same amendment, the substitution of (1 RAg/Ro)1/2 for (1 RAg/Ro), applies to kinetic as well as thermodynamic [8] characterizations of immunochemical reactions by kinetic exclusion assay is hardly surprising in that reacted site probability considerations [12–14] apply to both. In that regard, it is hoped that the current illustration of reacted site probability theory in action (Table 1 and Fig. 1) may help to convince immunochemists of its relevance to the characterization of antigen–antibody interactions. An obvious advantage of the KinExA approach to studying antigen–antibody reactions is its characterization of the interaction between the two reactants in solution, with this being an attribute that is not shared by conventional Biacore methodology, which quantifies the interaction between one reactant and a chemically modified (immobilized) form of the other [16–18]. This deficiency of conventional Biacore protocol has been overcome [17] by according the biosensor surface the same role as that played by the affinity matrix in a KinExA, namely, a means of monitoring the free concentration of one reactant in a solution with defined total concentrations of antigen and antibody. Furthermore, by incorporating theory reported by Stevens [19], the Swiss group [17] clearly recognized the need to allow for bivalence of the antibody when it was the partitioning solute—a factor overlooked throughout the development of KinExA methodology despite the existence of many publications that had already overcome the ligand multivalence problem [20–26]. Fortunately, the current invalidity of KinExA programs for the kinetic and thermodynamic characterization of antigen–antibody interactions can be rectified by minor amendment of equations within the existing software that would then justify consideration of the KinExA as the procedure of choice for the rapid and accurate characterization of antigen–antibody interactions.
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[16]
[17]
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[22]
[23]
[24]
[25]
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Contents lists available at ScienceDirect
Analytical Biochemistry journal homepage: www.elsevier.com/locate/yabio
Allowance for antibody bivalence in the determination of association rate constants by kinetic exclusion assay Donald J. Winzor ⇑ School of Chemistry and Molecular Biosciences, University of Queensland, Brisbane, Queensland 4072, Australia
a r t i c l e
i n f o
Article history: Received 7 May 2013 Received in revised form 22 June 2013 Accepted 24 June 2013 Available online 12 July 2013 Keywords: Antigen–antibody interaction Antibody bivalence Association rate constant Kinetic exclusion assay Reacted site probability
a b s t r a c t This investigation completes the amendment of theoretical expressions for the characterization of antigen–antibody interactions by kinetic exclusion assay—an endeavor that has been marred by inadequate allowance for the consequences of antibody bivalence in its uptake by the affinity matrix (immobilized antigen) that is used to ascertain the fraction of free antibody sites in a solution with defined total concentrations of antigen and antibody. A simple illustration of reacted site probability considerations in action confirms that the square root of the fluorescence response ratio, RAg/Ro, needs to be taken in order to determine the fraction of unoccupied antibody sites, which is the parameter employed to describe the kinetics of antigen uptake in the mixture of antigen and antibody with defined initial composition. The approximately 2-fold underestimation of the association rate constant (ka) that emanates from the usual practice of omitting the square root factor gives rise to a corresponding overestimate of the equilibrium dissociation constant (Kd)—a situation that is also encountered in the thermodynamic characterization of antigen–antibody interactions by kinetic exclusion assay. Ó 2013 Elsevier Inc. All rights reserved.
Of the chromatographic methods available for the thermodynamic characterization of immunochemical reactions, the kinetic exclusion assay (KinExA)1 affords a direct and hence rapid approach for quantifying the composition of a solution comprising an equilibrium mixture of a univalent antigen, bivalent antibody, and complexes thereof [1–7]. Basically, the procedure entails assessment of the extent of complex formation in such mixtures from the decrease in antibody uptake by an affinity matrix bearing a high concentration of immobilized antigen. However, whereas the difference between the consequent fluorescence response (RAg) and that for the same total concentration of antibody in the absence of antigen (Ro) has been taken to reflect the concentration of antibody sites occupied by antigen [1–7], it actually reflects the concentration of antibody sites present as the fully saturated complex AbAg2 [8]. This leads to a situation where the KinExA results for equilibrium mixtures need to be analyzed according to an expression in which [1–(RAg/Ro)]1/2 describes the rectangular hyperbolic dependence of response on free antigen concentration [8,9]. A similar problem pervades use of the KinExA approach to determine the association rate constant for an antigen–antibody interaction from time dependence of the decrease in RAg/Ro [7,10]. This investigation explores the changes required to convert the current invalid kinetic expression based on antibody univa-
lence into one that takes into account the bivalence of antibody (immunoglobulin G, IgG) in its interaction with immobilized antigen on the affinity matrix. Theoretical considerations In the determination of an association rate constant by KinExA, there are two time courses to consider. First, there is the time course of antibody site occupancy by antigen in the reaction mixture with defined total antigen and antibody concentrations ([Ag]tot and [Ab]tot, respectively). Second, there is the time course of KinExA response, which monitors the decrease in antibody uptake from the mixture by immobilized antigen (Ag⁄) on the affinity matrix. Previous determinations of rate constants by KinExA have entailed the assumption that the ratio of fluorometric responses reflecting matrix-bound antibody for the same total antibody concentration in the presence and absence of antigen reflects the fraction of unoccupied antibody sites in the reaction mixture [7,10]. However, this assumption is untenable because the two 1:1 Ab–Ag complexes still retain affinity for immobilized antigen; only the Ag–Ab–Ag complex exhibits no affinity for the affinity matrix [8]. The two events are considered in turn. Time course of complex formation in reaction mixture
⇑ Fax: +61 7 3365 4699. 1
E-mail address: [email protected] Abbreviation used: KinExA, kinetic exclusion assay.
0003-2697/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ab.2013.06.020
The time course of antigen uptake by antibody in the reacting solution can be described by a 1:1 interaction between antigen
Rate constant determination by KinExA / D.J. Winzor / Anal. Biochem. 441 (2013) 214–217
(presumed univalent) and antibody sites, with the relevant differential rate equation being 0
0
0
0
d½Ab Ag=dt ¼ ka ½Agt ð½Ab tot ½Ab Agt Þ kd ½Ab Agt ;
ð1Þ
0
where [Ab ]tot denotes the total antibody site concentration (= 2[Ab]tot), [Ab0 Ag]t denotes the concentration of complexed antibody sites ([–Ab–Ag] plus [Ag–Ab–]) at time t, and ka and kd are the respective rate constants for complex formation and dissociation. For illustrative purposes, it suffices to consider a situation in which the concentration of antigen in the mixture greatly exceeds that of antibody sites ([Ag]tot >> [Ab0 ]tot)—a simplification that allows the replacement of [Ag]t in Eq. (1) by [Ag]tot for all t. Eq. (1) then becomes a pseudo-first-order kinetic expression, 0
0
0
0
d½Ab Ag=dt ¼ ka ½Agtot ð½Ab tot ½Ab Agt Þ kd ½Ab Agt ;
ð2Þ
which is readily integrated. In the current context, it is convenient to use the following integrated form of the expression for fractional saturation of antibody sites, Fs = [Ab’Ag]t/[Ab0 ]tot,
F s ¼ ðka ½Agtot =kobs Þ½1 expðkobs Þt
ð3aÞ
kobs ¼ ka ½Agtot þ kd ;
ð3bÞ
which is the counterpart of that introduced by O’Shannessy and coworkers [11] for analysis of the adsorption stage of Biacore sensorgrams. The corresponding expression for the fraction of unoccupied antibody sites, therefore, becomes
ð1 F s Þ ¼ 1 ðka ½Agtot =kobs Þ½1 expðkobs tÞ
ð4Þ
Time course of corresponding KinExA response ratio As noted above, the current assumption [7,10] that (RAg)t/Ro can be substituted for (1 Fs) in Eq. (4) is clearly incorrect because the experimental parameter being monitored is the time dependence of [Ag–Ab–Ag]t instead of [Ab0 Ag]t. From reacted site probability theory [12–14], the concentration of AbAg2 is P 2Ab [Ab]tot, where PAb is the probability that an antibody site is occupied by an antigen molecule. On the basis that the fluorometric response ratio is thus monitoring ð1 F 2s Þ, the KinExA equivalent of Eq. (4) becomes 2
½ðRAg Þt =Ro ¼ 1 fðka ½Agtot =kobs Þ½1 expðkobs tÞg
ð5Þ
Although Singer [15] introduced the concept of reacted site probability theory into immunochemistry nearly 50 years ago, immunologists still seem reluctant to accept its validity. Confirmation of the inference that [1 (RAg)t/Ro]1/2 is the experimental counterpart of the fraction of unoccupied antibody sites, [Ab0 ]t/ [Ab0 ]tot, in the reaction mixture has thus been provided by the following illustrative example of its application. Relative contribution of AbAg2 to fractional antibody saturation To illustrate the reacted site probability approach, the proportion of antigen bound as AbAg2 is first determined for each successive antigen attachment in the pathway to saturation of all antibody molecules in the system. For simplicity, it is assumed that ka[Ag]tot kd to allow the disregard of complex dissociation (Eq. (3b)). Consider initially a system with only five bivalent acceptor (antibody) molecules A. For the first attachment of a univalent ligand (antigen) molecule S, there are clearly 10 possible ways of forming a 1:1 complex AS, all of which are equally probable (first line of Table 1). The first opportunity for AS2 formation occurs during attachment of the second ligand molecule. However, such formation of S–A–S is restricted to the attachment of ligand to only 1 of the 9 unoccupied acceptor sites; the alternative distribution
215
with S attached to sites on 2 different acceptor molecules is thus 8 times more likely (Table 1). For 20% acceptor site saturation (2 sites occupied), the probability of AS2 formation is thus 1/9 (i.e., 0.111); this probability, denoted as P(2,2) in Table 1, follows a general terminology P(i,j), where i refers to the total number of ligand molecules bound and j refers to the number of molecules bound as AS2. During the third ligand attachment, there are two more opportunities for AS2 formation by its location on either of the 2 AS molecules in the lower distribution for second ligand attachment. Values of P(3,j) for the three possible ligand distributions are listed in the penultimate column of Table 1. This availability of P(i,j) values for each possible ligand distribution then allows the determination of the fraction of bound antigen present as AS2, Fs(AS2), from the expression
F s ðAS2 Þ ¼
X ðj=iÞPði; jÞ
ð6Þ
which has been used to calculate the value of this parameter listed in the final column of Table 1. The results of similar considerations for the subsequent steps to saturation of the 5 acceptor molecules (10 sites) are summarized in the remaining lines of Table 1. On the grounds that those values of Fs(AS2) refer specifically to the system with 5 acceptor molecules, the whole exercise has been repeated for systems with 10, 15, 20, and 40 molecules of A in order to ascertain the likely variation of this parameter with number of acceptor sites. That variation in Fs(AS2) for 20, 40, 60, and 80% site saturation is shown in Fig.1, where the abscissa is expressed as the reciprocal of the number of acceptor sites to facilitate extrapolation to the ordinate intercept; the value in the limit of an infinite number of acceptor molecules should be more applicable to the experimental situation in KinExA studies, where the antibody concentration (typically 20– 60 pM) requires the consideration of systems with 1013 to 1014 as the order of magnitude for the number of acceptor molecules. Although the extent of that extrapolation is clearly underemphasized in Fig. 1, the results conform with the conclusion that the fraction of ligand (antigen) bound as AS2 is numerically equal to Fs, the fractional saturation of acceptor (antibody) sites. As foreshadowed above, the inference from Fig. 1 that Fs(AS2) is a measure of F 2s is merely an illustrative manifestation of reacted site probability considerations [12–15] that has been included to provide reassurance that the ratio of fluorometric responses in a kinetic experiment is, indeed, monitoring the square of the fractional saturation of antibody sites—the condition incorporated into Eq. (6). Experimental ramifications The consequences of the above theoretical considerations are illustrated by employing Eq. (4) to calculate the time dependence of fractional site saturation at 10-min intervals for a mixture with [Ab]tot = 10 pM, [Ag]tot = 500 pM, ka = 3 106 M–1 s–1, and kd = 1.5 10–4 s–1 (an equilibrium dissociation constant Kd of 50 pM). In that regard, the restriction of [Ab]tot (which does not appear in the calculation) to 10 pM (20-pM sites) ensures a decrease of less than 4% in [Ag]t and thus provides reasonable support for the approximation inherent in the substitution of [Ag]tot for [Ag]t that renders the system amenable to pseudo-first-order kinetic analysis. Delineation of the time course for the decrease in the fraction of unoccupied antibody sites, (1 Fs) = [Ab’]t/[Ab’]tot (shown as open symbols in Fig. 2), allows the calculation of F 2s and hence the corresponding decrease in (RAg)t/Ro (closed symbols in Fig. 2). The current practice [7,10] of regarding (RAg)t/Ro as (1 Fs) rather than (1 F 2s ) clearly leads to underestimation of kobs, or of ka if ka[Ag]tot kd, as well as overestimation of Kd from the time-indepen-
216
Rate constant determination by KinExA / D.J. Winzor / Anal. Biochem. 441 (2013) 214–217
Table 1 Deduction of probability of AS2 formation after each ligand (S) addition to 5 molecules of a bivalent acceptor A.
a b
Possible distributions of species
Probability of distributiona
Value
Fs(AS2)b
1st addition S–A– –A– –A– –A– –A–
P(1,0) = 10/10
2nd addition S–A–S –A– –A– –A– –A– S–A– S–A– –A– –A– –A–
P(2,2) = 1/9 P(2,0) = 8/9
0.1111 0.8889
0.1111
3rd addition S–A–S S–A– –A– –A– –A– S–A– S–A– S–A– –A– –A–
P(3,2) = (8/8)P(2,2) + (2/8)P(2,0) P(3,0) = (6/8)P(2,0)
0.3333 0.6667
0.2222
4th addition S–A–S S–A–S –A– –A– –A– S–A–S S–A– S–A– –A– –A– S–A– S–A– S–A– S–A– –A–
P(4,4) = (1/7)P(3,2) P(4,2) = (6/7)P(3,2) + (3/7)P(3,0) P(4,0) = (4/7)P(3,0)
0.0476 0.5714 0.3810
0.3333
5th addition S–A–S S–A–S S–A– –A– –A– S–A–S S–A– S–A– S–A– –A– S–A– S–A– S–A– S–A– S–A–
P(5,4) = (6/6)P(4,4) + (2/6)P(4,2) P(5,2) = (4/6)P(4,2) + (4/6)P(4,0) P(5,0) = (2/6)P(4,0) 0.1270 0.4444
0.2381 0.6349
6th addition S–A–S S–A–S S–A–S –A– –A– S–A–S S–A–S S–A– S–A– –A– S–A–S S–A– S–A– S–A– S–A–
P(6,6) = (1/5)P(5,4) 0.0476 P(6,4) = (4/5)P(5,4) + (3/5)P(5,2) P(6,2) = (2/5)P(5,2) + (5/5)P(5,0)
0.5714 0.3810 0.5556
7th addition S–A–S S–A–S S–A–S S–A– –A– S–A–S S–A–S S–A–S–A– S–A–
P(7,6) = (4/4)P(6,6) + (2/4)P(6,4) P(7,4) = (2/4)P(6,4) + (4/4)P(6,2)
0.3333 0.6667
0.6667
8th addition S–A–S S–A–S S–A–S S–A–S –A– S–A–S S–A–S S–A–S S–A– S–A–
P(8,8) = (1/3)P(7,6) P(8,6) = (2/3)P(7,6) + (3/3)P(7,4)
0.1111 0.8889
0.7778
9th addition S–A–S S–A–S S–A–S S–A–S S–A–
P(9,8) = (2/2)P(8,8) + (2/2)P(8,6)
1.0000
0.8889
10th addition S–A–S S–A–S S–A–S S–A–S S–A–S
P(10,10) = (1/1)P(9,8)
1.0000
1.0000
P(i,j) denotes the probability that j of the i bound ligand molecules are present as AS2. Calculated from Eq. (6).
Fig.1. Dependence of the fraction of ligand (antigen) bound as AS2, Fs(AS2), on the reciprocal of the number of acceptor (antibody) sites used to calculate the most probable distribution of species (A, AS, AS2) at each successive ligand addition. Lines denote the linear extrapolation of Fs(AS2) values for the indicated extents of acceptor site saturation (Fs) to demonstrate the identity of Fs and Fs(AS2) for mixtures with an experimentally realistic number of acceptor molecules.
dent asymptote [8]. For example, nonlinear least squares analysis of the time dependence of (RAg)t/Ro ( , Fig. 2) in terms of Eq. (4) instead of Eq. (5) yields values (± standard deviations [SD]) of 1.77 (± 0.08) 106 M–1 s–1 for ka and 1.06 (± 0.05) 10–3 s–1 for kobs, which on substitution into Eq. (3b) gives rise to a kd of 1.77 (± 0.09) 104 s1. The consequent apparent value of 100 (±10) pM for Kd, which encompasses that of 102 pM obtained from the time-independent asymptote in Fig. 2, thus overestimates the input equilibrium constant by a factor of approximately 2—the con-
Fig.2. Time dependence of the fraction of unoccupied antibody sites (e), (1 Fs), calculated by means of Eq. (4) for an antigen–antibody mixture with [Ag]tot = 500 pM and [Ab]tot = 10 pM for which complex formation is governed by respective association and dissociation rate constants of 3 106 M–1 s–1 and 1.5 10–4 s–1 (an equilibrium dissociation constant of 50 pM). Closed symbols ( ) denote the corresponding dependence of fluorometric response ratio, (RAg)t/Ro, calculated from Eq. (5), with additional points having been used to ascertain the form of the dependence during the first 10 min.
clusion reached previously from considerations of equilibrium KinExA data [8]. In an experimental context, the time dependence of (RAg)t/Ro is the starting point for rate constant determination; but the expressions incorporated into the KinExA software are in terms of [Ab0 ]t/ [Ab]tot because of the above-mentioned misinterpretation of the fluorometric response ratio. Rectification of this mistake merely entails the substitution of [1 (RAg)t/Ro]1/2 for [1 (RAg)t/Ro] in
Rate constant determination by KinExA / D.J. Winzor / Anal. Biochem. 441 (2013) 214–217
the KinExA program, which can accommodate not only the pseudo-first-order kinetic situation considered above ([Ag]t [Ag]tot) but also the more general second-order kinetic situation where allowance must be made for the progressive decrease in free antigen concentration from [Ag]tot to [Ag]e, its equilibrium value associated with the limiting value of (RAg)t/Ro as t ? 1. However, such allowance for variation in [Ag]t is at the expense of that for reversibility of complex formation; it necessitates the assumption that kd ka[Ag]t, which renders the reaction essentially irreversible over the time scale of the experiment. The magnitude of kd can then be determined as the product of ka and the equilibrium constant Kd [7,10]. Discussion This investigation completes the amendment of theoretical expressions for the characterization of antigen–antibody interactions by kinetic exclusion assay—an endeavor that has been marred previously by inadequate allowance for the consequences of antibody bivalence in its uptake by the affinity matrix (immobilized antigen Ag⁄) that is used to ascertain the fraction of free antibody sites in a solution with defined total concentrations of antigen and antibody [1–7,10]. The fact that the same amendment, the substitution of (1 RAg/Ro)1/2 for (1 RAg/Ro), applies to kinetic as well as thermodynamic [8] characterizations of immunochemical reactions by kinetic exclusion assay is hardly surprising in that reacted site probability considerations [12–14] apply to both. In that regard, it is hoped that the current illustration of reacted site probability theory in action (Table 1 and Fig. 1) may help to convince immunochemists of its relevance to the characterization of antigen–antibody interactions. An obvious advantage of the KinExA approach to studying antigen–antibody reactions is its characterization of the interaction between the two reactants in solution, with this being an attribute that is not shared by conventional Biacore methodology, which quantifies the interaction between one reactant and a chemically modified (immobilized) form of the other [16–18]. This deficiency of conventional Biacore protocol has been overcome [17] by according the biosensor surface the same role as that played by the affinity matrix in a KinExA, namely, a means of monitoring the free concentration of one reactant in a solution with defined total concentrations of antigen and antibody. Furthermore, by incorporating theory reported by Stevens [19], the Swiss group [17] clearly recognized the need to allow for bivalence of the antibody when it was the partitioning solute—a factor overlooked throughout the development of KinExA methodology despite the existence of many publications that had already overcome the ligand multivalence problem [20–26]. Fortunately, the current invalidity of KinExA programs for the kinetic and thermodynamic characterization of antigen–antibody interactions can be rectified by minor amendment of equations within the existing software that would then justify consideration of the KinExA as the procedure of choice for the rapid and accurate characterization of antigen–antibody interactions.
[2] [3]
[4] [5] [6]
[7]
[8]
[9]
[10]
[11]
[12] [13] [14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
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