Analytical Biochemistry 456 (2014) 38–42
Contents lists available at ScienceDirect
Analytical Biochemistry journal homepage: www.elsevier.com/locate/yabio
Confirmation of the validity of the current characterization of immunochemical reactions by kinetic exclusion assay Thomas R. Glass a,⇑, Donald J. Winzor b a b
Sapidyne Instruments Inc., 700 West Diamond Street, Boise, ID 83705, USA 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 19 December 2013 Received in revised form 8 April 2014 Accepted 11 April 2014 Available online 18 April 2014 Keywords: Antigen–antibody interactions Antibody bivalence Kinetic exclusion assay
a b s t r a c t Prior observations that questioned the validity of kinetic exclusion assays were based on the mistaken assumption that the assays quantified the fraction of those antibody molecules that had unoccupied binding sites. Instead, the standard KinExA assay quantifies the fraction of total antibody binding sites that are unoccupied, regardless of the number of unoccupied sites on each antibody molecule. Although the standard KinExA analysis assumes that there is only a small probability of antibody-site capture by the affinity matrix, the results of numerical simulations demonstrate the reliability of dissociation constants obtained by the standard KinExA analysis for capture probabilities as high as 30%. This finding further strengthens the potential of kinetic exclusion assays as the procedure of choice for the rapid and accurate characterization of immunochemical reactions that forms part of screening processes in the search for therapeutic antibodies. Ó 2014 Elsevier Inc. All rights reserved.
The kinetic exclusion assay (KinExA)1 has great potential for the accurate quantification of antigen–antibody interactions [1–7]—an important aspect of the screening process in the search for therapeutic antibodies. It comprises a heterogeneous immunoassay in which the concentration of unoccupied antibody sites in equilibrium mixtures of antigen and antibody is assessed by affinity chromatography on a microcolumn. After brief exposure of each equilibrated reaction mixture to an affinity matrix with a high concentration of immobilized antigen to capture a representative sample of antibody molecules bearing unoccupied binding sites, the amount of captured antibody is monitored by adding fluorescently labeled anti-immunoglobulin G as an antibody detector. The concentration of antibody molecules bearing unoccupied binding sites is then quantified as the ratio of the fluorescence response at the column midpoint for a mixture (RAg) to that (Ro) for the same total antibody concentration in the absence of antigen. Although the method has obvious potential for accurate measurement of the affinity constant, the validity of the analytical expressions incorporated into the KinExA software has been questioned recently [8–10]. This communication identifies the error in the derivation of the latter quantitative expressions [9,10], which were based on the premise that RAg monitors the concentration ⇑ Corresponding author. Fax: +1 208 345 5251. 1
E-mail address: [email protected] (T.R. Glass). Abbreviations used: KinExA, kinetic exclusion assay; BSA, bovine serum albumin.
http://dx.doi.org/10.1016/j.ab.2014.04.011 0003-2697/Ó 2014 Elsevier Inc. All rights reserved.
of antibody molecules bearing unoccupied sites rather than the concentration of unoccupied sites. By providing an alternative means of deriving the basic KinExA quantitative expression this investigation reinforces the validity of the currently used approach for the characterization of immunochemical reactions by kinetic exclusion assay.
Theoretical considerations KinExA is a contraction of the Kinetic Exclusion Assay and a fundamental principle underlying its operation is that only the unoccupied binding sites present in a mixture of occupied and unoccupied sites are ‘‘assayed.’’ Consider the simplest case of a monovalent receptor R binding to a monovalent ligand L. In solution there will be a mixture of R, L, and RL complex. The KinExA assay is performed by exposing the mixture to a solid phase that has one of the binding partners, usually the ligand, immobilized on it. The contact time between the solution and the solid phase is kept brief enough that there is no significant dissociation of RL complex during contact and thus the R existing as RL in the original solution is ‘‘kinetically excluded’’ from interacting with the solidphase affinity ligand, designated L⁄. The solid-phase binding which does occur then represents the binding of unliganded solution R to L⁄. Because the effective concentration of L⁄ is much greater than [R] the binding reaction will be pseudo first order and the resultant
Evaluation of Kd by kinetic exclusion assay / T.R. Glass, D.J. Winzor / Anal. Biochem. 456 (2014) 38–42
formation of RL⁄ will be directly proportional to [R], the equilibrium concentration of free receptor in the applied solution. The solid phase on a KinExA instrument is held in a microcolumn with an interstitial volume of about 4 ll [3] while typical sample volumes range from a from 250 ll to 5 ml. Thus for a typical flow rate of 250 ll/min it takes 1 to 20 min for the entire sample volume to traverse the solid phase. However, these relatively lengthy times do not represent the kinetic exclusion contact time for RL dissociation in the solution phase. If the applied sample is at equilibrium (typical for a Kd measurement by KinExA) the concentrations [R], [L], and [RL] are static. This remains true up until the moment a given 4 ll bolus of the sample contacts the solid phase. At 250 ll/min each bolus is in contact with the solid phase for approximately 1 s. By the time that the bolus reaches the column midpoint (the position monitored subsequently) the solution phase antibody concentration [R]tot has been reduced by the amount of R captured; but over the 0.5-s time interval this decrease is insignificant from the viewpoint of composition reequilibration within the solution phase. Furthermore, there is continual replacement of the bolus by another with the equilibrium composition of the mixture. Subject to this reasonable assumption that the dissociation of RL is insignificant over the small time interval of 0.5 s, the concentration of receptor captured at the detection point column is proportional to the equilibrium concentration of R in the applied mixture [R]. Because the same situation applies to each successive bolus of solution, the concentration of captured receptor (RL⁄) at the column midpoint increases but remains proportional to [R] as long as [L⁄] >> [R]tot (a situation incorporated into the design of a KinExA experiment). The most widespread application of KinExA has been to characterize the interaction between a bivalent antibody (Ab) and a small monovalent antigen (Ag). The quantitative expression incorporated into the KinExA analysis [4,5] assumes that antibody uptake by immobilized antigen (Ag⁄) is proportional to unoccupied antibody binding sites. The validity of this assumption (challenged in two recent publications [9,10]) will be established theoretically for the conditions pertaining to a kinetic exclusion assay. Data analysis The first indication of error in the proposed ‘‘correction’’ to the standard KinExA analysis is that the proposed change results in a poorer description of experimental data. The previous publication [9] included reanalysis of several sets of previously published data, but the reanalysis was limited to subsets of the data where fit errors were less apparent. For example, data published by Drake and co-workers [13] is very well fit by the standard KinExA model (see Fig. 1a) while the proposed alternative (Fig. 1b) shows a clear systematic fit error. While the fact that the standard model provides an excellent fit to the data does not establish it as the correct model, the fact that the proposed alternative model cannot fit the data, even with optimized parameters, does indicate its inapplicability. The proposed alternative model was fitted to all of the data cited in the previous publication [9] together with additional data sets with results similar to those shown in Fig. 1. One further piece of experimental evidence is that measurements on monovalent Fab have been found to agree with the corresponding whole IgG [14] which should not occur if the proposed change were correct.
39
related to its corresponding free concentration ([Ab]) and that ([Ag]) of free antigen by [11,12]
½Abtot ¼ ½Ab þ 2½Abð½Ag=K d Þ þ ½Abð½Ag=K d Þ2 ¼ ½Abð1 þ ½Ag=K d Þ2 ;
ð1Þ
where concentrations are expressed in molar terms and Kd is the intrinsic dissociation constant (the reciprocal of its association constant counterpart K). In Eq. (1) the three terms correspond to doubly unliganded antibody, singly liganded antibody, and doubly liganded antibody, respectively. Inasmuch as the uptake of antibody molecules by the immobilized antigen is restricted to those bearing at least one unoccupied site, RAg is a response to the sum of the first two terms, or [Ab](1 + 2[Ag]/Kd). As the doubly unliganded and singly liganded antibody bind to the solid phase, their concentrations in solution decrease. In principle these decreases are tempered by dissociation of soluble antigen–antibody complex(es) but as discussed above such dissociation should be negligible over the approximately half-second duration of each bolus’s exposure to the affinity matrix. Under this condition rate equations for the concentrations of the two species in the flowing solution-phase bolus can be written
d½Abdt ¼ 2k½Ag ½Ab
ð2aÞ
d½AbAg=dt ¼ k½Ag ½AbAg
ð2bÞ
⁄
where k[Ag ] is a pseudo-first-order association rate constant for interaction between immobilized antigen and an unoccupied antibody site on the basis of assumed constancy of [Ag⁄] because of an experimental design such that [Ag⁄]tot >> [Ab]tot. The factor of 2 in Eq. (2a) accounts for the fact that an unliganded Ab molecule may bind by either of its binding sites. For respective initial concentrations [Ab]0 and [AbAg]0 of doubly free and singly liganded antibody the integration of Eqs. (2a) and (2b) gives
½Abðt 0 Þ ¼ ½Ab0 expð2k½Ag t 0 Þ
ð3aÞ
½AbAgðt 0 Þ ¼ ½AbAg0 expðk½Ag t0 Þ
ð3bÞ
as the respective concentrations of doubly unliganded and singly liganded antibody–antigen complex in the solution phase by the time (t0 ) that the solution reaches the column midpoint. For a typical microcolumn with an interstitial volume of about 4 ll [3] operated at flow rate F (ll/s) the time t0 appropriate to Eqs. (3a) and (3b) is 2/F, whereupon the expressions for captured antibody concentration, [Ab]cap, at the column midpoint after that small time interval become
½Abcap ¼ 2k½Ag ½Ab0 expð4k½Ag =Fg
ð4aÞ
½AbAgcap ¼ k½Ag ½AbAg0 expð2k½Ag =FÞg:
ð4bÞ
In a KinExA experiment a large number of these boluses (all with the same initial composition) may be considered to flow sequentially over the solid phase, which thus accumulates the above concentrations of antibody in each small time increment t0 during the time t taken to apply the complete sample. Provided that no dissociation of captured antibody occurs during that overall time interval t, the concentration of captured antibody is thus N{[Ab]cap + [AbAg]cap}, where N = t/t0 . Because the fluorescence response RAg only monitors the concentration of captured antibody at the detection point, the measured response at total time t becomes
A kinetic derivation of the basic KinExA expression for a bivalent antibody
RAg ðtÞ ¼ Nf2k½Ag ½Ab0 expð4k½Ag =FÞ þ k½Ag ½AbAg0
For an equilibrium mixture of univalent antigen, Ag, and bivalent antibody, Ab, the total concentration of antibody ([Ab]tot) is
whereas its counterpart for uptake from an antigen-free solution with the same total antibody concentration is
expð2k½Ag =FÞg
ð5Þ
40
Evaluation of Kd by kinetic exclusion assay / T.R. Glass, D.J. Winzor / Anal. Biochem. 456 (2014) 38–42
Fig.1. Demonstration of the superiority of the regular KinExA analysis [4] over the erroneously amended version [9]. Panel (a) shows data originally published by Drake et al. [13] subjected to the standard KinExA analysis. The optimized (least squares) parameters for the fit are Kd = 24.7 pM; ligand activity = 63.8%; maximum signal (Sig100) curve 1 = 1.56 V, curve 2 = 1.60 V; nonspecific binding signal (NSB) curve 1 = 0.18 V, curve 2 = 0.09 V. Residual error after fitting = 1.65%. Panel (b) shows the same data subjected to the proposed alternative analysis. The optimized (least squares) parameters for the fit are Kd = 22 pM; ligand activity = 109%; maximum signal (Sig100) curve 1 = 1.51 V, curve 2 = 1.56 V; nonspecific binding signal (NSB) curve 1 = 0.17 V, curve 2 = 0.16 V. Residual error after fitting = 4.91%.
R0 ðtÞ ¼ Nf2k½Ag ½Abtot expð4k½Ag =FÞg
ð6Þ
It therefore follows that 2
RAg =R0 ¼
2½Ab0 fexpð2k½Ag =FÞg þ ½AbAg0 expð2k½Ag =F 2
2½Abtot fexpð2k½Ag =FÞg
ð7Þ The fraction of the doubly unliganded Ab that is captured in the first half of the column (a parameter straightforwardly measured, as described below) is given by
P0 ¼ f1 expð4k½Ag =FÞg;
ð8Þ
which allows Eq. (7) to be written as
RAg =R0 ¼ f2½Ab0 þ ½AbAg0 ð1 p0 Þ
Making the approximation that (1–p0 )1/2 1 and substituting from Eq. (1) into Eq. (9), recognizing that [Ab]0 and [AbAg]0 are the solution equilibrium concentrations of doubly unliganded and singly liganded antibody respectively, Eq. (9) reduces to
1=2
g=2½Abtot :
ð9Þ
RAg =R0 ¼ 1=ð1 þ ½Ag=K d Þ;
ð10Þ
which is equivalent to the expression derived previously [4] and incorporated into the KinExA analysis software. The dispute over the correctness of Eq. (10) [9,10] reflected a misconception about the parameter being monitored by the response RAg in that its description as the concentration of unoccupied binding sites [4,5] was misconstrued as the concentration of antibody with affinity for Ag⁄. Adoption of that stance leads to the conclusion that the only species concentration failing to be
Evaluation of Kd by kinetic exclusion assay / T.R. Glass, D.J. Winzor / Anal. Biochem. 456 (2014) 38–42
41
registered in RAg is the doubly-liganded complex AbAg2, in which case the expression analogous to Eq. (10) becomes [9]
ited to cases where the dissociation is either insignificant or the same for both singly liganded and unliganded antibody.
RAg =Ro ¼ ð1 þ 2½Ag=K d Þ=ð1 þ ½Ag=K d Þ2
Practical considerations in KinExA analysis
ð11Þ
It is now clear that for low capture percentages a KinExA experiment monitors the sum of the concentration of doubly unliganded antibody sites and half of that of singly liganded AbAg complexes because the rate of uptake of doubly unliganded antibody is twice that of 1:1 AbAg complexes. However, although only half of the unoccupied antibody sites on 1:1 AbAg complex in the solution are being monitored, the fluorescence response also reflects an equal concentration of occupied sites on captured AbAg. The concentration of those removed antigen-occupied antibody sites thus corresponds exactly with the concentration of extra unoccupied sites on AbAg that remain uncaptured from the equilibrium mixture. On those grounds RAg does become a rigorous monitor of the concentration of unoccupied sites, despite antigen occupancy of some of the antibody sites reflected in the measured fluorescence response.
Neglect of captured antibody dissociation from the solid phase Although the neglect of dissociation effects inherent in Eqs. (2a) and (2b) is certainly justified because of the small time period t0 required for a bolus of equilibrium mixture to reach the column midpoint, the above argument leading to Eqs. (5) and (6) as descriptions of RAg and R0 after the much longer time taken to apply the whole sample neglects the consequence of any dissociation of antibody from the solid phase. To the extent that loss of captured Ab is the same for AbAg⁄ and Ag⁄AbAg, it is irrelevant since such loss reduces the overall signal without changing its proportionality to the unliganded Ab binding sites. Furthermore, for sub-nanomolar equilibrium dissociation constants this loss will often be insignificant even over the time scale (minutes) of a typical KinExA experiment. For weaker binders, if the dissociation is significantly different for the two species it will show up in measurement data as relative washout slopes increasing montonically with soluble ligand concentration (because the fraction of the unliganded sites present on singly liganded Ab molecules increases with total soluble ligand). We are not aware of any KinExA data where this was a significant effect and the present analysis is lim-
Measurement of the antibody-site capture probability Published techniques for the measurement of antibody-site capture probability p [3,4] have relied on the use of fluorescently labeled primary antibodies that are not otherwise required for KinExA analysis. For several years Sapidyne Instruments has advised users of its KinExA equipment to adopt the following procedure for estimating the magnitude of p0 . A high capture probability gives rise to a significant gradient in the amount of antibody captured along the length of the microcolumn as a result of the progressive decrease in unoccupied antibody-site concentration. This effect can be exacerbated by doubling the length of the bead pack. The capture probability is then taken as one minus the square root of the ratio of fluorescence responses for the double-length and normal-length bead packs. In the unlikely event that the estimated capture probability exceeds 30%, a lower value of p0 can be achieved by applying the antibody solution to the microcolumn at a higher flow rate to lower the solid-phase contact time. It should be noted that the capture probability estimated by the following procedure is the capture probability p0 across the first half of a normal-length bead pack, whereas the previously published method [3,4] measured capture p across the entire bead pack. For an exponential gradient along the bead pack [Eq. (4)] the relationship between these two probabilities is
p0 ¼ 1 ð1 pÞ1=2
ð12Þ
In present terminology the highest published capture probability p of 0.42 for the entire bead pack thus corresponds to a capture probability p0 of 0.24 across the first half of the bead pack. On the basis of measurements made regularly in the Sapidyne laboratory for several years, this represents an unusually high capture. Effect of increasing the probability of antibody-site capture Use of Eq. (10), which relies on the approximation that (1 – p0 )1/2 1, is certainly justified in situations where p0
Contents lists available at ScienceDirect
Analytical Biochemistry journal homepage: www.elsevier.com/locate/yabio
Confirmation of the validity of the current characterization of immunochemical reactions by kinetic exclusion assay Thomas R. Glass a,⇑, Donald J. Winzor b a b
Sapidyne Instruments Inc., 700 West Diamond Street, Boise, ID 83705, USA 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 19 December 2013 Received in revised form 8 April 2014 Accepted 11 April 2014 Available online 18 April 2014 Keywords: Antigen–antibody interactions Antibody bivalence Kinetic exclusion assay
a b s t r a c t Prior observations that questioned the validity of kinetic exclusion assays were based on the mistaken assumption that the assays quantified the fraction of those antibody molecules that had unoccupied binding sites. Instead, the standard KinExA assay quantifies the fraction of total antibody binding sites that are unoccupied, regardless of the number of unoccupied sites on each antibody molecule. Although the standard KinExA analysis assumes that there is only a small probability of antibody-site capture by the affinity matrix, the results of numerical simulations demonstrate the reliability of dissociation constants obtained by the standard KinExA analysis for capture probabilities as high as 30%. This finding further strengthens the potential of kinetic exclusion assays as the procedure of choice for the rapid and accurate characterization of immunochemical reactions that forms part of screening processes in the search for therapeutic antibodies. Ó 2014 Elsevier Inc. All rights reserved.
The kinetic exclusion assay (KinExA)1 has great potential for the accurate quantification of antigen–antibody interactions [1–7]—an important aspect of the screening process in the search for therapeutic antibodies. It comprises a heterogeneous immunoassay in which the concentration of unoccupied antibody sites in equilibrium mixtures of antigen and antibody is assessed by affinity chromatography on a microcolumn. After brief exposure of each equilibrated reaction mixture to an affinity matrix with a high concentration of immobilized antigen to capture a representative sample of antibody molecules bearing unoccupied binding sites, the amount of captured antibody is monitored by adding fluorescently labeled anti-immunoglobulin G as an antibody detector. The concentration of antibody molecules bearing unoccupied binding sites is then quantified as the ratio of the fluorescence response at the column midpoint for a mixture (RAg) to that (Ro) for the same total antibody concentration in the absence of antigen. Although the method has obvious potential for accurate measurement of the affinity constant, the validity of the analytical expressions incorporated into the KinExA software has been questioned recently [8–10]. This communication identifies the error in the derivation of the latter quantitative expressions [9,10], which were based on the premise that RAg monitors the concentration ⇑ Corresponding author. Fax: +1 208 345 5251. 1
E-mail address: [email protected] (T.R. Glass). Abbreviations used: KinExA, kinetic exclusion assay; BSA, bovine serum albumin.
http://dx.doi.org/10.1016/j.ab.2014.04.011 0003-2697/Ó 2014 Elsevier Inc. All rights reserved.
of antibody molecules bearing unoccupied sites rather than the concentration of unoccupied sites. By providing an alternative means of deriving the basic KinExA quantitative expression this investigation reinforces the validity of the currently used approach for the characterization of immunochemical reactions by kinetic exclusion assay.
Theoretical considerations KinExA is a contraction of the Kinetic Exclusion Assay and a fundamental principle underlying its operation is that only the unoccupied binding sites present in a mixture of occupied and unoccupied sites are ‘‘assayed.’’ Consider the simplest case of a monovalent receptor R binding to a monovalent ligand L. In solution there will be a mixture of R, L, and RL complex. The KinExA assay is performed by exposing the mixture to a solid phase that has one of the binding partners, usually the ligand, immobilized on it. The contact time between the solution and the solid phase is kept brief enough that there is no significant dissociation of RL complex during contact and thus the R existing as RL in the original solution is ‘‘kinetically excluded’’ from interacting with the solidphase affinity ligand, designated L⁄. The solid-phase binding which does occur then represents the binding of unliganded solution R to L⁄. Because the effective concentration of L⁄ is much greater than [R] the binding reaction will be pseudo first order and the resultant
Evaluation of Kd by kinetic exclusion assay / T.R. Glass, D.J. Winzor / Anal. Biochem. 456 (2014) 38–42
formation of RL⁄ will be directly proportional to [R], the equilibrium concentration of free receptor in the applied solution. The solid phase on a KinExA instrument is held in a microcolumn with an interstitial volume of about 4 ll [3] while typical sample volumes range from a from 250 ll to 5 ml. Thus for a typical flow rate of 250 ll/min it takes 1 to 20 min for the entire sample volume to traverse the solid phase. However, these relatively lengthy times do not represent the kinetic exclusion contact time for RL dissociation in the solution phase. If the applied sample is at equilibrium (typical for a Kd measurement by KinExA) the concentrations [R], [L], and [RL] are static. This remains true up until the moment a given 4 ll bolus of the sample contacts the solid phase. At 250 ll/min each bolus is in contact with the solid phase for approximately 1 s. By the time that the bolus reaches the column midpoint (the position monitored subsequently) the solution phase antibody concentration [R]tot has been reduced by the amount of R captured; but over the 0.5-s time interval this decrease is insignificant from the viewpoint of composition reequilibration within the solution phase. Furthermore, there is continual replacement of the bolus by another with the equilibrium composition of the mixture. Subject to this reasonable assumption that the dissociation of RL is insignificant over the small time interval of 0.5 s, the concentration of receptor captured at the detection point column is proportional to the equilibrium concentration of R in the applied mixture [R]. Because the same situation applies to each successive bolus of solution, the concentration of captured receptor (RL⁄) at the column midpoint increases but remains proportional to [R] as long as [L⁄] >> [R]tot (a situation incorporated into the design of a KinExA experiment). The most widespread application of KinExA has been to characterize the interaction between a bivalent antibody (Ab) and a small monovalent antigen (Ag). The quantitative expression incorporated into the KinExA analysis [4,5] assumes that antibody uptake by immobilized antigen (Ag⁄) is proportional to unoccupied antibody binding sites. The validity of this assumption (challenged in two recent publications [9,10]) will be established theoretically for the conditions pertaining to a kinetic exclusion assay. Data analysis The first indication of error in the proposed ‘‘correction’’ to the standard KinExA analysis is that the proposed change results in a poorer description of experimental data. The previous publication [9] included reanalysis of several sets of previously published data, but the reanalysis was limited to subsets of the data where fit errors were less apparent. For example, data published by Drake and co-workers [13] is very well fit by the standard KinExA model (see Fig. 1a) while the proposed alternative (Fig. 1b) shows a clear systematic fit error. While the fact that the standard model provides an excellent fit to the data does not establish it as the correct model, the fact that the proposed alternative model cannot fit the data, even with optimized parameters, does indicate its inapplicability. The proposed alternative model was fitted to all of the data cited in the previous publication [9] together with additional data sets with results similar to those shown in Fig. 1. One further piece of experimental evidence is that measurements on monovalent Fab have been found to agree with the corresponding whole IgG [14] which should not occur if the proposed change were correct.
39
related to its corresponding free concentration ([Ab]) and that ([Ag]) of free antigen by [11,12]
½Abtot ¼ ½Ab þ 2½Abð½Ag=K d Þ þ ½Abð½Ag=K d Þ2 ¼ ½Abð1 þ ½Ag=K d Þ2 ;
ð1Þ
where concentrations are expressed in molar terms and Kd is the intrinsic dissociation constant (the reciprocal of its association constant counterpart K). In Eq. (1) the three terms correspond to doubly unliganded antibody, singly liganded antibody, and doubly liganded antibody, respectively. Inasmuch as the uptake of antibody molecules by the immobilized antigen is restricted to those bearing at least one unoccupied site, RAg is a response to the sum of the first two terms, or [Ab](1 + 2[Ag]/Kd). As the doubly unliganded and singly liganded antibody bind to the solid phase, their concentrations in solution decrease. In principle these decreases are tempered by dissociation of soluble antigen–antibody complex(es) but as discussed above such dissociation should be negligible over the approximately half-second duration of each bolus’s exposure to the affinity matrix. Under this condition rate equations for the concentrations of the two species in the flowing solution-phase bolus can be written
d½Abdt ¼ 2k½Ag ½Ab
ð2aÞ
d½AbAg=dt ¼ k½Ag ½AbAg
ð2bÞ
⁄
where k[Ag ] is a pseudo-first-order association rate constant for interaction between immobilized antigen and an unoccupied antibody site on the basis of assumed constancy of [Ag⁄] because of an experimental design such that [Ag⁄]tot >> [Ab]tot. The factor of 2 in Eq. (2a) accounts for the fact that an unliganded Ab molecule may bind by either of its binding sites. For respective initial concentrations [Ab]0 and [AbAg]0 of doubly free and singly liganded antibody the integration of Eqs. (2a) and (2b) gives
½Abðt 0 Þ ¼ ½Ab0 expð2k½Ag t 0 Þ
ð3aÞ
½AbAgðt 0 Þ ¼ ½AbAg0 expðk½Ag t0 Þ
ð3bÞ
as the respective concentrations of doubly unliganded and singly liganded antibody–antigen complex in the solution phase by the time (t0 ) that the solution reaches the column midpoint. For a typical microcolumn with an interstitial volume of about 4 ll [3] operated at flow rate F (ll/s) the time t0 appropriate to Eqs. (3a) and (3b) is 2/F, whereupon the expressions for captured antibody concentration, [Ab]cap, at the column midpoint after that small time interval become
½Abcap ¼ 2k½Ag ½Ab0 expð4k½Ag =Fg
ð4aÞ
½AbAgcap ¼ k½Ag ½AbAg0 expð2k½Ag =FÞg:
ð4bÞ
In a KinExA experiment a large number of these boluses (all with the same initial composition) may be considered to flow sequentially over the solid phase, which thus accumulates the above concentrations of antibody in each small time increment t0 during the time t taken to apply the complete sample. Provided that no dissociation of captured antibody occurs during that overall time interval t, the concentration of captured antibody is thus N{[Ab]cap + [AbAg]cap}, where N = t/t0 . Because the fluorescence response RAg only monitors the concentration of captured antibody at the detection point, the measured response at total time t becomes
A kinetic derivation of the basic KinExA expression for a bivalent antibody
RAg ðtÞ ¼ Nf2k½Ag ½Ab0 expð4k½Ag =FÞ þ k½Ag ½AbAg0
For an equilibrium mixture of univalent antigen, Ag, and bivalent antibody, Ab, the total concentration of antibody ([Ab]tot) is
whereas its counterpart for uptake from an antigen-free solution with the same total antibody concentration is
expð2k½Ag =FÞg
ð5Þ
40
Evaluation of Kd by kinetic exclusion assay / T.R. Glass, D.J. Winzor / Anal. Biochem. 456 (2014) 38–42
Fig.1. Demonstration of the superiority of the regular KinExA analysis [4] over the erroneously amended version [9]. Panel (a) shows data originally published by Drake et al. [13] subjected to the standard KinExA analysis. The optimized (least squares) parameters for the fit are Kd = 24.7 pM; ligand activity = 63.8%; maximum signal (Sig100) curve 1 = 1.56 V, curve 2 = 1.60 V; nonspecific binding signal (NSB) curve 1 = 0.18 V, curve 2 = 0.09 V. Residual error after fitting = 1.65%. Panel (b) shows the same data subjected to the proposed alternative analysis. The optimized (least squares) parameters for the fit are Kd = 22 pM; ligand activity = 109%; maximum signal (Sig100) curve 1 = 1.51 V, curve 2 = 1.56 V; nonspecific binding signal (NSB) curve 1 = 0.17 V, curve 2 = 0.16 V. Residual error after fitting = 4.91%.
R0 ðtÞ ¼ Nf2k½Ag ½Abtot expð4k½Ag =FÞg
ð6Þ
It therefore follows that 2
RAg =R0 ¼
2½Ab0 fexpð2k½Ag =FÞg þ ½AbAg0 expð2k½Ag =F 2
2½Abtot fexpð2k½Ag =FÞg
ð7Þ The fraction of the doubly unliganded Ab that is captured in the first half of the column (a parameter straightforwardly measured, as described below) is given by
P0 ¼ f1 expð4k½Ag =FÞg;
ð8Þ
which allows Eq. (7) to be written as
RAg =R0 ¼ f2½Ab0 þ ½AbAg0 ð1 p0 Þ
Making the approximation that (1–p0 )1/2 1 and substituting from Eq. (1) into Eq. (9), recognizing that [Ab]0 and [AbAg]0 are the solution equilibrium concentrations of doubly unliganded and singly liganded antibody respectively, Eq. (9) reduces to
1=2
g=2½Abtot :
ð9Þ
RAg =R0 ¼ 1=ð1 þ ½Ag=K d Þ;
ð10Þ
which is equivalent to the expression derived previously [4] and incorporated into the KinExA analysis software. The dispute over the correctness of Eq. (10) [9,10] reflected a misconception about the parameter being monitored by the response RAg in that its description as the concentration of unoccupied binding sites [4,5] was misconstrued as the concentration of antibody with affinity for Ag⁄. Adoption of that stance leads to the conclusion that the only species concentration failing to be
Evaluation of Kd by kinetic exclusion assay / T.R. Glass, D.J. Winzor / Anal. Biochem. 456 (2014) 38–42
41
registered in RAg is the doubly-liganded complex AbAg2, in which case the expression analogous to Eq. (10) becomes [9]
ited to cases where the dissociation is either insignificant or the same for both singly liganded and unliganded antibody.
RAg =Ro ¼ ð1 þ 2½Ag=K d Þ=ð1 þ ½Ag=K d Þ2
Practical considerations in KinExA analysis
ð11Þ
It is now clear that for low capture percentages a KinExA experiment monitors the sum of the concentration of doubly unliganded antibody sites and half of that of singly liganded AbAg complexes because the rate of uptake of doubly unliganded antibody is twice that of 1:1 AbAg complexes. However, although only half of the unoccupied antibody sites on 1:1 AbAg complex in the solution are being monitored, the fluorescence response also reflects an equal concentration of occupied sites on captured AbAg. The concentration of those removed antigen-occupied antibody sites thus corresponds exactly with the concentration of extra unoccupied sites on AbAg that remain uncaptured from the equilibrium mixture. On those grounds RAg does become a rigorous monitor of the concentration of unoccupied sites, despite antigen occupancy of some of the antibody sites reflected in the measured fluorescence response.
Neglect of captured antibody dissociation from the solid phase Although the neglect of dissociation effects inherent in Eqs. (2a) and (2b) is certainly justified because of the small time period t0 required for a bolus of equilibrium mixture to reach the column midpoint, the above argument leading to Eqs. (5) and (6) as descriptions of RAg and R0 after the much longer time taken to apply the whole sample neglects the consequence of any dissociation of antibody from the solid phase. To the extent that loss of captured Ab is the same for AbAg⁄ and Ag⁄AbAg, it is irrelevant since such loss reduces the overall signal without changing its proportionality to the unliganded Ab binding sites. Furthermore, for sub-nanomolar equilibrium dissociation constants this loss will often be insignificant even over the time scale (minutes) of a typical KinExA experiment. For weaker binders, if the dissociation is significantly different for the two species it will show up in measurement data as relative washout slopes increasing montonically with soluble ligand concentration (because the fraction of the unliganded sites present on singly liganded Ab molecules increases with total soluble ligand). We are not aware of any KinExA data where this was a significant effect and the present analysis is lim-
Measurement of the antibody-site capture probability Published techniques for the measurement of antibody-site capture probability p [3,4] have relied on the use of fluorescently labeled primary antibodies that are not otherwise required for KinExA analysis. For several years Sapidyne Instruments has advised users of its KinExA equipment to adopt the following procedure for estimating the magnitude of p0 . A high capture probability gives rise to a significant gradient in the amount of antibody captured along the length of the microcolumn as a result of the progressive decrease in unoccupied antibody-site concentration. This effect can be exacerbated by doubling the length of the bead pack. The capture probability is then taken as one minus the square root of the ratio of fluorescence responses for the double-length and normal-length bead packs. In the unlikely event that the estimated capture probability exceeds 30%, a lower value of p0 can be achieved by applying the antibody solution to the microcolumn at a higher flow rate to lower the solid-phase contact time. It should be noted that the capture probability estimated by the following procedure is the capture probability p0 across the first half of a normal-length bead pack, whereas the previously published method [3,4] measured capture p across the entire bead pack. For an exponential gradient along the bead pack [Eq. (4)] the relationship between these two probabilities is
p0 ¼ 1 ð1 pÞ1=2
ð12Þ
In present terminology the highest published capture probability p of 0.42 for the entire bead pack thus corresponds to a capture probability p0 of 0.24 across the first half of the bead pack. On the basis of measurements made regularly in the Sapidyne laboratory for several years, this represents an unusually high capture. Effect of increasing the probability of antibody-site capture Use of Eq. (10), which relies on the approximation that (1 – p0 )1/2 1, is certainly justified in situations where p0
