Statistical analysis of fluorescence time-resolved data
As shown in , the rate of association of the excess DOTA(Tb) with 2D12.5 was taken to equal the rate of dissociation of ABD(Y) from 2D12.5 mAb. The terbium luminescence response at 545 nm for DOTA(Tb) bound to 2D12.5 mAb (8
) was monitored as a function of time. To assure that excess DOTA(Tb) was present, three different concentrations were used. Similar results were obtained in all cases, and all three sets of triplicate results were fitted to one curve. The koff
was obtained from a fit of the following equation that describes the pseudo-first order association kinetics: Y
), where Y
is the terbium luminescence intensity and the pseudo-first-order rate constant for binding DOTA(Tb) is k = koff
Competition experiment to measure koff by forming the luminescent DOTA(Tb) complex with antibody 2D12.5. First the antibody was saturated with ABD(Y), then it was mixed with a large excess of DOTA(Tb).
The dissociation rate constant of ABD(Y) with 2D12.5 antibody was determined from the plot of the fluorescence signal growth of 2D12.5-DOTA(Tb) complex with time (). Attempts to fit the data to more than one rate constant gave inferior results. Having measured koff and KD, we can calculate kon = (koff/KD) M−1s−1 ().
Kinetic parameters for the dissociation of ABD(Y) ligand from Antibody 2D12.5.b
With these results we are now able to model the behavior of this infinite affinity system, explaining previous observations in vitro
and in vivo.
Corneillie et al. (11
) studied the rate of formation of the covalent bond between the acryloyl group of AABD(Y) and the cysteine of the G54C mutant of antibody 2D12.5. As shown in , we were able to fit Corneillie’s results with the following set of kinetic equations:
In order to fit the complex kinetic behavior observed in reference 11
, it is necessary to assume that AABD(Y) can form either a reactive complex CR
(which can form the product P
) or a non- reactive complex CNR
with the G54C antibody binding site. This is consistent with the observation of two modes of binding in crystal structures of the parental antibody complex (12
). The fit in gives the value kirr
= 2.5 × 10−2
derived using the nonlinear least-squares method. The partitioning of complex formation between the non-reactive CNR
, with probability 1 − x
= 0.8, and the reactive CR
, with probability x
= 0.2, is also necessary to fit these data.
It is expected that the different binding modes should have different values for kon
. However, the values of kon
are not sufficiently different to be measured separately by the methods available to us. The crystal structures show two binding modes in which the antibody-DOTA interactions are almost identical but the DOTA side chain is oriented differently (12
). Based on the proximity of the DOTA side chain to residue 54 in one structure but not the other, these could plausibly correspond to the reactive and unreactive binding observed kinetically in reference 11
. The goodness of fit in justifies the use of this minimal set of rate constants to model the processes of tumor uptake discussed below.
We have found that the 2D12.5 G54C Fab expressed in D. melanogaster S2 cells (11
) is largely cysteinylated (has a cysteine monomer attached to Cys-54 by a disulfide bond), which protects the Cys-54 side chain from reaction with electrophiles such as AABD(Y) (13
). The experiments in reference 11
were carried out using an excess of radiolabeled AABD(Y) to detect the rate of covalent bond formation by Michael addition; thus the results were sensitive only to protein with a free thiol at G54C, and the value of kirr
is not affected by the presence of cysteinylated protein.
Reviewing the values of these new rate constants shows that the second-order association reaction controlled by kon
= 3.5 × 106
is rather rapid compared to the more typical values of kon
observed for antibody-protein association (14
). However, it is not as fast as the biotin-streptavidin association, for which the rate constant is reported to be kon
= 7.5 × 107
at 25 °C (15
). This comparison is important because the streptavidin-biotin system has been widely used for pretargeted delivery of small probes for imaging and therapy of cancer (16
). However, streptavidin is immunogenic in human patients (17
), so the irreversible antibody capture system may provide a desirable alternative: currently, making antibodies minimally immunogenic is less challenging than doing the same for streptavidin (5
The rate constant for reversible dissociation of nonelectrophilic ABD(Y) from parental antibody 2D12.5, koff
= (7.0 ± 0.7) × 10−3
, is quite fast compared to that for dissociation of biotin from streptavidin, which is reported to be koff
= (4.1± 0.3) × 10−5
at 37 °C under comparable conditions (18
). This rapid dissociation, combined with the effects of irreversible binding and the branching between reactive and non-reactive complexes, makes the behavior of the reactive AABD(Y) with the G54C binding site quite unusual. The results above imply that in a homogeneous solution approximately
of the AABD(Y) molecules will form permanent bonds with the receptor site the first time they bind, without ever dissociating, while the majority (≈ 80%) will dissociate, diffuse, and rebind. Because the results in vivo
will depend on additional factors such as effective target site concentration on cell surfaces and transport behavior of the probe in a tumor, it was important to observe this system in an animal model to see if it would exhibit biological properties useful for imaging and perhaps therapy.
Wei et al. (5
) prepared an engineered version of the 2D12.5 G54C antibody with infinite affinity, termed DAbR1, in which the binding site was expressed as part of an scFv-Fc fusion protein on the surface of U-87 human glioma cells implanted in the flanks of SCID mice. The behavior of AABD(86
Y) after injection into the tail vein was monitored by microPET imaging, with highly successful results; some experimental data from reference 5
are used in . Having determined values for kon, koff, kirr,
under physiological conditions in vitro
, we sought to use these numbers to fit the data for in vivo
tumor uptake in order to understand how this system behaves relative to reversible ligand-receptor pairs.
Figure 6 Experimental tumor uptake data (●±SD) from reference 5, showing the results of injection of AABD(86Y) into the tail veins of SCID mice bearing xenograft U-87 human glioma tumors expressing the DAbR1 reporter gene, plotted as average % (more ...)
It is convenient to describe the basic unit of a glioma tumor as a “cord,” a cylindrical array of cancer cells surrounding a blood capillary (19
), from which the tumor tissue receives nutrients and also small molecules such as the PET probe AABD(86
Y). Characterizing the behavior of cells around one capillary can lead to a useful description of the main features of the tumor (20
). Our goal here is to consider the major consequences of irreversible binding of the probe to its receptor, in order to gain insight into the essentials of this chemistry in a living organism.
The immediate requirement is the formulation and solution of a set of partial differential equations to describe not only the chemistry already considered in vitro
but also the diffusion of the AABD(Y) probe in vivo
to its receptors on the tumor cell surface. Assuming cylindrical symmetry and considering a small segment of a long cylinder reduces the geometry to one dimension: the radial distance r
from the center of the capillary (20
). The appropriate reaction-diffusion equations are given below as equations 5
; only the probe diffuses, but the concentration of each species depends on both r
We sought realistic solutions to these equations by fitting the in vivo
tumor uptake data from reference 5
, using nonlinear least-squares with the concentration of AABD(Y) at the periphery of the capillary as the only adjustable parameter, along with literature values for the typical radius of a capillary (10μ) and a tumor cord (≈100μ) (20
), the diffusion coefficient of a metal chelate in brain tissue (3.4 × 10−6
), and the observed ≈5 min circulatory half-life of the AABD(Y) probe (5
). The calculated behavior is compared to experimental data in ; it is important to emphasize that the values for kon, koff, kirr,
were those determined from in vitro
experiments described above, with no adjustments. An important underlying assumption throughout is that, consistent with the crystal structures of parental Fab-DOTA complexes (12
), the complex tolerates molecular changes at positions peripheral to the sites of van der Waals contact between protein and DOTA. Thus the most important interactions between receptor and ligand persist through genetic substitution of Cys for Gly at position 54, reformatting of the mouse IgG to chimeric G54C Fab or to G54C scFv, and chemical changes at the para
position of the DOTA side chain.
also illustrates the predicted effects of eliminating irreversible binding of the AABD(Y) probe in the tumor while maintaining reversible binding (setting kirr
→ 0), or eliminating probe binding altogether (setting kon
→ 0). The fit to the in vivo
experimental data in was made with the assumption that all the G54C binding sites expressed on cell surfaces have the side chain of Cys-54 free to react with the acryloyl group of AABD(Y), which would not be true for the highly cysteinylated antibody expressed in S2 cells (13
). Changing the calculation to assume a significant degree of blocked Cys-54 leads to curves that resemble the red curve (kirr
→ 0) in , diverging markedly from experimental observation. Thus the binding sites expressed on the surface of U-87 cells in vivo
exhibit little or no blockage of the Cys-54 side chain. Because there is no established way to change the oxidation state of the expressed G54C side chain on cell surfaces in vivo
, this information is crucial to the design of future experiments. It also illustrates the diversity of posttranslational modification in eukaryotic cells.
One of the central issues of molecular targeting is the transport of probe molecules, whether large or small, from the blood into the target tissue (21
). A number of interesting experiments have been carried out to demonstrate the effects of various parameters on this process (e.g., 23
). The results in (5
) demonstrate rapid, durable uptake in a xenograft tumor but lack the spatial resolution to provide information about whether the irreversible binding occurs at the edge of the capillary or throughout the tumor. The model developed here predicts that probe molecules present in the tumor 1 hr after administration will all be irreversibly bound to the engineered receptor, with a spatial distribution plotted in . While the average concentration of probe in the tumor at 1 hr was observed to be approximately 7%/g, the predicted concentration near the capillary is almost 12%/g, declining to slightly more than 6%/g near the periphery of the tumor cord.
Figure 7 Top: predicted concentration of the probe 60 min after injection, as a function of distance from just outside the boundary of the capillary (12μ) to the outer periphery of the tumor cord (110μ from the center of the coordinate system). (more ...)
It is informative to use the solutions of equations 5
to gain a sense of how the concentrations of the different probe-containing species vary with time and position in the tissue. The curves in imply that the free probe AABD
) is quickly incorporated into complexes CNR
, with the latter rapidly converted to P
. Because of the short lifetime of either complex, the product P
becomes the dominant species within a short period. Rapid clearance of the probe from the circulation leads to disappearance of the other species from the tumor before 60 minutes have elapsed. The behavior illustrated in for positions near the capillary and near the outer periphery of the tumor cord is predicted to be qualitatively similar at intermediate distances from the capillary, with no remarkable differences except that the final concentration of P
varies with r
according to , and the concentrations of all the species are predicted to change more slowly farther from the capillary.
The interplay between molecular transport and ligand-receptor binding provides an almost inexhaustible range of possibilities for molecular imaging probes. This is above all true when irreversible binding may take place, as in the case of antibodies with infinite affinity (3
). In the most extreme manifestation, an irreversible probe molecule would encounter its receptor and become permanently attached immediately after entering tumor tissue; in that case the curve in would be a narrow peak at the left side, falling to zero with increasing distance. In the case examined here, the probe has both reactive and non-reactive modes of binding to the receptor, and the latter is more probable; non-reactive binding and dissociation can serve to further spread the probe through the tissue before it becomes permanently fixed on a receptor.
If we compare this to purely reversible binding, we see the possibility that it can offer a useful middle ground encompassing the most desirable aspects of opposing extremes. Very strong (though reversible) binding such as provided by the streptavidin-biotin pair would be expected to yield a zone of stably bound probe molecules near the capillary, as long as streptavidin is in excess over biotinylated probe molecules. This nonuniform distribution of probes in tissue, referred to as the binding-site barrier, could be ameliorated if the concentration of avidin molecules were too low to efficiently capture the probes, or if the concentration of probe molecules were large enough to saturate the streptavidin sites, allowing other probe molecules to diffuse past. On the other hand, weaker reversible binding naturally allows small probe molecules to penetrate into tissue by virtue of repeated dissociation, diffusion, and association. This might eventually lead to uniform distribution in tissue, but since the binding is not durable, the concentration of the probe falls to zero in an inconveniently short time. The contour plot in implies a combination of durable binding and broad distribution that would be difficult to produce by other means.