The procedure described in this chapter for the determination of affinity constants and kinetic dissociation constants by band-shift assay refers to an ideal antibody fragment (e.g., a scFv or an Fab fragment) binding to a well-behaved protein antigen (pure, of well-defined oligomeric state, migrating as a single band in non-denaturing gel electrophoresis. As mentioned in the introduction, either the antibody or the antigen has to be labeled in these assays. In order to illustrate different labeling methods, we will use radiolabeled recombinant antibodies in the protocol for the determination of Kd, and fluorescently labeled antigen in the protocol for the determination of koff. Recombinant antibodies can conveniently be radiolabeled site-specifically and without loss of immunoreactivity using genetically introduced phosphorylation sites . This procedure is described in this Chapter. However, other radiolabeling methods can be used (e.g., radioiodination using Iodogen tubes), provided that immunoreactivity is preserved and that a sufficiently high specific activity is achieved. The fluorescent antigen labeling method presented in this Chapter features the use of N―hydroxysuccinimido ester derivatives of a commercially available cyanine dye (Cy5―NHS; Amersham Pharmacia Biotech). These reagents react with primary amino groups in the antigen molecule, and may impair the antigen’s ability to react with the antibody after labeling.

Phosphorylation of recombinant antibodies

1. Dissolve in buffer A the purified antibody carrying an engineered phosphorylation site (e.g., the sequence DDDSDDDD; at 1 mg/ml concentration (alternatively, exchange buffer by dialysis).
2. To 100 µl antibody solution, add 500 units casein kinase II (4 µl; New England Biolabs) and 1.6 mCi of [g-32P]ATP (12 µl, Cat. No. 35020 from ICN Biomedicals).
3. Incubate at room temperature for 1 hour.
4. Remove unreacted ATP using a disposable PD-10 column (Amersham Pharmacia Biotech), preblocked with bovine serum albumin dissolved in PBS (2 mg/ml) and extensively washed with PBS. This gel filtration procedure is performed by loading the 116 µl reaction mixture onto the column, waiting until the column is dry, adding 2.4 ml PBS (without collecting the liquid), waiting until the column is dry, and finally adding 2 ml of PBS, collecting the liquid from the column. This fraction of radiolabeled antibody can be used in band-shift assays.

Labeling the antigen with a red fluorophore

1. Dissolve in PBS the purified antigen at 1 mg/ml concentration (alternatively, exchange buffer by dialysis).
2. Add 1 ml to a dry tube containing an adequate amount of Cy5-NHS for the labeling of 1 mg protein (Cat. No. PA25001 ― monoreactive Cy5 dye pack from Amersham Pharmacia Biotech).
3. Incubate 30 min. at room temperature.
4. Remove unreacted Cy5 dye using a disposable PD-10 column (Amersham Pharmacia Biotech), extensively pre-washed with PBS (> 25 ml). This gel filtration procedure is performed by loading the 1 ml reaction mixture onto the column, waiting until the column is dry, adding 1.5 ml PBS (without collecting the liquid), waiting until the column is dry, and finally adding 2.5 ml of PBS, collecting the liquid from the column. A clear separation between incorporated and unincorporated dye should be visible. The collected blue-coloured fraction of fluorescently labeled antigen can be used in band-shift assays.

Native gel electrophoresis

1. Protein samples (4 µl) are diluted with 2 µl of gel mix immediately before the electrophoretic separation.
2. Apply protein samples onto a Gradient 8-25 Phast Gel (Amersham Pharmacia Biotech; a different gel percentage may be required, depending on the mobility of the proteins which have to be measured), equipped with native buffer strips in a Phast System separation module (Amersham Pharmacia Biotech) and run the gel according to the manufacturer’s instructions. Alternatively, vertical non-denaturing polyacrylamide gel electrophoresis can be performed, for example as described in .
3. Remove the gel from the Phast System and image with a suitable imaging technique (autoradiography with a PhosphorImager in the case of radioactive band-shifts; fluorescence imaging in the case of fluorescent band-shifts; .

Band-shift assay for the determination of Kd

1. Determination of the concentration of radiolabeled antibody

1. In parallel tubes, incubate a ‰ 1 µM radiolabeled antibody solution in PBS with increasing concentrations of antigen (e.g., ranging between 0.1 µM and 10 µM). At these high concentrations (> Kd), the antibody should rapidly and quantitatively bind to the available antigen.
2. After few minutes of incubation, run the reaction mixtures on a non-denaturing gel as described in the previous section.
3. Image the gel by autoradiography (preferably on a PhosphorImager). The concentration of antigen at which a "band shift" is observed (Figure 1) is equal to the concentration of radiolabeled antibody used in the assay (N.B.: it is NOT equal to the Kd, since we are working in a concentration range > Kd).

1. Determination of the dissociation constant

In these experiments, it is convenient to work at antibody concentrations lower than the dissociation constant. This range of concentrations is typically not known a priori, and is determined by performing band-shift experiments at low antibody concentrations (e.g., 0.1 ― 1 nM). The intensity of the bands is integrated with appropriate methods and fitted to the equation: Kd = [A][B]/[AB] (this procedure relies on the assumption that band intensity is proportional to the concentrations at the moment the sample is loaded on the gel). This fitting may yield an acceptable Kd value, or may indicate in which concentration range one should work to obtain more accurate results. Another important issue to consider is how long should antibody and antigen be allowed to react before analysing the reaction mixture on a native gel. In pseudo-first order conditions (e.g., when the antibody concentration is kept constant and << Kd) this reaction time is a function of the kinetic association constant kon and of the antigen concentration. The concentrations of antibody free and in complex will exponentially tend to a limit value, and one should wait at least t > 1 / (kon x [Antigen] total) before considering the reaction close to completion. Whenever the kon of the antibody is not known, it is important to measure affinity constants by band-shift with two different incubation times. If the reaction has reached equilibrium, the affinity values in the two experiments should not differ significantly. The protocol listed below is for a labeled antibody with Kd = 3 nM and kon = 106 s-1M-1.

1. In parallel tubes, incubate a 0.3 nM radiolabeled antibody solution with increasing concentrations of antigen (e.g., ranging between 0.3 nM and 30 nM) in PBS.
2. After 30 minutes incubation at room temperature, run the reaction mixtures on a non-denaturing gel as described in the previous section.
3. Image the gel by autoradiography (preferably on a PhosphorImager). The concentration of antigen at which a "band shift" is observed (Figure 1) is equal to the dissociation constant Kd. Alternatively, the Kd value can be obtained by fitting the intensity of the bands to the equation: Kd = [A][B]/[AB].
4. To be sure that the equilibrium is reached when the samples are applied onto the gel, repeat the procedure, with 2 hours incubation time in step 2.

Band-shift assay for the determination of k_off

A prerequisite of this technique is that the antibody-antigen complex can be detected in a non-denaturing gel (if possible as a single band). If the antigen is fluorescently labeled, it is essential to be able to resolve the band of the complex from the band of the free antigen. It is also worth mentioning that only negatively charged proteins (or protein complexes) migrate towards the anode in a standard non-denaturing polyacrylamide gel with Tris/Glycine buffer, pH 8.3 (see above). If one wishes to study positively charged proteins (or protein complexes), one needs to work with reverse polarity electrodes and possibly with buffer systems different from Tris/Glycine (see manufacturer’s instructions of the Phast System, Amersham Pharmacia Biotech). The following protocol is written for the negatively charged complex of a recombinant antibody with a fluorescently labeled antigen. Although the reaction conditions are here given in the micromolar range, much lower concentrations can also be used if a high-sensitivity luminescence analyser is available (see for example .

1. In separate reaction tubes, incubate a 1 µM solution of fluorescently labeled antigen with a 1 µM solution of antibody in PBS for few minutes at room temperature.
2. At different times (e.g., at time 0h, 24h, 48h, 60h, 66h, 69h, 71h, 71h30min, 71h40min; the choice of time intervals depends on the koff value) add unlabeled antigen to one of the tubes, reaching a final antigen concentration of 20 µM. By this procedure, in the different tubes the antibody-(labeled antigen) complex is allowed to compete with a molar excess of unlabeled antigen for different amounts of time.
3. At the end of the competition reactions (e.g., at 71h 41 min), run the reaction mixtures on a non-denaturing gel as described in a previous section.
4. Image the gel with a suitable fluorescence imager, making sure that even the strongest bands fall in the linear detection range of the imager .
5. Integrate the fluorescence intensity of the bands.
6. Plot band volume versus time (Figure 3). An exponential fit of the band volume decaying with time will yield the characteristic constant k_off.