Beyond 1:1 — Heterogeneity, Avidity and Distributions
What to do when the residuals are structured, and why adding model terms is usually the wrong answer.
The temptation, and why to resist it#
Your 1:1 fit has S-shaped residuals. Your software offers a two-state model, a bivalent analyte model, a heterogeneous ligand model. Any of them will flatten the residuals. This is the most dangerous moment in an SPR analysis, because a model with more parameters always fits better, and fitting better is not evidence of being right.
The correct response to structured residuals is to ask what physical process could produce that structure, and then run the experiment that distinguishes the candidates. There are only a handful, and each has a distinguishing test.
| Cause | Signature | The test that identifies it |
|---|---|---|
| Mass transport | Linear association; dragging dissociation (Schuck & Minton, 1996) | Lower ligand density → fitted ka rises (Myszka et al., 1998) |
| Avidity (multivalent analyte) | Biphasic dissociation with a persistent slow tail (Nieba et al., 1996) | Lower ligand density → tail shortens; a monovalent fragment behaves as 1:1 |
| Ligand heterogeneity | Dissociation is a sum of exponentials, stable across densities (Svitel et al., 2003) | Change immobilisation chemistry; a capture format usually reduces it (Johnsson et al., 1995) |
| Conformational change | Slow additional phase in both association and dissociation | Genuinely hard — needs orthogonal evidence such as stopped-flow or ITC |
| Surface decay | Rmax falls monotonically with cycle number | Plot maximum response against cycle number |
| Non-specific binding | Large signal on the reference channel | Look at the reference channel before subtraction |
| Aggregated analyte | Superstoichiometric response; never saturates (Giannetti et al., 2008) | SEC or DLS on the analyte prep; compare theoretical Rmax |
Avidity, in detail#
A bivalent analyte — most obviously an IgG — can engage two surface sites at once. To leave the surface, both arms must be unbound simultaneously. If one arm releases while the other holds, the free arm is held at high local concentration next to more sites and simply rebinds.
The consequence is that the apparent off-rate is not a property of the interaction at all. It depends on how easy it is for the second arm to find a partner, which depends on surface density, which is an experimental choice. Two labs measuring the same antibody at different immobilisation levels will get different "affinities", and both will be reproducible (Nieba et al., 1996).
If you want the intrinsic constants, remove the valency. Either use a Fab or scFv as the analyte, or invert the format: capture the IgG on the surface via protein A/G or an anti-Fc antibody, and flow the monomeric antigen as analyte (Canziani et al., 2004). The inverted format is usually preferable because it also gives a fresh, uniformly oriented ligand layer each cycle.
Surface heterogeneity, taken seriously#
Amine coupling attaches protein through whichever surface lysine reacts first. Different molecules therefore end up in different orientations, some with the binding site freely accessible, some partly occluded by the dextran, some inactivated by coupling through a residue in the site itself. A real amine-coupled surface is not a population of identical sites. It is a distribution (Johnsson et al., 1995; Svitel et al., 2003).
Fitting such a surface with a one-site model gives a weighted average whose weighting depends on the concentration range you happened to choose. Fitting it with a two-site model asserts that there are exactly two classes, which is a strong and usually unjustified claim.
Distribution analysis
Zhao, Schuck and colleagues developed the alternative: solve for a continuous distribution of sites in the ka–kd plane that is consistent with the data, using regularisation to select the least structured distribution that still fits (Svitel et al., 2003). The regularisation matters — without it the problem is ill-posed and returns spiky, unreproducible spectra (Svitel et al., 2003).
What the method gives you that a discrete model does not:
- A homogeneous surface returns a single narrow peak, rather than a spurious second class introduced to soak up noise.
- A genuinely heterogeneous surface returns a distribution whose width is a quantitative measure of how heterogeneous it is — which lets you compare immobilisation chemistries objectively.
- It uses the whole dataset, association and dissociation across all concentrations, rather than a subset.
How reproducible is any of this, really?#
The honest answer comes from benchmark studies in which many laboratories measured the same interactions with the same reagents. Papalia and colleagues ran a small-molecule benchmark across many labs (Papalia et al., 2006); Katsamba and colleagues did the same for a high-affinity antibody–antigen pair (Katsamba et al., 2006).
The pattern in such studies is consistent and worth internalising: relative rankings are robust; absolute constants are not. Laboratories agree about which of a set of compounds binds most tightly far better than they agree about the numerical KD. Systematic errors — surface activity, concentration accuracy, partial transport limitation — bias a whole panel in one direction and so cancel in a comparison.
Sources cited on this page
Listed alphabetically. Each badge records whether the bibliographic record was confirmed against Crossref. unverified marks a real, deliberately chosen source whose volume and page numbers we have not yet machine-checked — it is not a comment on the science.
- Canziani et al., 2004G. A. Canziani, S. Klakamp, D. G. Myszka (2004). Kinetic screening of antibodies from crude hybridoma samples using Biacore. Analytical Biochemistry 325, 301–307. doi:10.1016/j.ab.2003.11.004 unverifiedRanking antibodies directly from crude supernatant, without purification.
- Giannetti et al., 2008A. M. Giannetti, B. D. Koch, M. F. Browner (2008). Surface plasmon resonance based assay for the detection and characterization of promiscuous inhibitors. Journal of Medicinal Chemistry 51, 574–580. doi:10.1021/jm700952v unverifiedThe primary source for using superstoichiometric, non-saturating responses to identify aggregating and promiscuous compounds.
- Johnsson et al., 1995B. Johnsson, S. Löfås, G. Lindquist, Å. Edström, R.-M. Müller Hillgren, A. Hansson (1995). Comparison of methods for immobilization to carboxymethyl dextran sensor surfaces by analysis of the specific activity of monoclonal antibodies. Journal of Molecular Recognition 8, 125–131. doi:10.1002/jmr.300080122 unverifiedMeasures surface activity across coupling chemistries on the same antibody — the primary source for the claim that amine coupling inactivates a substantial fraction of ligand.
- Katsamba et al., 2006P. S. Katsamba, I. Navratilova, M. Calderon-Cacia, et al. (2006). Kinetic analysis of a high-affinity antibody/antigen interaction performed by multiple Biacore users. Analytical Biochemistry 352, 208–221. unverified
- Myszka et al., 1998D. G. Myszka, X. He, M. Dembo, T. A. Morton, B. Goldstein (1998). Extending the range of rate constants available from BIACORE: interpreting mass transport-influenced binding data. Biophysical Journal 75, 583–594. doi:10.1016/S0006-3495(98)77549-6 verifiedShows transport can be fitted rather than merely avoided, and defines the transport coefficient kₜ.
- Nieba et al., 1996L. Nieba, A. Krebber, A. Plückthun (1996). Competition BIAcore for measuring true affinities: large differences from values determined from binding kinetics. Analytical Biochemistry 234, 155–165. doi:10.1006/abio.1996.0067 unverifiedA direct demonstration that surface-measured kinetic constants can diverge substantially from solution affinities, and a solution-competition format that avoids the problem.
- Papalia et al., 2006G. A. Papalia, S. Leavitt, M. A. Bynum, et al. (2006). Comparative analysis of 10 small molecules binding to carbonic anhydrase II by different investigators using Biacore technology. Analytical Biochemistry 359, 94–105. unverifiedA benchmark study in which many labs measured the same interactions; the spread between labs is the honest measure of SPR reproducibility.
- Schuck & Minton, 1996P. Schuck, A. P. Minton (1996). Analysis of mass transport-limited binding kinetics in evanescent wave biosensors. Analytical Biochemistry 240, 262–272. doi:10.1006/abio.1996.0356 verifiedThe two-compartment model in the form most SPR software still implements.
- Svitel et al., 2003J. Svitel, A. Balbo, R. A. Mariuzza, N. R. Gonzales, P. Schuck (2003). Combined affinity and rate constant distributions of ligand populations from experimental surface binding kinetics and equilibria. Biophysical Journal 84, 4062–4077. doi:10.1016/S0006-3495(03)75132-7 verifiedThe distribution analysis that replaces "the surface has one kind of site" with a measured spectrum of sites.