Bivalent analyte evaluation

Dear Arnoud,
We were reading about evaluation of bivalent analytes, as we are interested in figuring out, by how much the affinity has been increased due to avidity. Therefore, a nanobody was cloned in a way, to create a homomeric dimer linked by a flexible linker. Unfortunately I wasn't able to understand all the literature on this matter.

1) As far as I could understand, kd1 and kd2 are interchangeable, because the binding of a subunit does not influences the dissociation of another one. Therefore, kd1 should be equal to kd2. In both cases the dissociation of the complex occours "randomly".
2) Ka1 and ka2 should be different, since the binding of one subunit should promote the binding of another subunit. However, the second ka gives off no signal, as there is no mass increase during the second binding.

3) When we try to fit our data with bivalent analyte model, we get two different ka and kd. However, we don't get an overall Kd. We tried to figure out how to do it, and one paper suggested that Kd is the product of Kd1 and Kd2. Which makes mathematically sense, if we assume that the Kd is derived from sum of the change of free Gibbs energy.

4) However, most paper report two separate Kd values, without giving us the affinity improved by avidity.
I am a biochemist and most differential equations are too complicated for me as I don't use them on a daily basis, and I wasn't able to get smarter from all the equations shown in papers and on this website.

I hope that you could once again help us, and I am very sorry for the barrage of questions that I am posting here.

Thank you,
Marton

1) correct
2) ka1 is reported in M-1s-1 and because ka2 does not give a response increase it is calculated in RU-1s-1 to keep the formulas consistent ?! To calculate ka2 (M-1s-1): ka2 (RU-1s-1) * Mranalyte * 100 = ka2 (M-1s-1). (1)
3) The problem with multiplying both KD's is that the result can get an absurd low affinity. Also the affinities KD1 and KD2 are not independent from each other since the first binding changes the chances of the second binding.
If you can reach steady state in your experiments you can try to use the steady state fit to determine the overall KD of the interaction. In the same way you can do a solid state affinity experiment (basically an ELISA; www.protocol-online.org/biology-forums/posts/35807.html ).
4) I find this paper very informative in concepts about bivalent analytes: (2)

1. Karlsson, R., Mo, J. A. and Holmdahl, R.; Binding of autoreactive mouse anti-type II collagen antibodies derived from the primary and the secondary immune response investigated with the biosensor technique. Journal of Immunological Methods (188) 1: 63-71; 1995.
2. Vauquelin, G. and Charlton, S. J.; Exploring avidity: understanding the potential gains in functional affinity and target residence time of bivalent and heterobivalent ligands. British Journal of Pharmacology 2013.

Suggested reading:
• Muller, K. M., Arndt, K. M. and Pluckthun, A.; Model and simulation of multivalent binding to fixed ligands. Analytical Biochemistry (261) 2: 149-158; 1998.
• Cooper, M. A. and Williams, D. H.; Kinetic analysis of antibody-antigen interactions at a supported lipid monolayer. Analytical Biochemistry (276) 1: 36-47; 1999.

And two more advanced/difficult papers:
• Tiwari, P. B., Wang, X., He, J., et al.; Analyzing surface plasmon resonance data: Choosing a correct biphasic model for interpretation. Review of Scientific Instruments (86) 3: 035001; 2015.
• Tiwari, P. B., Üren, A., He, J., et al.; Note: Model identification and analysis of bivalent analyte surface plasmon resonance data. Review of Scientific Instruments (86) 10: 106107; 2015.

Wow, thank you very much! This is very helpful