A crucial part of the assay is the antibody-to-antigen binding. In this section we try to analyse its behaviour and try to determine if the system is going to have the appropriate amount of sensitivity.

Consider and antibody 'A' and its complimentary antigen 'a'. We have looked at a simple model, which can be illustrated by the following equations^[1]:

$$\frac{d[A-a]}{dt}=k_a[A][a] \\ \frac{d[A]}{dt}=-k_a[A][a] \\ \frac{d[a]}{dt}=-k_a[A][a]$$

• where

  [A]	- concentration of antibody 'A'
  [a]	- concentration of antigen 'a'
  ka	- affinity of antibody and antigen
  [A-a]	- concentration of antibody 'A'

In our case, as the antibody and antigen concentrations do not change in time individually, we can simplify this system to the following:

$$ \frac{[A-a]}{dt}=k_{a}(A_{0}-[A-a])(a_{0}-[A-a])$$

where A₀ and a₀ are the initial concentrations of the antibody and antigen.

We can clearly see that this system is not linear, but we can simulate it and analyse it.

Fig.1 Antibody-to-Antigen binding time evolution curves, according to different affinity values

From this model we deduced several important points.

Firstly, the abundance of the antigen is a limiting factor. In fact, the less the antigen concentration is the slower the binding process is going to be. This can be partially counteracted by increasing the amount of antibodies available. However, this is generally a good sign, as that means the system is less prone to false positives.

Moreover, as initial antigen concentration increases the efficiency of the binding increases exponentially, thus making it much easier to detect the antigen.

Furthermore, on Fig.1 we can see how the binding lag is influenced by different affinities. From the highest to the lowest curve, the affinity differs by 1 order of magnitude, but we still get at least 30% binding after just an hour. This further confirms that our system is stable enough to handle different types of antibodies.

This is an important point as we are detecting for two different antigens with two different antibodies and are demanding similar behaviour from both.

References

[1] McFadden, R., Kwok, C. S., "Mathematical Model of Simultaneous Diffusion and Binding of Antitumor Antibodies in Multicellular Human Tumor Spheroids, CancerRes 1988; 48:4032-4037