# Team:Aberdeen Scotland/Modeling/Assay

Team:Aberdeen Scotland/Modelling - 2014.ogem.org

# Assay Model

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.

### 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