# Team:HZAU-China/Modeling/Immune responce

Immune response

### Aim:

To know how much glycoprotein could cause the immune response to generate adequate levels of antibody.

### Steps:

1. Build the model;

2. Determine the parameters;

3. Simulate and analyze the results.

### Background:

In mammals, when foreign materials are detected in the body, B cells transform into B lymphocytes that could secrete antibodies. B cells could also proliferate itself and antibodies are produced by the stimulation of specific antigens.

Most of the antigens cannot proliferate, so they will be cleaned by the immune system quickly. But here our glycoprotein is proliferative, and it is possible for it to ultimately reach a steady level.

Suppose N is the concentration of the antigen (glycoprotein) and A is the concentration of the antibody. Assume that the rate of B lymphocyte secreting antibody is f(N).If the amount of antigen is small, f(N) is approximate to βN where β is a parameter. Suppose that the growth rate of the antigen is αN and the mortality rate of the antigen is subject to mass interaction law with antiboby, then, the change rate of antigen with time can be expressed as:

N=αN-κAN,          (1)

where κ is parameter.

Suppose the mortality rate of the antibody is subject to mass interaction law with antigen and the self degradation rate of antibody is μA, then, the change rate of antibody with time is expressed as:

A= - κAN+βN-μA.          (2)

The values of the parameters are determined by experiments using data fitting and listed in the following table.

We used MATLAB to solve differential equations. The concentrations of antibody and antigen change over time and the figure is as follows.

Also we drew the relationship diagram between antibody and antigen

From the figures we could know that the amount of antigen gradually decreases and the amount of antibody gradually increases with the extension of time. Also we may find out the relationship between antibody and antigen. The amount of antibody would eventually reach a stable value. If the amount of antibody does not meet our requirements, the dogs could be immunized for many times.

### Reference:

1.Liu Qing. The analysis of the relationship between antibody and antigen in immune response[J]. Journal of Ningxia University:17(4):64-66 (In Chinese)

2.Yumin Xia, A.J. The constant region affects antigen binding of antibodies to DNA by altering secondary structure[J]. Molecular Immunology,2013, (11) : 28–37

3.Jane M., et al., An empirical modeling platform to evaluate the relative control discrete CHO cell synthetic processes exert over recombinant monoclonal antibody production process titer. Biotechnology and Bioengineering, 2011. 108(9), 2193 - 2204