On contacts of immunocompetent cells with antigen (Note on a probability model)

Abstract

This communication continues with the mathematical formulation and solution of some problems connected with the beginning of the immunological reaction. This paper generalizes some previous results concerning the probability of the event that such a contact of an immunocompetent cell (ICC) with antigen, which is followed by the differentiation of this cell towards the antibody forming cell (AbFC), takes place. Previous papers considered the onset of the differentiation of an ICC caused by the contact of this cell with at least a certain threshold amount of antigen (assuming that these contacts form a non-homogeneous Poisson process). This paper is based on the more general and more realistic assumption thatn such contacts are necessary for the stimulation of ICC differentiation (this new assumption has its origin in some new biological findings on the nature of the contact considered, and corresponds with the idea that the diffentiation of ICC does not begin until the antigen is bound on a certain fixed number of receptors on the cell surface). Under this assumption, the probability of the event that the differentiation of the cell begin at all (i.e. that at leastn contacts will take place), and the probability distribution of the epoch of the beginning of the differentiation (i.e. of the epoch at which thenth contact take place), are derived.— If necessary, these new forms of probability distributions may be used (instead of previous ones) as building stones in models of ICC differentiation.