The kinetics of aggregation phenomena I. Minimal models for patch formation on lymphocyte membranes

Abstract

Numerous biological processes involve the assembly of one or more monomers into aggregates or networks of interconnected units. In this paper we present the initial aspects of a mathematical theory for network formation on lymphocyte membranes. We assume the fluid mosaic membrane model is valid, that a lymphocyte possesses a homogeneous set of mobile but membrane bound receptors and that these receptors can form bimolecular complexes with antigen. We show that these complexes tend to aggregate and derive expressions for their size distribution as a function of time, antigen valence and concentration, and antigenreceptor affinity. At early times, the mass of the system (receptors plus antigen) is in very small aggregates. However under appropriate conditions, a critical time is reached at which they coalesce in such a way that the mass shifts, becoming concentrated predominantly in large aggregates. We assume that this coalescence (“patch” formation) is a necessary condition for lymphocyte triggering and briefly pursue the consequences. It is shown that the time required for patch formation is a sensitive function of affinity ( K), antigen valence and antigen concentration ( C), and that if KC is either too high or too low patch formation will not be possible. Moreover within the range of binding constants which can lead to patching, there will be an optimum value which leads to the fastest rate of triggering, and this optimum shifts to higher affinity as the concentration of free antigen surrounding the cell decreases. For optimum KC values we estimate times typically of the order of (10–100) seconds for patch formation. The theory also suggests that if antigen valence is too low, triggering will not be possible within times of interest, without introducing other factors. It thus leads naturally to a requirement for auxiliary cells which would tend to present low valence antigens in such a way that the B lymphocytes see an effective, increased valence. The theory, although primitive, thus meets some minimal requirements in that it distinguishes binding reactions from triggering reactions, makes predictions consistent with observations on affinity maturation and the nonresponsiveness to high doses and low doses of antigen, and suggests the need for helper cells (or their products) in order for low valence antigens to be effective in lymphocyte triggering.