A theory of coalescence of signaling receptor clusters in immune cells

Abstract

A theory of coalescence of signal receptor clusters in mast cells is developed in close connection with experiments. It is based on general considerations involving a feedback procedure and a time-dependent capture as part of a reaction-diffusion process. Characteristic features of observations that need to be explained are indicated and it is shown why calculations available in the literature are not satisfactory. While the latter involves static centers at which the reaction part of the phenomenon occurs, by its very nature, coalescence involves dynamically evolving centers. This is so because the process continuously modifies the size of the cluster aggregate which then proceeds to capture more material. We develop a procedure that consists of first solving a static reaction-diffusion problem and then imbuing the center with changing size. The consequence is a dependence of the size of the signal receptor cluster aggregate on time. A preliminary comparison with experiment is shown to reveal a sharp difference between theory and data. The observation indicates that the reaction occurs slowly at first and then picks up rapidly as time proceeds. Parameter modification to fit the observations cannot solve the problem. We use this observation to build into the theory an accumulation rate that is itself dependent on time. A memory representation and its physical basis are explained. The consequence is a theory that can be fit to observations successfully.

Keywords: Coalescence via diffusion; Random walk aggregation.

Fig. 1.

Time-laps of receptor cluster coalescence. Total internal reflection fluorescence microscope images of mast cell loaded with fluorescent IgE-receptor complexes pipette-pressed onto a fluid lipid POPC bilayer with 25 mol% DNP lipid after 12 s, 77 s, and 192 s of initial contact. Large-scale reorganization of fluorescent receptors into a synaptic-like structure occurs within 3 min at 37 °C. Scale bar represents 5 μm.

Fig. 2.

Failure of the simple theory. Comparison of data to a coalescence theory for finite coalescence probability given in Eq. (13) with inverse-Laplace transforming Eq. (13) for ξ = 0 (instantaneous coalescence for C₂ → ∞), best fit ξ = 0.007, and ξ = 0.1. The bars represent the error obtained from the propagation of errors using the standard error of the mean obtained from five individual cell data sets.

Fig. 3.

The success of the theory when a memory-dependent capture is incorporated and extraction of system parameters C₂ and α. Best fit procedure with Eq. (1) results in α = 33 × 10⁻¹²/s and αC₂ = 0.12 m²/s². The bars represent the error obtained from the propagation of errors using the standard error of the mean obtained from five individual cell data sets.

Fig. 4.

Examination of the validity of the half-Markoffian approximation. The exact expression is I(t) as in Eq. (20), see text (black solid line). The approximate expression (red dots), obtained by pulling the slow factor out of the integrand but maintaining the upper limit of integration as t, and doing the integral of the fast factor, is Eq. (21). The main figure shows the case of large β = 5 (time variations of the two factors of the integrand disparate). The approximation is reasonable both at short and long times. Insets show the cases β = 1/5 which means the variation is disparate but the fast one is taken out of the integration. This case shows the approximation to be bad. In our analysis we use the approximation in reverse: to replace a product of time functions by a convolution. See text.