A random walk model related to the clustering of membrane receptors

Abstract

In a cellular medium, the plasmic membrane is a place of interactions between the cell and its direct external environment. A classic model describes it as a fluid mosaic. The fluid phase of the membrane allows a lateral degree of freedom to its constituents: they seem to be driven by random motions along the membrane. On the other hand, experimentations bring to light inhomogeneities on the membrane; these micro-domains (the so-called rafts) are very rich in proteins and phospholipids. Nevertheless, few functional properties of these micro-domains have been shown and it appears necessary to build appropriate models of the membrane for recreating the biological mechanism. In this article, we propose a random walk model simulating the evolution of certain constituents-the so-called ligands-along a heterogeneous membrane. Inhomogeneities-the rafts-are described as being still clustered receptors. An important variable of interest to biologists is the time that ligands and receptors bind during a fixed amount of time. This stochastic time can be interpreted as a measurement of affinity/sentivity of ligands for receptors. It corresponds to the sojourn time in a suitable set for a certain random walk. We provide a method of calculation for the probability distribution of this random variable and we next determine explicitly this distribution in the simple case when we are dealing with only one ligand and one receptor. We finally address some further more realistic models.