  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.
