Vesicles in haptotaxis with hydrodynamical dissipation.

We analyze the problem of vesicle migration in haptotaxis (a motion directed by an adhesion gradient), though most of the reasoning applies to chemotaxis as well as to a variety of driving forces. A brief account has been published on this topic. We present an extensive analysis of this problem and provide a basic discussion of most of the relevant processes of migration. The problem allows for an arbitrary shape evolution which is compatible with the full hydrodynamical flow in the Stokes limit. The problem is solved within the boundary integral formulation based on the Oseen tensor. For the sake of simplicity we confine ourselves to 2D flows in the numerical analysis. There are basically two regimes (i) the tense regime where the vesicle behaves as a "droplet" with an effective contact angle. In that case the migration velocity is given by the Stokes law. (ii) The flask regime where the vesicle has a significant (on the scale of the vesicle size) contact curvature. In that case we obtain a new migration law which substantially differs from the Stokes law. We develop general arguments in order to extract analytical laws of migration. These are in good agreement with the full numerical analysis. Finally we mention several important future issues and open questions.