  We study theoretically vesicle locomotion due to haptotaxis.
  Haptotaxis is referred to motion induced by an adhesion gradient on a
substrate. The problem is solved within a local approximation where a
Rayleigh-type dissipation is adopted. The dynamical model is akin to the Rousse
model for polymers. A powerful gauge-field invariant formulation is used to
solve a dynamical model which includes a kind of dissipation due to bond
breaking/restoring with the substrate. For a stationary situation where the
vesicle acquires a constant drift velocity, we formulate the propulsion problem
in terms of a nonlinear eigenvalue (the a priori unknown drift velocity) one of
Barenblat-Zeldovitch type. A counting argument shows that the velocity belongs
to a discrete set. For a relatively tense vesicle, we provide an analytical
expression for the drift velocity as a function of relevant parameters. We find
good agreement with the full numerical solution. Despite the oversimplification
of the model it allows the identification of a relevant quantity, namely the
adhesion length, which turns out to be crucial also in the nonlocal model in
the presence of hydrodynamics, a situation on which we have recently reported
[I. Cantat, and C. Misbah, Phys. Rev. Lett. {\bf 83}, 235 (1999)] and which
constitutes the subject of a forthcoming extensive study.
