A physico-chemical model of a self-maintaining unity or protocell is constructed on the basis of reaction and diffusion processes. The surface motion of the protocell is taken into account explicitly by a so-called Stefan condition, which leads to a nonlinear feedback to the reaction and diffusion processes. The spatio-temporal dynamics in the neighbourhood of the steady states is investigated in the framework of linear stability analysis with the use of an expansion in terms of spherical harmonics Ylm. It is shown that modes with l greater than or equal to 2 become successively unstable with increasing nutrient supply. The leading instability with l = 2 initiates a process of the nonlinear dynamics which is interpreted as the onset of division. A stabilizing effect of surface tension is also discussed.