The growth and dissolution of a dislocation-free simple cubic (SC) crystal in the vicinity of its apex are studied by computer simulation. The dissolution kinetics with respect to concentration were non-linear and the exact functional dependence of rate on distance from equilibrium was not established. The growth and dissolution kinetics are highly asymmetrical with a very fast dissolution rate, in contrast to that found for growth and dissolution on an infinitely flat (100) surface. During dissolution, the apex appears to develop a (111) face, which is full of kink sites and in its vicinity, (110) faces. The crystal was also found to continue to dissolve even at a small supersaturation and the rate was found to depend on the size of the simulation surface. On the other hand, crystal growth near the previously undissolved apex follows the usual two-dimensional (2D) nucleation. Only partial restoration of a previously dissolved apex was observed when it was subjected to the same number of elementary events and the growth rate of this surface is smaller compared with dissolution at the same distance from equilibrium.