We have investigated the shape, size, and motility of a minimal model of an adherent biological cell using the Monte Carlo method. The cell is modeled as a two dimensional ring polymer on the square lattice enclosing continuously polymerizing and depolymerizing actin networks. Our lattice model is an approximate representation of a real cell at a resolution of one actin molecule, 5 nm. The polymerization kinetics for the actin network are controlled by appropriate reaction probabilities which correspond to the correct experimental reaction rates. Using the simulation data we establish various scaling laws relating the size of the model cell to the concentration of polymerized and unpolymerized actin molecules and the length of the enclosing membrane. The computed drift velocities, which characterize the motility of the cell, exhibit a maximum at a certain fraction of polymerized actin which agrees with physiological fractions observed in experiments. The appearance of the maximum is related to the competition between the polymerization-induced protrusion of the membrane and the concomitant suppression of membrane fluctuations. (C) 2004 American Institute of Physics.