In eukaryotic cells, microtubules and their associated motor proteins can be organized into various Large-scale patterns. Using a simplified experimental system combined with computer simulations, we examined how the concentrations and kinetic parameters of the motors contribute to their collective behavior. We observed self-organization of generic steady-state structures such as asters, vortices, and a network of interconnected poles. We identified parameter combinations that determine the generation of each of these structures. In general, this approach may become useful for correlating the morphogenetic phenomena taking place in a biological system with the biophysical characteristics of its constituents.