A physically-motivated genetic algorithm (GA) and full enumeration for a tile-based model of self-assembly (JaTAM) is implemented using a graphics processing unit (GPU). We observe performance gains with respect to state-of-the-art implementations on CPU of factor 7.7 for the GA and 2.9 for JaTAM. The correctness of our GA implementation is demonstrated using a test-bed fitness function, and our JaTAM implementation is verified by classifying a well-known search space $S_{2,8}$ based on two tile types. The performance gains achieved allow for the classification of a larger search space $S^{32}_{3,8}$ based on three tile types. The prevalence of structures based on two tile types demonstrates that simple organisms emerge preferrably even in complex ecosystems. The modularity of the largest structures found motivates the assumption that to first order, $S_{2,8}$ forms the building blocks of $S_{3,8}$. We conclude that GPUs may play an important role in future studies of evolutionary dynamics.