The detailed pattern of spatial structures for the reaction-diffusion system involving substrate-inhibited reactions on immobilized uricase enzyme was studied. Depending on the governing parameters, three basic solutions may exist and there are two kinds of possible branching, either successive primary bifurcation from a basic trivial branch or consecutive secondary bifurcation. In both cases the branching follows the sequence of symmetric --> asymmetric --> symmetric, and so forth. The emergence of subsequently more complex spatial structures with the increasing length of systems suggests a close similarity to gradual buildup of complex morphogenetic patterns in developmental biology.