Recently, both microfluidic and fluorescence correlation spectroscopy experiments have revealed that diffusion coefficients of active biological catalysts (enzymes) rise proportionately to their catalytic rate. Similar effects have also been observed for active material catalysts, such as platinum nanocatalysts in hydrogen peroxide solution. While differences in diffusion coefficients have recently been cleverly exploited to spatially separate active from inactive catalysts, here we investigate the consequences of these novel findings on the spatiotemporal organization of catalysts. In particular, we show that chemical reactions-such as coupled catalytic reactions-may drive effective attraction or repulsion between catalysts which in turn drives their spatiotemporal organization. This, we argue, may have implications for internal cell signaling.