Coordination is an important aspect in developing distributed systems. While in silicon-based agents, i.e., mechanical robotics, designing individual-level behavior that may emerge into one global function is a typical approach to such systems, in DNA-based agents, programming of each individual's behavior still remains a challenge, as they are based on chemical reactions. These reactions occur immediately after all reactants have been mixed into a solution, which introduces challenges in logical control. In this work, we report a design strategy for coordinated eventdriven DNA-based systems by using a Petri Nets model. First, computational primitives based on DNA strand displacement reaction are introduced. Second, their molecular implementation is abstracted by Petri Nets for high-level design. Third, as our main contribution, we propose the model of interacting multi-agent systems based on DNA-only reactions. We verify our design via in silico simulation and show initial experiments of Petri Nets operators. From the obtained results, we argue that our design strategy is feasible for coordinating interaction of distributed DNA-based systems.