The energy consumption of a multi-access edge computing (MEC) system must be reduced to save operational costs. Determining a set of active MEC servers (MECSs) that can minimize the energy consumption of the MEC system while satisfying the service delay requirements of the tasks is an NP-complete problem. To solve this problem, we take a bio-inspired approach. We note that the sleep control problem of the MECS differentiates the operational mode among neighboring MECSs. Therefore, by mimicking the cell differentiation process in a biological system, we designed a distributed sleep control method. Each MECS periodically gathers the utilization and delta levels of the neighboring MECSs. Subsequently, by using the gathered information and the Delta–Notch inter-cell signaling model, a MECS autonomously decides whether to sleep. We evaluated the performance of our method through extensive simulations. Compared with a conventional method, the proposed method reduces energy consumption in a MEC system by more than 13% while providing a comparable service delay. In addition, our method reduces the variations in the service delay by more than 35%.