The neurotransmitter is a chemical substance that transmits information between neurons. Its metabolic process includes four links: synthesis, storage, release and inactivation. As one of the important chemical components of neurotransmitters, acetylcholine is synthesized under the catalysis of acetylcholine coenzyme A and choline acetylase. Inspired by the biological fact that enzymes exist in neurons and that enzymes are involved in neurotransmitter synthesis, we propose spiking neural P systems with enzymes (SNPE). Different from the previous spiking neural P systems and their variants, each neuron of SNPE contains two classes of objects, and each spiking rule has the participation of enzymes. In addition, the number of spikes and enzymes in a neuron can also serve as a consumption condition for controlling whether a reaction (rule execution) occurs. When the number of enzymes meets the requirements of a specific biochemical reaction, the number of occurrences of the reaction can also be controlled. As number generation and acceptance devices, the proposed SNPE systems are proved to be Turing universal. In addition, 61 neurons are used to construct an SNPE system that realizes function computation, which proves the Turing universality in this mode. Finally, we also explore using a uniform SNPE model to solve the subset sum problem and compare it with the standard SN P and its several variants.