Final answer:
A postsynaptic cell can open chemically gated ion channels for fast responses, close ion channels via G proteins for slow responses, and regulate protein synthesis, affecting its metabolic activities in response to neurotransmitter binding.
Step-by-step explanation:
In response to the binding of a neurotransmitter, a postsynaptic cell can engage in multiple actions. It can open chemically gated ion channels, causing graded potentials known as fast synaptic potentials, like what happens when acetylcholine or glutamate triggers an excitatory postsynaptic potential (EPSP). It can also close ion channels via G proteins and second messenger systems, leading to slower responses, such as when GABA induces an inhibitory postsynaptic potential (IPSP). Additionally, neurotransmitter binding can regulate protein synthesis and affect the metabolic activities of the postsynaptic cell.
Therefore, the correct answer is E): The postsynaptic cell can open chemically gated ion channels causing fast synaptic potentials, close ion channels via G proteins and second messenger systems producing slow responses, and regulate protein synthesis affecting the metabolic activities of the postsynaptic cell.