How is the enzyme glycogen synthase regulated in regards to glycogen synthesis? I think I understand that phosphorylation decreases its activity (through glycogen synthase kinase?), but what role do molecules like insulin, glucagon (or epinephrine), protein kinase A, and Protein phosphatase 1 (PP1) play? Do they activate or inhibit glycogen synthesis, and how so?
The answer to this question can easily be found in standard biochemistry texts, e.g. one available online (albeit an old edition): http://www.ncbi.nlm.nih.gov/books/NBK22444/. Presumably this is why this question has not been answered previously. However I think it worth providing a summary answer in order to stress something that I feel perhaps does not receive enough emphasis in texts, and is ignored in the question.
This is that the regulation of glycogen metabolism (and indeed other aspects of carbohydrate metabolism) differs in different tissues because of the different situations in which its synthesis and breakdown is appropriate in relation to the integrated metabolism of the organism. If we just consider skeletal muscle and liver, the muscle stores glycogen to use itself when it needs to generate ATP to power contraction, whereas the liver stores glycogen so that it can provide other tissues with glucose when they require it either for work (e.g. the skeletal muscle) or in starvation (e.g. erythrocytes and brain).
The appropriate integration is provided in the first instance by the hormones mentioned (so-called first messengers), and the presence or absence of receptors in the tissues so that they will respond to these hormones or be refractory towards them, as appropriate.
Both liver and skeletal muscle have receptors for epinephrine (adrenaline) which is secreted as a stress response. This results in the breakdown of glycogen — used internally by the skeletal muscle and released as glucose by the liver, as already mentioned.
Liver also has receptors for glucagon, the hormone which is secreted in starvation or fasting, and also causes the breakdown of glycogen. The skeletal muscle does not have glucagon receptors because the nutritional state is irrelevant to its energy requirements.
Both liver and skeletal muscle have receptors for insulin, the hormone of the fed state, and respond to this by increasing the synthesis of glycogen to build up supplies to be available when needed.
How do these hormones work?
Glucagon and epinephrine (adrenaline) both work by activating adenyl cyclase (a so-called second messenger) in a G-protein-coupled manner to generate cyclic-AMP. This acts by a phosphorylation cascade to activate glycogen phosphorylase and inhibit glycogen synthase as described in any standard text. (Protein phosphatase 1 is also involved.) Glycogen breakdown is this enhanced.
Insulin acts differently. Binding to its receptor activates the protein's intrinsic tyrosine kinase, and there is a chain of subsequent protein self-phosphorylations culminating in the phosphorylation of protein phosphatase 1. This causes it to dephosphorylate both glycogen synthase and glycogen phosphorylase leading to the activation of the former and the inactivation of the latter. Glycogen synthesis is thus enhanced.
It is important to understand the molecular mechanisms of metabolic control, but it is also important to see them in the context of integrated tissue metabolism. So I will end with a question for the reader. The brain (which is absolutely dependent on glucose for part of its metabolism) stores a small amount of glycogen. How would you expect the brain to respond to the three hormones mentioned, and what would you expect to be the fate of the product of glycogen breakdown? If you have understood the principles above for liver and muscle you should be able to make rational predictions for brain (even should the facts turn out to be more complex).