Glycogen synthesis:

To synthesize glycogen three enzymes are needed:

1.  UDP-glucose   pyrophosphorylase catalyzes the synthesis of UDP-glucose from UTP and glucose 1-phosphate:

The   pyrophosphate    (PP_i)    is   immediately    hydrolyzed    through   inorganic pyrophosphatase, releasing energy.  Therefore the whole reaction is very exergonic and essentially irreversible.

2.  Glycogen synthase now transfers the glucosyl residue from UDP-glucose to the C4 OH group at the nonreducing end of a glycogen molecule, forming a α1-4 glycosidic bond. Amusingly, glycogen synthase can only extend an existing chain.  Therefore  it  needs  a  primer;  this  is  a  protein  known as glycogenin.  Glycogenin  holds  eight  glucosyl  units  linked  via α1-4  link- ages,  which  are  added  to the  protein  through itself  for instance  Autocatalysis.  It is this molecule   which   glycogen   synthase   then   extends.   Every   glycogen   granule contains only a single glycogenin molecule at its core. The reality that glycogen synthase is fully active only when in contact with glycogenin limits the size of the glycogen granule.

Figure:  The UDP-glucose pyrophosphorylase reaction.

3.  Branching    enzyme    [amylo-(1-4→1-6)   transglycosylase]   is   a   variant enzyme from glycogen-debranching enzyme. Units have been joined as a straight chain with α1-4 linkages and branching enzyme breaks one of the α1-4 bonds and transfers a block of residues (generally about seven) to a more interior site in the glycogen molecule after a number of glucose, reattaching these by creating a α1-6 bond. The branches are important because the enzymes which degrade and synthesize glycogen (glycogen synthase and glycogen phosphorylase, correspondingly) work only at the ends of the glycogen molecule. Therefore the existence  of  several  termini  permits  a  far  more  rapid  rate  of  degradation and synthesis  than would be possible with a nonbranched polymer.

Figure:   Synthesis of glycogen by glycogen synthase.