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Hexokinase is the enzyme that starts off glycolysis in eukaryotes. It phosphorylates glucose on the 6 carbon to create glucose-6-phosphate. This requires the input of 1 ATP molecule to drive the reaction forward.
However, if glycogen is the starting reactant of glycolysis, a different enzyme breaks down glycogen into glucose-1-phosphate units (starch phosphorylase), and then hexophosphomutase isomerizes the molecule to glucose-6-phosphate. This does not require the use of ATP.
While hexokinase is normally used in almost all cells in our body, a competing enzyme, glucokinase, is used by the liver to eventually create glycogen from glucose. So which enzyme is uses, and when?
Due to the fact that the Km of hexokinase is around 1 micro-molar and glucokinase has a Km about 100 times that, hexokinase works best at low glucose concentrations (which makes sense- we want our glucose to be utilized for cellular respiration), while at high glucose concentrations glucokinase kicks in (more glucose=storage as glycogen).