How does hydrolysis of atp drive transport?

The exergonic process of ATP hydrolysis is used to drive an endergonic process
ATP drives transport and mechanical work:transport work: ATP phosphorylates tansport proteinsmechanical work: ATP binds noncovalently to motor proteins and then is hydrolyzed-ATP hydrolysis causes changed in the shapes and binding affinities of proteins. 
It can occur directly through phosphorylation as seen for a transport protein or in most instances of mechanical work via non-covalent binding of .ATP

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