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Do carrier proteins require energy to change shape?

Why don’t carrier proteins require ATP to change shape in facilitated diffusion?
But require ATP to change shape in active transport?

Also, I know that a carrier protein changes shape twice in active transport (1 when ATP is broken into ADP and Pi, 2 when Pi is released), but do they (membrane pumps) always transport 2 different molecules (e.g. potassium-sodium pump)?

so when 1st molecule bind to the carrier protein, 1st shape change is triggered (ATP break down into ADP and Pi), 1st molecule leave;

and when 2nd molecule bind to the carrier protein, 2nd shape is triggered (Pi is released), 2nd molecule leave.

Is there membrane pump which only transport one molecule? If so, how does it work (i.e. how is 2nd shape change triggered in this case if there isn’t a 2nd molecule)?

Thanks a lot!!

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