ELI5 Donor Organs - do donor organs age according to the donor’s age or the new host’s age?

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If all the cells in a person’s body have their individual DNA imprinted in them, then what happens after an organ transplant? Does the organ eventually adapt the DNA of the recipient or is the donor’s DNA permanently a part of the recipient? Do donated organs age according to the donor´s age or do they adapt to the age of the new body? Why do some countries have much higher percentages of people donating their organs compared to others?

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